North-Holland Mathematical Library Board ofAdvisory Editors:
M. Artin, H. Bass, J. Eells, W. Feit, P. J. Freyd, F. W. Gehring, H. Halberstam, L. V. Hormander, M. Kac, J. H. B. Kemperman, H. A. Lauwerier, W. A. J. Luxemburg, F. P. Peterson, I. M. Singer and A. C. Zaanen
VOLUME 28
NORTH-HOLLAND PUBLISHING COMPANY AMS T E RDAM. NEW YORK . OXFORD
Graded Ring Theory c. NASTASESCU
University of Bucharest Bucharest, Rumania and
F. VAN OYSTAEYEN University of Antwerp Wilrijk, Belgium
d
NORTH-HOLLAND PUBLISHING COMPANY A M S T E R D A M . NEW YORK . OXFORD
Worth-Holland Publishing Company, 1982 All rights reserved. N o part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, without the prior permission of the copyright owner.
ISBN: 0 4 4 4 8 6 4 8 9 ~
Published by: NORTH-HOLLAND PUBLISHING COMPANY AMSTERDAM . NEW Y O R K . O X F O R D
Sole distributors for the U.S.A.and Canada: ELSEVIER SCIENCE PUBLISHING COMPANY, INC. 52, VANDERBILT A V E N U E NEW YORK, N.Y. 10017
PRINTED IN T H E NETHERLANDS
FOREWORD This book is aimed to be a ‘technical’ book on graded rings. By ‘technical’ we mean that the book should supply a kit of tools of quite general applicability, enabling the reader to build up his own further study of non-commutativerings graded by an arbitrary group. The body of the book, Chapter A, contains: categorical properties of graded modules, localization of graded rings and modules, Jacobson radicals of graded rings, the structure thedry for simple objects in the graded sense, chain conditions, Krull dimension of graded modules, homogenization, homological dimension, primary decomposition, and more .....
One of the advantages of the generality of Chapter A is that it allows direct applications of these results to the theory of group rings, twisted and skew group rings and crossed products. With this in mind we have taken care to point out on several occasions how certain techniques may be specified to the case of strongly graded rings. We tried to write Chapter A in such a way that it becomes suitable for an advanced course in ring theory or general algebra, we strove to make it as selfcontained as possible and we included several problems and exercises.
.
Other chapters may be viewed as an attempt to show how the general techniques of Chapter A can be applied in some particular cases, e.g. the case where the gradation is of type Z. In compiling the material for Chapters B and C we have been guided by our own research interests. Chapter 6 deals with commutative graded rings of type 2 and we focus on two main topics: artihmeticallygradeddomains, and secondly, local conditions for Noetherian rings. In Chapter C we derive some structural results relating to the graded properties of the rings considered. The following classes of graded rings receive special attention: fully bounded Noetherian rings, birational extensions of commutative rings, rings satisfying polynomial identities, and Von Neumann regular rings. Here the basic idea i s to derive results of ungraded nature from graded information. Some of these sections lead naturally to the study of sheaves over the projective
vi
Foreword
spectrum Proj(R) of a positively graded ring, but we did not go into these topics here. We refer to [125] for a noncommutative treatment of projective geometry, i.e. the geometry of graded P.I. algebras. Chapter D presents the theory of filtered rings and modules and we focus in particular on the gradation associated to a filtration. In the sections entitled 'Comments, References, Exercises' we included references to interesting related topics that have not been treated in this book.
A remark about references: only references to a chapter different from the chapter containing the reference contain the chapter's indication A, B, C or D; references within the same chapter only mention the section numbers.
ACKNOWLEDGEMENT The authors thank the department of mathematics of the co-author's institution for i t s hospitality while much of this work has been conceived and written. Francine did a nice job typing this book and we thank her for the quality of her work as well as for her patience.
1
CHAPTER A: SOME GENERAL TECHNIQUES IN THE THEORY I: Graded Rings and Modules
OF GRADED RINGS
I . 1. Graded Rings and the Category of Graded Podules
Let G be a m u l t i p li cat i v e group with i d en t i t y element e . A ring R i s said to be a graded ring of type C- i f there i s a family of R , say {Ru, u c G } , such t h a t R
=
8
oCG
o f additive subgroups
R u a n d RoRT c RUT f o r a l l
where we denoted by RoRT the s e t of a l l f i n i t e sums of products r
C
U,T
G,
with
c Ru, rITc RT. The elements of U Ro ar e called homogeneous elements
of R, a nonzero r
oEG
6
Ru i s said t o be homogeneous of degree o, and we write
deg r = u. Any nonzero r
C
R has a unique expression as a sum o f homogeneous
elements, r = 2 ro, where r o i s nonzem f o r a f i n i t e number of u in G . of G
The nonzero elements ro in the decomposition of r are called the homogeneous components of r. Let R be a graded ring of type G. I f then t h e ring S = R with gradation
TI
: G
+
Sa = 8CICN
ring o f type H . This r i n g wi l l be denoted by
H i s an epimorphism of groups
'(a)
R
Ro,
f o r a l l a C H , i s a graded If
(H)'
e
morphism of groups, the ring R ( H ) with gradation ( R ( H ) ) u 0
c H i s a graded ring of type H. I n case G = Z , H
i s denoted by R ( d ) i . e . R(d) =
@
ncZ
=
: H =
+
F i s a mono-
Re(u)for a l l
{nd, n C Z } then R ( H )
Rnd.
I f G i s commutative and R i s a graded ring o f type G then the ring S = R
with gradation defined by S graded ring.
o
= R
-1 f o r a l l u
u
c
G , i s called the opposite
1.1.1. Proposition I f R i s a graded ring of type G then Re i s a subring of R and 1 c Re.
2
A. Some General Techniques in the Theory of Graded Rings
Proof : Let
From ReRe c Re i t follows immediately t h a t Re i s a s u b r i n g of R .
1 =
z r
u ~ G
be the homogeneous decomposition of 1
and h, F RT, then h rUiT
u
=
, with rUXT c RUT.
1.X = 2 r X UCG
-
e R.
Pick
T
e
G
Consequently
c7
0 holds f o r a l l u # e in 6 . I t follows t h a t rUX = 0 f o r a l l
# e i n G and f o r a l l X e R . Therefore 1 = re e Re.
1 . 1 . 2 . Examples
1. Let G be a group, A a r i n g . The group r i n g R = A [ G I i s a graded
r i n g o f type G where Ro = A.
for all
u F
G.
2. Any r i n g R may be considered a s a graded r i n c o f type G , f o r any
group G, by p u t t i n g Re
=
R , Ro
0 for
=
u
# e i n G . Such a r i n g i s
s a i d t o be t r i v i a l l y G-Graded.
3. Let R be a graded r i n g of type G . Let R" be the o p p o s i t e r i n g f o r R i . e . R" has t h e same underlying a d d i t i v e group a s R b u t m u l t i p l i c a t i o n i n R" i s defined by the r u l e x =
O
y
=
yx f o r x , y R".
Putting ( R 0 ) d
R -1 makes R" i n t o a graded ring o f type G. 0
4. Let R be any r i n g and l e t
9 :
R
--f
The r i n g o f t w i s t e d polynomials S a variable X t o
r
6
R
R an i n j e c t i v e r i n g homomorphism =
R [ x , , ~ ] i s obtained by adding
R and d e f i n e m u l t i p l i c a t i o n by ,o(r)X
=
Xr f o r a l l
and extend i t t o polynomialsin X ( w r i t i n g c o e f f i c i e n t s on t h e
l e f t hand s i d e o f the powers of X ) . Obviously S i s graded of type i
w i t h Si
=
0 if i
c
0 and Si = {a X i , a
c R
}
i f i z 0. I f
automorphism of R then i t i s c l e a r t h a t R I X , X-',,O
'0
i s an
1 may be c o n s t r u c t e d
i n a s i m i l a r way. 5. Let k be a f i e l d , X an i n d e t e r m i n a t e , and l e t k ( X ) be the f i e l d o f r a t i o n a l f u n c t i o n s i n X over k . Although k ( X ) can only be graded of type 2 i n t h e t r i v i a l way, i t i s very well p o s s i b l e t o d e f i n e nont r i v i a l g r a d a t i o n s of type Z/nZ on k ( X ) , f o r every n e N. Indeed;
3
1. Graded Rings and Modules
I t i s e a s i l y seen t h a t , i f i a n d j have the same image mod n then
k ( X ) : = k ( X ) ? , hence the gradation i s well defined. Straightforward J
1
v e r i f i c a t i o n learns t h a t k ( X ) i s graded of type Z/nZ.
For a ring R we write R-mod f o r the category of l e f t R-modules whereas mod-R denotes the category of r i g h t R-modules.
c R-mod i s said t o be a graded
Let R be a graded ring o f type G , an En l e f t R-module i f there i s a family [Ma,
F! such t h a t M = Q Mu ot G
h(M) =
U
OF G
OF
G} of additive subgroups of
for a l l
and ROYT c Ma-,
T
0,
F G . Elements of
Ma are called homogeneous elements of M. I f m # 0 i s i n \IO
f o r some o t G then m i s homogeneous of degree
o
a n d we write : deg m = a.
Any nonzero m F M has a unique decomposition m = c
o6G
f i n i t e number of the mu a r e zero.
mO where a l l b u t a
Unless otherwise s t a t e d module will mean l e f t module. A submodule f l of M i s a graded submodule i f N =
(N
Q
n
Mu),
or equivalently, i f f o r any
O G
x c N the homogeneous components of x are again i n H .
1.1.3. Example. Let R be a graded ring, M a graded R-module and l e t N be a submodule of M. Let ( N )
9 I t i s easily verified that ( N )
are graded submodules of M.
be the submodule of M generated by N
4
i s maximal amongst submodules of N which
Consider graded R-modules M and P I . An R-linear f : M
a graded morphism of degree Graded morphisms of degree
T, T
T
+
F G, i f f(F?,) c Mu-,
N i s said t o be
for all a
C
build an additive subgroup H O M R ( M , N ) T
HomR(M,N). I t i s c l e a r t h a t HOMR(M,N) = 8 H O M R ( M , N ) T abelian group of type G.
n h(M).
TFG
i s a graded
G. of
4
A. Some General Techniques in the Theory of Graded Rings
Composition of a graded morphism f : En morphism g : N degree
m.
+
P of degree
T
+
N of
degree
o
C C- with a graded
G y i e l d s a graded morphism g o f of
c
I f N i s a submodule of En then P/N may be made i n t o a graded
module by putting : ( t 4 / N ) 0 R-linear projection M
--f
=
Mn
t
N/N.
With t h i s d e f i n i t i o n , the canonical
M/N i s a graded morphism of degree e ( l e f t a n d
r i g h t ! ) . The category R-grG consists of graded R-modules of type G and the morphisms a r e taken t o be the graded morphisms of degree e . I n the sequel we s h a l l write R-gr i f there i s no ambiguity possible about G. The category R-gr possesses d i r e c t sums a n d products. Furthermore i f (Mu, f a $ ; a , p c I ) i s some inductive system, resp. projective system
of objects in R-gr, then the module li+m Mg c R-mod, resp. lim Ma c R-mod a
a
may be graded by putting ( 1 L m M a ) u a
for a l l
u
=
l i m (Ma)4, resp. ( l i m Enna)o = lim(Va)a a
a
a
c G.
If M, N c R-gr t h e n we write HomR-gr(rVI,N) f o r the graded morphisms o f degree e from
M t o N . For any f c HomR-gr(M,N),
b o t h Ker f a n d Coker f
a r e i n R-gr. Indeed, since Im f i s a graded submodule of N i t follows
t h a t Coker f = N/Im f =
@
( N O + Im f)/Imf i s the cokernel o f f i n R-gr.
IJCG
I t i s not hard t o verify t h a t R-gr i s an abelian category which s a t i s f i e s Grothendieck's axioms Ab 3, Ab 4 , Ab 3* and Ab 4*, cf [1021. Since a l s o Ab 5 i s e a s i l y v e r i f i e d , we w i l l say t h a t
R-gr i s a Grothendieck
category. The forgetful functor, U : R-gr
-f
R-mod, ( U f o r "ungrading", unusual
perhaps, b u t preferable t o "degrading") associates t o M ungraded R-module.
the
underlying
The forgetful functor U has a r i g h t a d j o i n t which
associates t o M 6 R-mod the graded R-module $4 = 0 where each uc G "M i s a copy of M written tux, x 6 MI, with R-module s t r u c t u r e defined by : r
*
Tx = uc(rx) f o r each r c Ro. If f : ).I
+
N i s R-linear t h e n
5
1. Graded Rings and Modules
Gf :
GM
+
N' i s given by G f ( O x )
= uf(x),
and i t i s c l e a r t h a t Gf has
degree e 6 G. The f u n c t o r M
+
GM, from R-mod
be a d i r e c t sum o f copies o f
G R-gr i s exact. Note t h a t U( M) need n o t
--t
M
s i n c e t h e component
ON,
u
e
C-,
i s not
G an R-submodule o f M ( b u t i t i s an Re-submodule o f course). 1.1.4.
Notation
. b u t we o m i t t h i s f o r R s i n c e i t w i l l always be c l e a r
We w i l l w r i t e U(M) =
from the c o n t e x t whether R i s being considered as a graded r i n g o r n o t . One o f t h e standard problems concerning R-gr t h a t we w i l l be
P as an o b j e c t o f R-gr,
w i t h may be phrased as f o l l o w s : i f F4 has p r o p e r t y i s i t t r u e then t h a t
dealing
has p r o p e r t y P i n R-mod. The converse problem i s
u s u a l l y f a r more e a s i e r t o handle.
I f M 6 R-grG then f o r u C G we d e f i n e t h e 0-suspension M(u) o f Y t o be t h e graded module a f u n c t o r To : R-gr
by p u t t i n g M ( u ) ~ = PITU. So we d e f i n e d
o b t a i n e d from +
R-gr, To(M)
=
M(u)
and t h i s f u n c t o r
may be
c h a r a c t e r i z e d by :
1" TOoT,
=
, for
Tm
all u,-rCG
2" T o T -1= T - 1 , ~ T = Id, f o r a l l 0
0
0
o
c G
U
3" U o Ta = U, f o r a l l
c G
o
I t f o l l o w s t h a t , f o r any
c 6,
CT
Tu d e f i n e s an equivalence o f c a t e g o r i e s .
I n a s t r a i g h t f o r w a r d way one proves t h a t {R(o), u
c
F) i s a s e t o f
generators o f R-gr and t h a t R-gr i s a Grothendieck category w i t h enough i n j e c t i v e o b j e c t s , c f . [ 102 J
.
An F c R-gr i s s a i d t o be o r - f r e e i f F has a b a s i s o f homogeneous elements, o r e q u i v a l e n t l y any
M c
F
8 R(u), where S i s a subset o f G. Since acs R-gr i s isomorphic t o a q u o t i e n t o f a g r - f r e e o b j e c t i n R-or and
A. Some General Techniques in the Theory of Graded Rings
6
s i n c e t h e l a t t e r a r e c e r t a i n l y p r o j e c t i v e as o b j e c t s o f R-gr,
i t follows
t h a t R-gr has enough p r o j e c t i v e o b j e c t s .
1.1.5.
Example. L e t H be a subgroup o f 6 and l e t S cG be a s e c t i o n f o r
G modulo
H i . e . a s e t o f r e p r e s e n t a t i v e s f o r t h e r i g h t H-cosets o f G.
L e t R be a graded r i n g o f t y p e G and M M("'H)
0
=
Mho.
hCH
Then M(''H)
c
F o r any u c S we p u t
R-yG.
i s a graded R(H)-module w i t h g r a d a t i o n :
( M ( " y H ) ) h = MhUfor a l l h c H. I t i s c l e a r t h a t M = any
M
S,
0
+
M(OYH) d e f i n e s a f u n c t o r : R-gr
i f G = Z and H = {nd, n c 7) w i t h d For 0 5 k 1.1.6.
d - 1 we w r i t e R ( k '
5
Remark
tJ
all u
+
{O,l,
=
..., d - l } .
d, = R ( k y ').
I f R i s a graded r i n g o f t y p e 6 t h e n
N i s s a i d t o be graded o f degree
c G.
R(H)-gr. I n p a r t i c u l a r
0 t h e n we may choose S
c a t e g o r y gr-R o f graded r i g h t R-modules. f :
+
0 M(OYH and f o r ocs
Note t h a t t h e s i d e on w h i c h
T
I f V, N T,
T
c
c 6,
we can d e f i n e t h e gr-R t h e n an R- i n e a r i f f(MU)
c
F?Tu f o r
i s w r i t t e n i s d e t e r m i n e d by
t h e f a c t t h a t f s h o u l d be r i g h t R - l i n e a r . The o-suspension o f M s h o u l d t h e n be d e f i n e d as f o l l o w s (u)M i s d e t e r m i n d e d by g r a d a t i o n ( ( u ) M ) ~ = FIUT; a g a i n , t h e s i d e on w h i c h
0
c
c
gr-R
mod-R w i t h
a c t s i s determined
by t h e f a c t t h a t ( u ) M s h o u l d now be a graded r i g h t R-module ! Consequently
if f i s b o t h a l e f t and r i g h t y a d e d morphism o f t w o - s i d e d graded Rmodules t h e n f must have c e n t r a l degree i . e .
deg f c Z ( G ) .
7
I. Graded Rings and Modules
1 . 2 . Elementary Properties of Graded Modules
Throughout t h i s section R i s a y-aded ring of type G f o r some fixed g r o u p G , so we omit notational references t o P i n w h a t follows.
1.2.1. Lemma. Let M; N, P
c R-or and consider the following commutative
diagram of R-morphisms :
where f has degree e . I f g ( r e s p . h ) has degree e then there e x i s t s a graded morphism h ' ( r e s p . 4 ' ) of d e y e e e , such t h a t f f
=
9 o h ' (resp.
g' o h).
=
Proof
We give the proof in case deg g = e .
I _
Pick x c Mo. From h ( x )
=
h ( x ) u i t follows t h a t h ' : M
2
+
N may be
UCC-
defined by putting h ' ( x ) : h ( x ) o . 1 . 2 . 2 . Corollary.
Consider P 6 R-gr; then P i s a projective
in R-gr i f a n d only i f
p
i s a projective R-module.
I f P i s a projective in R-gr then i t i s a d i r e c t summand of a
Proof
gr-free object F. Therefore suppose
p
f
p entails o g
=
g' : P
F
+
f
--t
P
i s projective in R-mod. Conversely,
i s projective i n R-mod. Since P c R-gr there i s a or-free
object F c R-or such t h a t F Of
object
+
f
P
+
0 i s exact in R-gr. P r o j e c t i v i t y
the existence o f an R-linear map
a
:
p+5
such t h a t
l p . According t o Lemma 1.2.1. there e x i s t s a graded morphism -
F
of degree e such
+o,
t h a t f 0 g ' = I,,. Thus the exact sequence
s p l i t s and therefore P i s a projective object of R-gr.
We will sometimes r e f e r t o projective objects of R-gr as gr-projective graded modules. So by the above corollary gr-projective i s nothing b u t
A. Some General Techniques in the Theory of Graded Rings
8
graded p l u s p r o j e c t i d e . 1.2.3.
Corollary. L e t 0
+
N
sumnand o f M i f and o n l y i f
-
M be e x a c t i n R - o r . Then N i s a d i r e c t i s a d i r e c t summand o f
F.
A graded l e f t R-submodule I o f R i s c a l l e d (an homogeneous o r ) a graded l e f t i d e a l o f R. I f I i s two-sided then i t i s c a l l e d a graded i d e a l o f R.
1.2.4.
Lemna. L e t E c R-gr. The f o l l o w i n g statements a r e e q u i v a l e n t :
1. E i s an i n j e c t i v e o b j e c t o f R-gr. 2 . The f u n c t o r HOMR(-, E) i s exact.
3. For every graded l e f t i d e a l I o f R, t h e canonical i n c l u s i o n i : I - R gives r i s e t o a s u r j e c t i v e morphism HOMR(i, l E ) : HOVR(R,E) Proof. 1
_ I _
+
HOMR(I,E)
2. By d e f i n i t i o n E i s i n j e c t i v e i n R-gr i f and o n l y i f t h e
f u n c t o r HomR-gr(-,E)
= HOMR(-, E ) o i s exact. Moreover E i s i n j e c t i v e i n
R-gr i f and o n l y i f E(u) i s i n j e c t i v e f o r every u € G. Since a morphism f : M
+
E o f degree u may be viewed as a morphism o f degree e, M
+
E(o),
t h e f o r e g o i n g remarks prove t h e d e s i r e d equivalence.
2 = 3. Obvious. 3 = 1. The p r o o f i s s i m i l a r t o t h e p r o o f o f Baer’s theorem i n t h e ungraded case. 1.2.5.
C o r o l l a r y . I f E E R-gr i s such t h a t
E is
an i n j e c t i v e R-module
then E i s i n j e c t i v e i n R-gr. I n j e c t i v e o b j e c t s i n R-Gr w i l l be s a i d t o be g r - i n j e c t i v e . 1.2.6.
Remarks
1. I f E i s g r - i n j e c t i v e then E need n o t be i n j e c t i v e . For example, l e t
k be any f i e l d and consider R = k [ X, X-’]
where X i s a v a r i a b l e
commuting w i t h k and w i t h g r a d a t i o n g i v e n by Ri
=
{ a X’,
a
c
k } for
9
1. Graded Rings and Modules
a11 F 2. Check t h a t R
i s i n j e c t i v e i n R - g r b u t n o t i n R-mod
2. I f L t R-gr i s such t h a t ! = i s a f r e e R-module t h e n L need n o t be g r - f r e e . F o r example, t a k e R = Z x Z w i t h t r i v i a l g r a d a t i o n . L e t L be R b u t endowed w i t h t h e f o l l o w i n g g r a d a t i o n : Lo = Z x { O } , and Li = 0 f o r i # 0, 1. O b v i o u s l y
L
The p r o j e c t i v e d i m e n s i o n o f an R-module
If
M
o f C o r o l l a r y 1.2.2. 1.2.7.
101 x Z
w i l l be denoted by p . d i m R ( y . M i n t h e category R-Cr w i l l
We have t h e f o l l o w i n g d i r e c t consequence
:
M c
Corollary. If
1.2.8. Lemna. L e t M
c
R-gr t h e n gr.o.dimR(M)
R-gr,
= p.dimR
(M)
N a graded submodule o f M.If N i s e s s e n t i a l
i n M as a s u b o b j e c t o f P i n R-gr,
Proof. Recall
=
i s not gr-free.
F R-gr t h e n t h e p r o j e c t i v e dimension o f
be denoted by gr.p.dimR(M).
L1
then
I? i s
essential i n
1.
t h a t a s u b o b j e c t o f M i s s a i d t o be an e s s e n t i a l s u b o b j e c t
o r M i s an e s s e n t i 3 1 e x t e n s i o n o f N, i f N i n t e r s e c t s a l l nonzero s u b o b j e c t s o f M non-trivially.
Hence, t h e f a c t t h a t N i s g r - e s s e n t i a l
in
M
means t h a t
f o r each x # 0 i n h(M) t h e r e e x i s t s an a F h(R) such t h a t ax 6 N - t o } . If x
c El i s nonzero t h e n we may decompose x as x
a r e t h e homogeneous components o f x o f degree o i n d u c t i o n on n, e s t a b l i s h i n g t h a t f o r each x t such t h a t a x c
E.
= xo
. The
1
+. . .+xo
n
where t h e xu
p r o o f goes by
t h e r e i s an a t h(R)
For n = 1 t h i s i s obvious from g r - e s s e n t i a l i t y o f N
i n M. P u t y = xu t... t x . S i n c e xo # 0 t h e r e i s an a c h(R) such 1 “n- 1 n
# 0. T h e r e f o r e a x - a x o = a y has a t most n - 1 “n n homogeneous components i n M. Now, i f ay = 0, t h e n a x c N. I f ay # 0 t h e n that,
ax
6 Nand
x
“n
t h e i n d u c t i o n h y p o t h e s i s p r o v i d e s us w i t h a b F h(R) such t h a t b a y c PI and b a y # 0. Thus b a x = b a xo
n
t
b a y t N. B u t b a x = 0 would
10
A. Some General Techniques in the Theory of Graded Rings
e n t a i l b a xu
n
= b a y = 0, c o n t r a d i c t i n g t h e choice o f b, t h e r e f o r e
b a x # O . 1.2.9. M
Remark.
(mi s
c
Let M
R-gr. and l e t N be a small graded submodule of
t h e n o t i o n dual t o e s s e n t i a l ) , t h e n N need n o t be small i n
For example, i n t h e graded r i n g R = k [ X I , where k i s a f i e l d ,
M.
(X) is
small i n t h e graded sense b u t n o t i n R-mod.
f
c
c
Lemna. L e t M, N
1.2.10.
HornR
(tj,fj)
R-gr. The group HOMR(M,N) c o n s i s t s o f a l l
f o r which t h e r e e x i s t a f i n i t e subset o f G, F say, such
that :
(*) f(Mu) c
z
'pcF
Proof.
Noo,
f o r a l l u 6 6.
Suppose t h a t f
<
HOMR(M,N) then t h e r e e x i s t ol, u2,..., on C G
+...+ f o
such t h a t f = f
n
and f G F HOMR(M,N)G-, i = 1,..., n. C l e a r l y ,
i
1
f s a t i s f i e s (*). Conversely, suppose f E HornR(!, some f i n i t e s e t F c G. Look a t an xo 6 Mo =
C,
'p cF
F No,$,
yG,o,Q f o r unique elements
d e f i n e f (xu) = y thus f
c
0 , OP
.
,
fj)
s a t i s f i e s (*) f o r
by (*) i t f o l l o w s t h a t f ( x o ) = t F. For any '0 < F we
I t i s obvious t h a t f,o c HOMR(M,N)
and f = 'P
E
'p CF
f,o
HOMR(M,N).
1.2.11. C o r o l l a r y .
We have t h a t HOMR(M,N)
= HornR(tj,N)in each o f t h e
f o l l o w i n g cases : 1" The y o u p G i s f i n i t e M i s f i n i t e l y generated. 2" Proof. __
1" T r i v i a l 2" L e t t h a t ml
be generated by rnl,..
,..., m
.,
mr.
I t i s n o t r e s t r i c t i v e t o suppose
a r e homogeneous and nonzero,
o f degree al ,...,
resp.
11
1. Graded Rings and Modules
N),
Consider f F HornR(!, with n o ,
lj
F N
then : f o r i = 1,..., r , f(m i) =
, 0..c G. For each i , 1 5 i
Oij
5
1J
,
-1 o i l , a -1 Fi = {a. . oi2,.
. . , a -1 i
We will now check t h a t f(Mu) c
uiti} C
No$
‘P~F
r we p u t
c ti
j =1
no
ij
:
r and F = U F i . i=l for all u c H
we may write : - C r rimi with ri c h(R) and, deg r . = uai-1 . Therefore i t follows - i=l 1
For xo E Mo
that f(xo)
=
Zr r.f(m.) =
i=1 l
‘i C rino. However deq rinOo. = ouTaT1 i i = l j=1 1.j ij
.
zr
ui
and therefore : where
‘inO.. F No$ 1J Remark
‘0
= u f l uij €
F for all 1
5
i
5
r.
If M i s not f i n i t e l y generated, then i t may happen
HOMR(M,N)
# H o m R ( M , N ) . For example, l e t R
=
0
ncZ
that
Rn be a graded ring o f
type Z such t h a t P,, # 0, V n c Z. Then there e x i s t s an element (a n)nFZ
n R which i s not i n 8 R P u t M = R(’) and define f F HomR(M,R) by neZ ncZ putting f ( ( x ) ) = C x i a i . By Lemma 1.2.10, c le a rly f f H O M R ( M , R ) . icz If R = 8 R i s a araded r i n g , then R i s cal l e d l e f t fr-Noetherian i f oC6 every l e f t graded ideal i s f i n i t e l y generated ( f o r d e t a i l s see A . I I . 3 . ) . 6
The derived functors of the l e f t exact functors HornR(. ,.) resp. H O M R ( . ,.) by E x t i ( ( . , . ) resp. EXT:
(.,.).
Then we have :
1.2.12. Corollary. I f R i s a graded l e f t Noetherian rino a n d M i s a f i n i t e l y generated graded R-module, then, f o r every n
3
0 :
EXTi(M,N) = Exti(M,N) f o r every graded R-module N .
-
Proof. Since R i s g r . l e f t Noetherian, M has a f i n i t e resolution :
_L
...
F2
+
F1
Fo
+
M
-
0
where the Fi a r e f r e e graded R-module and al s o f i n i t e l y generated.
j
A. Some General Techniques in the Theory of Graded Rings
12
Now C o r o l l a r y 1.2.11. f i n i s h e s t h e p r o o f . Remark.If t h e group G i s f i n i t e , we have EXTi((F1, N) = Ext:(M, n
?
N) f o r e v e r y
0, and M, N C R-gr. t h i s s e c t i o n we s t u d y t e n s o r p r o d u c t s o f praded modules
To c o n c l u d e
Throughout R =
Ro i s a oraded
8
r i n g o f t y p e G; w h e r e v e r z - g r
i s being
OCG
c o n s i d e r e d we assume t h a t Z i s e q u i p p e d w i t h t h e t r i v i a l G - g r a d a t i o n .
c
Definition. Let M
gr-R,
c
by p u t t i n g (M @ N)O, o R elements x bo y w i t h x
c
R-gr. The a b e l i a n group M Q N may be graded R G, equal t o t h e a d d i t i v e subgroup p e n e r a t e d by N
c Ma,
e NP
y
such t h a t aB = u
.
The o b j e c t o f Z - g r
hence d e f i n e d w i l l be c a l l e d t h e Graded t e n s o r p r o d u c t
of M and N. An
N c R-gr i s s a i d t o be g r - f l a t i f t h e f u n c t o r - bo N : pr-R - 2 - g r R
is
exact ( i t i s i n general o n l y r i g h t e x a c t ) . L e t us l i s t some e l e m e n t a r y p r o p e r t i e s ,
the p r o o f o f which i s s i m i l a r t o
t h e p r o o f o f t h e c o r r e s p o n d i n q s t a t e m e n t s i n t h e ungraded case. P r o p o s i t i o n . L e t R and S be graded r i n g o f t y p e 6 . Consider
1.2.13. M
c
gr-R,
N C R-gr-S,
then
M
6E N i s a graded r i n h t S-module.
R
i . 2 . 1 4 . P r o p o s i t i o n . L e t M c gr-R,
P
c
N
c
R-pr-S,
S-gr, M
c
S-gr-R,
gr-S,
then t h e r e i s
a n a t u r a l graded isomorphism : HOMS(M 8 N,P)
h
R
HOMR ( W , HOMS(N,P)).
P r o p o s i t i o n . L e t P c gr-R,
1.2.15.
N
c
then there i s
a c a n o n i c a l graded morphism @, @ : P 8
R
HOMS(M,N)
+
d e f i n e d by + ( p 8 f ) g
c
HOMR (P, M).
an isomorphism.
HOMS(HOMR(P,M), N),
(c)
= ( f o
g) (p), f o r p
F P,f
C
HOMS(M,N),
I f P i s f i n i t e l y g e n e r a t e d and p r o j e c t i v e t h e n
+
is
13
1. Graded Rings and Modules
Proposition. I f P
1.2.16.
c
N c S-qr, Y c R-gr-S,
gr-R,
then t h e r e
e x i s t s a canonical graded morphism Y, 'Y : HOMR(P,M) @
S
d e f i n e d by
N
HOMR (P, M 60 S
-c
N), I f P i s f i n i t e l y generated and
Y ( f 6 D n ) ( p ) = f ( p ) 8 n.
p r o j e c t i v e , then 'Y i s an isomorphism.
M(o) 8
R
c
Remark. I f G i s a b e l i a n then f o r any M
1.2.17.
N(T)
= (M 8
R
1.2.18.Proposition.
N)
( m ) f o r o, T
Let M
c
N c R-gr we have
qr-R,
c G.
then M i s a r - f l a t i f and o n l y i f M is
R-or,
f l a t i n R-mod. __ P r o o f . That f l a t n e s s o f
Conversely, as i n [ 5
M i s obvious.
e n t a i l s gr-flatness o f
1, i t i s easy
t o show t h a t M i s t h e i n d u c t i v e
l i m i t i n R-gr of g r - p r o j e c t i v e modules. Then, by C o r o l l a r y I.2.2., i s t h e i n d u c t i v e l i m i t o f p r o j e c t i v e modules, hence 1.2.19.
C o r o l l a r y . The g r - f l a t dimension o f
f o r any
+
MI+
M
+
N c cjr-R t h e sequence 0
i n 2 - gr. 1.2.20.
which i s denoted by
wdimRs
0
An e x a c t sequence
i s flat.
M c R-or, denoted by qr-wdimRM
( d e f i n e d as t h e correspondincj uncraded concept wdimR), i s equal t o
P
-
M"
+
0 i n R-gr i s s a i d t o
N
60
PI
+
R
C o r o l l a r y . The e x a c t sequence 0
+
+
M'
N
60
R
+
M
M
+
+
M"
N
be Q r - p u r e if
60 M"
+
R
+
0
gr-pure i f and o n l y i f t h e corresponding e x a c t sequence 0
0 i s exact
i n R-vr i s +
Ma+
M-x
+
0
i n R-mod i s a pure sequence. Proof . result
The i d e a of [62]
o f the p r o o f i s modeled upon
e n t a i l s t h a t the or-purity o f 0
unoraded techniques i . e . a +
M'
+
M
+
M"
t h a t t h i s sequence i s an i n d u c t i v e l i m i t o f a f i l t e r e d s e t o f
+
0
exact
implies
A. Some General Techniques in the Theory of Graded Rings
14
s p l i t t i n g sequences of the form : 0 + M '
+
M ' 8 Pfi)+
P(i)
--f
0 where
each P ( i ) i s f i n i t e l y presented i n R-gr. The properties of the l a t t e r sequence imply the corresponding ungraded analogues f o r
0
+
5'
+
M' 8 ,(i) -
-f
,(i) -
--f
0.
Therefore by the r e s u l t of l o ~ .c i t . the l a t t e r sequence i s pure i n R-mod a n d the same will hold f o r the l i m i t sequence 0
+
r'
+
C -
--f
Conversely, purity in R-mod obviously implies g r - p u r i t y in R - g r .
M" -
+
0
15
1. Graded Rings and Modules
1 . 3 . Strongly Graded Rings 8 Ru of type G i s s a i d t o be a s t r o n q l y araded r i n o OEG i f RoRT = R m f o r a l l cr, T c G . I f G = z then a z-graded r i n q
A graded r i n g R =
of type G
i s s t r o n g l y graded i f and only i f RR1 = R , i f and only i f R I R = R . Note a l s o t h a t a commutative r i n g can only be s t r o n g l y graded by an a b e l i a n group. 1 . 3 . 1 . Examples. a . Let G be a group, A a r i n g . The group a l g e b r a R = A [GI i s a s t r o n g l y graded r i n g of type G .
R i s a s t r o n g l y graded r i n g of type G and i f n : G
b. I f
+
H i s an
epimorphism t h e n R(ti) i s s t r o n g l y graded. I f H I i s a subgroup of G then R ( H ' ) i s a l s o s t r o n g l y Graded.
c . I f R i s s t r o n g l y graded then so i s R/I, f o r every graded i d e a l I of R . 1.3.2. Lemma.
The G-graded r i n g R i s s t r o n c l y Graded if and only i f
1 c R R -1 f o r a l l u E G . u u
Proof.
Suppose 1 c R R
u u
holds f o r a l l o c F. For any
T
c F i t follows
then t h a t : =
l.RoT
c RuRo-lRm
c
RORT,
hence Rm
=
RORT
The converse i s even more obvious.
&
: It
i s c l e a r t h a t t h e foregoing may be s t r e n a t h e n e d t o : R i s
s t r o n g l y graded i f 1 E R o , O T 1 f o r some s e t of g e n e r a t o r s o i , i
C
I,
Of
1 1
1.3.3. Corollary. I f
R i s s t r o n g l y graded then, f o r a l l
o E
G, Ro i s
f i n i t e l y generated and p r o j e c t i v e in R -mod and a l s o i n mod-R n e u t r a l element of G ) .
e ( e the
G.
A. Some General Techniques in the Theory of Graded Rings
16
Proof. From R o - l Ro = Re i t f o l l o w s t h a t Ro c o n t a i n s a f i n i t e l y generated Re-module
Lo
y i e l d s : Lu
such t h a t R o - l = Ro ;
hence
Lu
i s finitely
Ru
Lu
Then Ru R o - l
= Re.
= Ru y i e l d s
= RR,e
9enerated
in
R -mod ( s i m i l a r l y as a r i g h t Re-module). We now e s t a b l i s h t h a t Rois a
p r o j e c t i v e l e f t Re-module. I f 1 =
c
uivi
w i t h ui
c Ru-l,
r i g h t m u l t i p l i c a t i o n by ui d e f i n e s an Re-morphism f o r each r F R
we have r = zi pi(r)vi.
vi
F Ru then
: Ru * Re such t h a t
pi
Therefore by t h e well-known
"Basis Lemma", Ru i s p r o j e c t i v e i n Re-mod ( s i m i l a r i n mod-Re). For a graded r i n g R o f type G we may d e f i n e a f u n c t o r R
Oa
-
: Re-mod * R-gr
Re as f o l l o w s : f o r an M F Re-mod, d e f i n e (R
M)u = Ro
60
Re
bo
M f o r a l l u F G.
Re
On t h e o t h e r hand one can d e f i n e t h e r e s t r i c t i o n f u n c t o r ( - ) e : R-or by p u t t i n g (M), = fe : Me
1.3.4.
= Me w h i l e a morphism f : M
+
+
Re-mod
N i n R-gr r e s t r i c t s t o (f),
* N,.
Theorem. I f R i s a graded r i n g o f type G then t h e
f o l l o w i n g statements
a r e e q u i v a l e n t t o one another :
1. R i s a s t r o n g l y graded r i n o . 2. Every graded R-module i s s t r o n g l y graded. 3. The f u n c t o r s R
-
60
and ( - ) e define an equivalence between t h e
Re Grothendieck c a t e g o r i e s R-gr and Re-mod. Proof. -
=
I n p r o v i n g 1 a 2 we o n l y have t o e s t a b l i s h
we have : Mm
= ReMm
= RoRo-l
Mm c Rcr
MT, hence
1 = 2. For u, Mm=
Ro
T
<
G
MT, f o r any
M c R-gr. 3 i t w i l l s u f f i c e t o e s t a b l i s h 2 = 3.
Again, i n p r o v i n g 2
Take M c R-gr and l e t f i M = R ca Me Re given by 6M(r 8 me)
=
r me,
--t
M be t h e canonical R - l i n e a r morphism
r c R, me c Me and n o t e t h a t i t i s graded o f
17
1. Graded Rings and Modules
degree e . I t i s e a s i l y seen t h a t 6M i s a n a t u r a l transform of t h e composition f u n c t o r ( R 60 - ) o ( - ) e t o t h e i d e n t i t y f u n c t o r on R-gr. I t i s R
e
c l e a r t h a t SM i s an epimorphism. I f K = Ker of R
8
Re
then K i s a oraded submodule
hM
~ ) ~ (6Fl)e : Re Me and Ke = K e r ( € ~ where
66
Re
M e i s an isomorphism,
R . K = 0 , i . e . K = 0. o e Conversely, i f M c Re-mod l e t aM : M -, ( R ~a M ) e be given by a M ( x ) = 160 x , Re
hence Ke = 0 and a l s o K O
x
6
=
M. I t i s c l e a r t h a t a M d e f i n e s a n a t u r a l transform between t h e i d e n t i t y
functoron R
e
-mod and the composition f u n c t o r ( - )
e
o (R
@J
-).
Re
1 . 3 . 5 . C o r a l l a r y . Let R be s t r o n o l y graded r i n q of type G and l e t M , N C R-gr.
If f : M
+
N i s a morphism i n R-gr, then f i s a monomorphism, epimorphism,
isomorphism i f and only i f f o : M o + N
i s r e s p . a monomorphism, epimorphism
isomorphism i n Re-mod f o r some u C G .
Proof. Since
the f u n c t o r To : R-gr
--f
R-or i s an equivalence i t follows
t h a t f i s monic, e p i c , isomorphism i f and only i f To(f) : M ( c r )
+
N(o)
is
r e s p . monic, e p i c , isomorphism. Now M ( u ) ~ = Mo,
N(u), = No,
hence a p p l i c a t i o n of Theorem 1 . 3 . 4 . y i e l d s
the r e s u l t .
An R-bimodule RMR i s i n v e r t i b l e i f t h e r e e x i s t s N = R , N 69 M
an R-bimodule R N R such
that M
@
1.3.6.
P r o p o s i t i o n . The following p r o p e r t i e s hold f o r a s t r o n 9 l y graded
R
R
s
R a s R-bimodules.
ring R : 1. For every M
c R-or and a l l
u,
T
c G t h e canonical morphism R
60
Ma
Re d e f i n e d by r
2. For every
@J
C,T
x
+
rx i s an isomorphism.
c G , t h e canonical morphism Ro
morphism of Re-bimodules.
6n
Re
RT
+
ROT i s an i s o -
+
M(a)
ia
A. Some General Techniqus in the Theory of Graded Rings
3. F o r e v e r y u c G,
Ru i s an i n v e r t i b l e R -bimodule. e
Proof. ___ 1. D i r e c t f r o m Theorem 1.3.4.
c?
2. By t h e f i r s t p a r t , R
RT
+
R(T) i s an isomorphism, hence (R 69 RT)a =
Re
Re
3. Immediate f r o m 2. 1.3.7. only i f 1.3.8.
M
C o r o l l a r y . L e t R be s t r o n g l y p-aded,
Mo
=
0 f o r some u
c
C R-gr,
then
M
=
0 i f and
G.
Corollary.
I n a s t r o n g l y craded r i n c l R e v e r y araded l e f t i d e a l I i s g e n e r a t e d as a l e f t i d e a l by Ie. __ P r o o f By Theorem I . 3 . 4 . , thus : I = R I 1.3.9.
2.,
I i s s t r o n g l y graded, hence I G = R o I e and
e'
C o r o l l a r y : L e t '0 : R
+
S be a graded r i n a morphism, o f dearee e,
r i n g s R and S. Then
between s t r o n g l y graded
'0
i s injective (surjective,
,oe : Re
b i j e c t i v e ) i f and o n l y i f t h e r e s t r i c t i o n
+
Se i s i n j e c t i v e
(surjective, bijective). Proof.
If '0 i s i n j e c t i v e t h e n by C o r o l l a r y 1.3.8. e
i s i n j e c t i v e too. I f entails
' D ~i
Se = ,o(Ro-l)
and t h u s So = S
u
S
e
=
s s u r j e c t i v e t h e n ,n(RU-])
O(RO)
S o - l 'o(Ro).
'R
c Su-1 and 'c(Re) =
se
T h e r e f o r e Su-l ,o(Ro) = Se
S S -1 ' D ( R ~ ) y i e l d i n g
u u
i t follows that
So
=
' ~ ( 1 7 ~f )o r u C C-.
L e t us now f o c u s on some methods f o r c o n s t r u c t i n g s t r o n g l y graded rings L e t A be any rincj; w r i t e Z(A) f o r t h e c e n t e r o f A, U(A) f o r t h e g r o w s of units
o f A,
A u t ( A ) f o r t h e g r o w o f r i n n automorqhisms o f A and
19
1. Graded Rings and Modules
I n n ( A ) f o r t h e subgroup of A u t ( A ) c o n s i s t i n g o f i n n e r automorohisms o f A. C o n s i d e r an A-bimodule M and a, p c A u t ( A ) . L e t aM
B
be t h e A-bimodule
: e q u i p t h e a d d i t i v e grouo u n d e r l y i n g M w i t h an
d e f i n e d as f o l l o w s
A-bimodule s t r u c t u r e g i v e n by a . x = a ( a ) x ,
c
x.a = x p ( a ) f o r x 6 V, a
A.
l 1. I d e n t i f y M and P 1.3.10.
Lemma. I f a, p, y
1. The map x
-t
y ( x ) d e t e r m i n e s an isomornhism o f A-bimodules
2. The map x c3'y
lAp +
IAa
t Aut(A) then :
+
x a(y)
: aAP
+
yaAyp.
d e t e r m i n e s an isomorohism o f A-bimodules :
IAap
c
3. There i s e q u i v a l e n c e between ap-' 4. I f 6 : A u t ( A )
+
I n n ( A ) and lAa 2 ]Ag as A-bimodules
P i c ( ? ) i s g i v e n by ~ ( a )= [ 1A
1
I
then t h e fo'ilowina
sequence o f groups i s e x a c t : 1
Inn(A)
--f
+
Aut(A)
Pic(A)
--f
6
Proof.
1. and 2 a r e o b v i o u s .
3. Suppose t h a t f : lAa
--f
A
i s an A-bimodule isomorphism. L e f t
I P
A - l i n e a r i t y o f f y i e l d s t h a t f ( x ) = xu where u = f ( 1 ) f(.(a))
= f(1.a)
=
L e t us denote u x u - l by a u ( x ) , A and a = au o p, t h u s a p - l =
uau
f o r s c ~ eu
c
c
then
aU
=
u p(a)u-'.
i s an i n n e r automorphism o f
I n n ( A ) . Conversely, s u m o s e t h a t & - ' ( a )
U(A).
I f we d e f i n e f : lAa
by f ( x ) = xu f o r a l l x lAg c l e a r t h a t f i s an A-bimodule isomorphism. +
c
4. A d i r e c t conseauence o f 2. and 3.
1.3.11. Lemna. L e t P be an i n v e r t i b l e A-bimodule, 1. A
1
EndA (PA) and A
U ( A ) . Moreover
u p ( a ) . On t h e o t h e r hand f ( u ( a ) ) = f ( a ( a ) . l ) = a ( a ) u .
Hence i t f o l l o w s t h a t a ( a ) u = @ ( a ) o r .(a)
-1
C
c
EndA(AP)o.
then :
lAa,
then i t i s
=
A. Some General Techniques in the Theory of Graded Rings
20
( P ) , w h e r e b y ~ n d ~ - ~ (wpe )d e n o t e t h e g r o u o o f a l l A - b i m o d u l e A-A h o m o m o r p h i s m s o f P.
2. Z(A)%End
3. I f P
2
A as l e f t A-modules t h e n P
n
lAaas A - b i m o d u l e s f o r s o m e a
C
Aut(A).
4. P i s l e f t ( r i g h t ) f i n i t e l y g e n e r a t e d p r o j e c t i v e A - m o d u l e .
5 . T h e d u a l P* P @A P
=
HornA ( P A , A ) i s a n i n v e r t i b l e A - b i m o d u l e a n d
P*
~1
A,
A as A - b i m o d u l e s .
h
6. I f M , N c A-mod., ap :
P
HornA (M,N)
t h e n t h e homomorphism --f
HornA ( P
N), g i v e n by ap(f)
M, P
=
1 63 f i s a n
isomorphism. 7. I f M , N are A - b i m o d u l e s , t h e n t h e h o m o m o r p h i s m a p : HomA-A(M,N)
--f
HomA-A ( P
N),given by
A M, P
@
a p ( f ) = 163
f, i s
an isomorphism. Proof ___ 1. By a s s u m p t i o n t h e r e e x i s t s a n A - b i m o d u l e Q a n d t h e A - b i m o d u l e i s o m o r p h i s m s
P BAQ2 A,
A. L e t 6 : A
Q mAP
L e t x . c P, y 1. c Q , i = 1, 2 ,..., n = c '@(Xi @ Y i ) . i=l n If e ( a )
E n d ( P A ) be d e f i n e d by e ( a ) ( x ) = a x . n n be s u c h t h a t 1 = '0 ( Z x i 60 y i ) = i=l
+
0 t h e n a = a . 1 = c , P ( a x i ~ y i =) 0, t h e r e f o r e i =1
=
i n j e c t i v e h o m o m o r p h i s m . On t h e o t h e r f
@
l g : P @A Q
--f
8
i s an
h a n d , i f f c End (PA) t h e n
P @A Q i s r i g h t A - l i n e a r .
T h e r e e x i s t s a.
F A such
t h a t t h e following diagram is commutative. 0
P @ A Q - - - ~ A
f'
I f@
@AQ
i
4a0)
IQ
'D
____j
A
w h e r e s ( a o ) ( a ) = aoa. I f w e d e f i n e 0 : P
--f
P by p u t t i n g g ( p ) = aop f o r
21
1. Graded Rings and Modules
a l l p 6 P, i t i s c l e a r t h a t f 60 1
Q -- g e l 9 .
r.
P @A A
5
From ( P @ A Q ) ~ ~ A P % P B ARAP) (Q
P i t f o l l o w s t h e n t h a t f = q and c o n s e q u e n t l y 0 i s s u r j e c t i v e .
2. We d e f i n e : e Z ( A )
e
statement I . ,
--f
EndA-A(P),
x C P . By t h e
e ( a ) ( x ) = ax, a 6 Z ( A ) ,
i s i n j e c t i v e . L e t f t EndA-A(F); t h e r e e x i s t s a f A such
t h a t f ( x ) = a x f o r a l l x t P. As f i s a l e f t A-homomorphism t o o , we have n f ( b x ) = b f ( x ) , o r ( a b ) x = ( b a ) x f o r a l l x 6 P. S i n c e 1 = Z q ( x i boyi) n n i= 1 t h e n ab = C q ( a b xi ca yi) = 2 Q (ba xi 60 yi) = ba. Hence a 6 Z ( A ) . i=l i=l
A i s a l e f t A - l i n e a r isomorphism and l e t a 6 A. We d e f i n e -1 a l e f t A - l i n e a r isomorphismg:P+P by p u t t i n g ~ ( p )= f (f(p)a) for a l l
3. Suppose f : P
-f
p C P. S i n c e A
5
EndA (AP)o, t h e r e e x i s t s an a ( a )
c
A such t h a t ~ ( p )= p a(a),
i n o t h e r words f ( p a ( a ) ) = f ( p ) a f o r a l l p C P . E v i d e n t l y a 6 A u t ( A 1 and t h e f o r e q o i n g may t h e n be p h r a s e d as f o l l o w s : f : Pla
+ A i s an A-bimodule
isomorphism. 4,5.First
u,
+ A such t h a t t h e diacrams
we p r o v e t h a t we may choose t h e A-bimodule isomorphisms P
and Q @A P
Q +A
+
a r e commutative ( h e r e a, p,
a',
p'
a r e t h e o b v i o u s maps)
Indeed, suppose we want t h e f i r s t diapram
t o commute. S i n c e a l l maps a r e
bimodule isomorphisms, t h e r e e x i s t s an A-bimodule automorphism u : P such t h a t p o (Ip 60 g ) = u o a o ( f
60
+
P,
lp). By t h e s t a t e m e n t 2. t h e r e e x i s t s
c C Z(A) such t h a t u ( x ) = c x f o r a l l x C P. i s an i n v e r t i b l e element. Hence we r e p l a c e
As u i s an automorohism t h e n c Q
by c . q and t h e f i r s t square
22
A. Some General Techniques in the Theory of Graded Rings
i s commutative.
!k now p r o v e t h a t t h e second d i a g r a m comrnutes,i.e.a'o(Ji
P.
Indeed, l e t q, q ' C Q , p C
(p'o(lQ
c Q.
(q
8 cp))
Q O ~ Q O9
Let
Q
6 0 ~P
have q ( q ( q
60
p) q '
d
lk
TJ
and p '
A
60
p') =
Q - p'o(lo
@
cp)=O
Then (a'0(J,6OlQ))( q C 4 p w q ' ) = Ji(q Oa o l q ' ,
') = q
+
1 )
@
(p
60
c P.
q ' ) . We denote d = i ( q @ o ) q ' - q 4 P @ q ' ) ;
( q bo p ) . q ( q '
~
i s an A-bimodule morphism, we
Since $
60 D ' )
t h e f i r s t diagram i s commutative, t h e n : p J , ( 9 '
= $(q 60
60
p . $ ( q ' 60 p ' ) ) . S i n c e
p ' ) = ~ ( 60p 4 ' ) p '
and t h e r e f o r e :
q(Ji(a
Qp
P) q '
P') = c(q
@
IP(P
@
S ' ) P ' ) = j i ( q . v ( p 6O 4 ' )
P')
@
Hence i t i s c l e a r t h a t q ( d bo p ' ) = 0 and hence d = 0.
Q -
T h e r e f o r e a'o($ rn 1 )
p'o(1
q) = i 0 Q@
v
Now we p r o v e t h e s t a t e m e n t 4. S i n c e P 6 0 ~Q n 1 = c G (pi 69 qi) where pi c P, qi 6 Q. i=1
s i n c e t h e diagram
--f
A is an isomorphism t h e n
(1) i s commutative. Thus P i s f i n i t e l y g e n e r a t e d and
projective.
We p r o v e t h e s t a t e m e n t 5 . Note t h a t cp i n d u c e s a bimodule homomorphism
Q
2 HomA(PA,A),
p c P.
Since 1 =
n
=
e ( q ) ( p ) = $ ( q QO p ) . If e ( q ) = 0 t h e n $ ( q
c q(q i=l
M
pi)qi
c"
q(pi i=l =
0
60
qi),
pi
c P, qi
C
@
Q, t h e n q =
P) = 0 f o r a11 n z 9.9(pi @ qi)
i=l
( as t h e d i a g r a m ( 2 ) i s commutative). Hence
@
is
i n j e c t i v e . To e s t a b l i s h s u r j e c t i v i t y l e t f C Hom (P ,A). I f p C P t h e n n n A * n 2 f ( P i ) $ ( q i €t P) = @ Pj) = f ( p ) = f ( c Q(Pi @ qi)P) = f ( 2 .Pi.c(qi i=1 i=l ;=I n n = c Ji(f(Pi).qi 6O n) = C( 2 f(Pi).qi @ P). i=1 i=1
=
I. Graded Rings and Modules
23
I f we denote y =
n 1 f(pi)qi, =1
The s atements 6
and 7. f o l l o w f r o m t h e f a c t t h a t t h e f u n c t o r M
t h e n f ( p ) = q ( q 60 p ) = e ( q ) ( p ) and hence a(q)=f
+
P
AM
@
i s an e q u i v a l e n c e . R e c a l l t h a t P i c ( A ) i s t h e g r o u p o f isomorphism c l a s s e s [ P I o f i n v e r t i b l e A-bimodules.
The group l a w i s : [ P I . [ ? ] = [ P tzA 3 1 . By lemma 1.7.11.
we have t h a t [ P I
-'
=
[HomA(PA,A)l
=
[HO~~(~P,A)].
L e t A be a r i n g . There e x i s t s a c a n o n i c a l group homomorphism
1.3.12.
Lemma
Pic(A)
2 Aut(Z(F)).
Proof.
L e t [ P I C P i c ( A ) and c C Z ( A ) . We have t h e homomorohism o f bimodules
P
--f
P, x
Since A
h
+
xc.
End A ( P A ) o (Lemma 1.3.11.1).
a u n i q u e element
There e x i s t s
a ( c ) such t h a t a ( c ) x = x c f o r a l l x F P. C l e a r l y t h a t aP : Z(A) P P
i s an automorphism and e : P i c ( A )
+
Aut(Z(A)),
e([Pl)
--f
Z(A)
= up, i s a aroup
homomorphism. : 6 * P i c ( A ) a aroup
L e t G be a group ( n o t n e c e s s a r y commutative) and homomorphism. P u t +(u) = [ P o l f o r a l l o
__ to 9
we mean a f a m i l y f = tfo,T;
U,T
i s a n isomorphism o f bimodules f o r a l l
F
c
G. By a f a c t o r s e t a s s o c i a t e d
GI where O,T
6
fo,T;
Po
c+
u,T,~
--f
6 and t h e f o l l o w i n g
diagrams ( 3 ) :
a r e commutative f o r e v e r y
PT
F G,
where a : Re * A i s a g i v e n A-bimodule isomorphism.
Y
Pa
24
A. Some General Techniques in the Theory of Graded Rings
We denote by Fs(+) t h e s e t o f f a c t o r s e t a s s o c i a t e d t o I f f c Fs(+) t h e n we denote by A c f, 0, G > =
0
w i t h m u l t i p i c a t i o n o f elements i n A < f, 0, G >
Po
ocG d e f i n e d by t h e r u l e :
x.y = f
,J,T
c
(x ay) for x
1.3.13. P r o p o s i t i o n . A < f, and A c f ,
+,
c
Po, y
+,
G >
PT
i s a s t r o n g l y graded r i n g w i t h i d e n t i t y
c o n t a i n s a s u b r i n g i s o m o r p h i c t o A; i . e .
G >
P
e
‘k
A
(e i s the i d e n t i t y o f G). Conversely, i f R =
8
Ro i s a s t r o n g l y graded r i n g o f t y p e G t h e r e e x i s t
UCG
a group homomorphism R - R
e
@
: G
+
P i c ( R e ) and a f a c t o r s e t f
c Fs (+)
such t h a t
c f , + , G > .
Proof.
+,
G > i s a rino. Since
: Pe mAPe + Pe i s an e ,e A- b i m o d u l e isomorphism, Pe i s a s u b r i n a o f A c f, 0, G > . S i n c e + ( e ) = =
Clearly A < f,
-
[ A ] = [ P e l , we have t h a t Pe
A as A-bimodules.
c A.
t h a t Pe = Ap = pA and aD = pa f o r a l l a we can w r i t e f bimodule,
e,e
then c
c
( p bo p ) = cp f o r some c
c
f
There e x i s t s p t Pe such
S i n c e fe,e i s an isomorphism A. S i n c e P
e
i s an i n v e r t i b l e
Z ( A ) and c i s a u n i t . I f we p u t p o = c
( p o @ po) = po. and t h e map A
+
Pe, a
apo
--f
-1 p, t h e n fe,e
i s a r i n a isomorphism.
F u r t h e r m o r e , po i s t h e i d e n t i t y o f A < f, 9, G >. Indeed, i f x F exists y = fe,JPo
c
Po such t h a t x = f 60 fe,o
e ,o
0
S i m i l a r l y we have x.p
0
c
A c f,
there
( p o boy) and t h e n fe,o ( p o 63 x ) =
(Po m y ) ) = fe,o(fe,e(Po
Hence p . x = x f o r a l l x
Po
+,
‘53
Po) boy) = feJP0
G >.
= x f o r every x
c
A < f,
T h e r e f o r e , po i s t h e i d e n t i t y e l e m e n t o f A
c
f,
a,
G >.
a,
G
P.
The second s t a t e m e n t f o l l o w s f r o m t h e p r o p o s i t i o n 1 . 3 . 6 .
boy) = x .
25
1. Graded Rings and Modules
L e t 0, + ' : G
+
+,
two f a c t o r s e t s a s s o c i a t e d t o
, u
[ Pia ]
=
A morphism
< Fs(+),
P i c ( A ) two groups homomorphisms and f respectively t o
+'.
f ' F Fs(+')
P u t +(o) = [ P o ] , + ' ( o ) =
C G.
f i n f ' i s a f a m i l y a = ( a o ) o c G where a .
Of
: Po
PL
+
are
A-bimodule homomorDhisms and t h e f o l l o w i n g diaaram
i s commutative f o r e v e r y
1.3.14. P r o p o s i t i o n . A c f
C G.
O,T
The map
0, G > + < f ' , a ' , G
'1
=
:
8 . a ocfi
i s a graded r i n g homomorphism
2.
I f ae i s s u r j e c t i v e t h e n a(1) = 1. Moreover f uo i s an isomorphism f o r e v e r y u
only Proof =
Ifx f Po, y
< P
ao,T ( f o , T ( ~ QO Y ) )
and t h e r e f o r e
a
i s an isomorphism i f and
< 6.
then a ( x y ) = a(xy) = a ( f
= f ' o,T
( a o ( x ) 6~
.,(Y))
0 3 1
( x my) =
= a,(x).a,(y)
=
a(x).c(y),
i s a r i n p homomorphism. I f po i s t h e i d e n t i t y element
a
f o r A c f , a, 6 > t h e n
fe,e ( p o 60 po) = p,
apo = p a f o r e v e r y a c A. We denote p', 0
and Pe = Apo = poA and
= a(p
0
diagram ( 4 ) i s commutative, f ' ( p o l R p o l ) = p',. then PA = Apo' = pt; A . C l e a r l y p',
) = ae(po). Since the Since
i s surjective e i s the i d e n t i t y o f A < f ' , G 7. a
+,
The second s t a t e m e n t i s o b v i o u s . 1.3.15. A
c
that
f,
Remark. From t h i s p r o D o s i t i o n we have i n p a r t i c u l a r t h a t t h e r i n g
+,
G > = [
i s i n d e p e n d e n t o f t h e c h o i c e o f t h e f a m i l y ( P o ) o C G such
Po]
A. Some General Techniques in the Theory of Graded Rings
26
A-bimodule isomorphisms
Indeed; i f +(u) = [ P I D ] t h e n t h e r e e x i s t a
: P',
+
f = ( f
U,T
)
Po.
a,,'
I f we denote by f ' f
C
Fs(+)
1.3.12
o
f
U,T
we deduce t h a t A
i s a factor set. <
f, 4 , C- >
h
A
<
f',+, G
2.
I . By Lemma
P i c ( A ) be a y o u p homomorphism. P u t @(u)= [ P
+
where
a,),
o(a,c%
then c l e a r l y f =
By t h e p r o p o s i t i o n 1.3.14. Let 9 : G
U,T
-1 = a m
we have t h e homomorphisms
2 P i c ( A ) 2 A u t (Z(A) Hence + d e f i n e s an a c t i o n o f
6 on U(Z(A). I f u
d e f i n e O ( C ) such t h a t
xc
G
=
O(C)X
C
G, c C Z(A), t h e n we
f o r e v e r y x C Pa.
2 Then we can c o n s i d e r t h e y o u p Z (G, U ( Z ( A ) ) ( t h e group o f c o c y c l e s o f second degree) i . e .
2 t h e group o f f a c t o r s e t s o v e r G. An e l e m e n t q C Z (G,U(Z(
i s a f u n c t i o n q : GxG
+
U ( Z ( A ) ) , q(u,.)
( t h e group of c o b o u n d a r i e s ) , i . e .
d : G
+
U(Z(A)), q
( 0 , ~ ) =
U ( Z ( A ) ) , d ( u ) = d,
--f
.
such t h a t,,,q,
~(q,,~) =
2 f o r e v e r y o,-c,e 6 G . A l s o we can c o n s i d e r t h e group B (G,U(Z(A))
= qm,e.qo,,
q : G x G
= ,,,q,
t h e group o f a l l f u n c t i o n s
qu,T3 f o r w h i c h t h e r e i s a f u n c t i o n
such t h a t ,,q,
=
-1 -1 dm du o(dT) .
2 We have t h a t B2(G, U(Z(A)) i s a subgroup o f Z (G, U ( Z ( A ) ) ) and we denote by
H
2
1.3.16.
i,
(G,
U ( Z ( A ) ) ) t h e f a c t o r Croup
Theorem. L e t 9 : G
If
= (fo,T)o,T
then q f = (q
+
c G'
u , ~ '' o , ~1u , 6~ G
Z2(G, U(Z(A)))/B2(G,
P i c ( A ) a group homomorphism.
Fs(')
and q = (qu,T)u,T
U(Z(A))). P u t @ ( u )= [ p J , u C (
6 G
,
c z~(G,u(z(A)
i s a f a c t o r s e t a s s o c i a t e d t o 9.
2 ii)C o n v e r s e l y , i f f, g C F s ( 0 ) , t h e n t h e r e e x i s t s q C Z (G, U ( Z ( A ) ) ) such t h a t g = q f . 2 F u r t h e r m o r e , q C B (G, U ( Z ( A ) ) ) i f and o n l y i f t h e r e e x i s t s a graded r i n g A-isomorphism from A < f, 4 , G > t o A < q f ,
@,
G >
.
27
1. Graded Rings and Modules
Proof.
w
i ) We o n l y have tQv e r f f y t h a t t h e diagram 160 g T , , Po 60 PT 60 P,
ii)
If
qn,,
=
Fs(+).
f 3
go,,
fitT
.
=
C l e a r l y , qo,,
P o @ PT. By Lemma 1.3.11.2. q %,T
G and 9 = (!?o,T)o,T
(f,,,C)o,T
C U(Z(A)). Hence
i s the m u l t i p l i c a t i o n by an element
0,T
.
go,T = qT ,
Suppose now t h a t q c B d : G
+
fu,,
qo,,
2
.
Because t h e diagram ( 5 ) i s an element from
( G , U ( Z ( A ) ) ) . Then t h e r e e x i s t s a f u n c t i o n = d
U(Z(A)), d ( o ) = do such t h a t q
d e f i n e a : A < f,
=
W@ D U t
U(Z(A))).
Z2(G,
If x
G
i s a A-bimodule automorphism o f
i s commutative, we o b t a i n t h a t q = (q,,T)u,T
X C ?
c
+,
G >
+
A < qf,
+,
'd,
o(dT)-'.
We
03,
G > by a ( x ) = do x f o r every
o.
c
Po, y C PT then a ( x y ) = d
do
4dT)
-1
fo
,(x@Y)
~
= qo,T
fo,T ( X 60y) = fo,T
(do x @ d T y ) = a(x).a(y).
A. Some General Techniques in the Theory of Graded Rings
28
On t h e o t h e r hand, i f x t Po, a C A , we have a ( a x ) a i s A-linear.
=
a a ( x ) and hence
C l e a r l y a i s an isomorphism, hence a graded r i n g isomorphi
Conversely, we suppose t h a t t h e r e e x i s t s a graded rino A-isomorphism a : A < f,
*,
can w r i t e
e =
G > 8
+,
A < qf,
+
G >
, where
2
q t Z (6, U(Z(A))).
i s t h e A-bimodule automorphism of Po.
a, where a
tJCG
There e x i s t do t U ( Z ( A ) ) such t h a t a,(x)
= dox f o r every
a commutes with m u l t i p l i c a t i o n we o b t a i n t h a t q,,T =
o r q0,.,
1 dm d, a(dT)
=
1.3.17. C o r o l l a r y . +
'.
x c Po. Because
do a ( d T ) f o , T ( x m y )
( x s y ) f o r every x t Po, y t PT. Hence ,,q,
doT f O , T
Then we
do u (dT)
=
dm
2
Hence q t B (G, U ( Z ( A ) ) ) .
Let f t F s ( + ) . Then t h e map q
--f
qf of Z L ( G , U(Z(A)))-
F S (a) i s b i j e c t i v e .
Proof.
D i r e c t l y from Theorem 1.3.16, i )
If
G
@ :
+
Pic(A) i s a group homomorphism, we can c o n s i d e r t h e s e t of
s t r o n g l y graded r i n g s
P
= {
A
<
t h e s e t P/z, where D, D ' c P, D
f , a, F
> }f
Fs(+). W e denote by C ( A , + :
D ' i f and only i f D and D ' a r e graded
A-isomorphic. 1.3.18. C o r o l l a r y . --f
Let f t Fs (a). Then the map 9
--f
qf of H
2
( G , U(Z(A)))
C ( A , a) i s b i j e c t i v e .
e apply Theorem I . 3 . 1 6 . , i i ) . Proof. W 1 ile denote by Z ( G , U ( Z ( A ) ) )
the s e t o f maps d : G Let
+
: G
S
A
<
=
+
U ( Z ( A ) ) such t h a t u d
(T)
d(o)
d(m)-'
i.e. =
1.
Pic(A) be a group homomorphism and f t F s ( + ) . We denote
f , 9, G >. Let
such t h a t Re
+
t h e s e t o f cocycles of f i r s t degree,
R =
0 OCG
R, be a s t r o n g l y graded r i n q of type C
A . We denote by AlgrA (R,S) the set of G-graded r i n g
homomorphisms which a r e a l s o A - l i n e a r .
29
1. Graded Rings and Modules
1.3.19. Proposition. Let
i)
c A1grA(R,S).
9
i s an isomorphism. 1 i i ) The map Z ( G , U ( Z ( A ) ) ) 1 = (du)ocG, d F Z ( G , U ( Z ( A ) ) ) tp
+
AlarA(R,S), given by d
+
(where d =
and ( d . , D ) ( x ) = do .(x)
f o r every x c SD,
d.,p
e G, i s b i j e c t i v e . iii)
Z1(G,U(Z(A))
c-
AlgrA ( S , S ) .
Proof. i ) Suppose t h a t .(x) Since x y 6 Re
e
0, x
=
A and
‘p
xi
C
Ro-l
Ru
,o(Ra-l), yi c
C
and then (yxi)yi c Hence y
=
Ru. Let y c R o - l
i s invertible, x
we have 1 F $9 ‘9
Z(yxi) yi i
@
=
C
@
UC
G
=
‘p
0 a n d therefore D i s i n j e c t i v e .
, R ( R ~ ) and
such t h a t 1 = 7 xiyi. I f i (Ro).(,p i s A-linear).
e
,p(Ro),
or
= $ ‘9
--f
,p(RD)
d.,D
-1
= So.
hence there e x i s t s y
Therefore
then yxi c S e
So
C
‘3
i s surjective.
i s injective.
C A 1 c A ( S , S ) . By the
statement i ) ,
where du i s a n A-bimodule isomorphism of So.
there e x i s t s an element do c U ( Z ( A ) ) such t h a t the map multiplication
= ,p(x).~~(y)=O.
(xy) = (xy)rp(l)=(xy).l.
(RJ
AlgrA (R,S) then d Bu,
(Ro-l)
‘0
i i ) I t i s c l e a r t h a t the map d
If -$
. Then ,o(xy)
i s A-linear we have 0 =
Hence xy = 0. Since R o - l Since 1
C
By Lemma 1.3.11.
aa
i s just
by the element dD. Since a i s a rino homomorphism, the
system d = (do)oCG i s an element of
Z’(G,U(A)))
i i i ) We apply i i )
Now consider the following commutative diagram
Pic(A) Where G, G ’ a r e groups and u , F, F ’ a r e aroup homomorphisms
A. Some General Techniques in the Theory of Graded Rings
30
If
( f ' o l , T l ) o , , T # c i s a n element of F s ( F ' ) , then i t i s c l e a r
f' =
t h a t the system f = ( f ' u ( o ) , u ( T i ) o , T
i s a f a c t o r s e t associated t o
ct
F; i . e . f F F,(F). 1.3.20. Examples. 1) Let T be a ring a n d A c T a subring of T , having the same i d e n t i t y
element. We denote by I n v ( T , A ) the group of a l l i n v e r t i b l e two-sided A-submodules of T, where a two-sided A-submodule X of T i s i n v e r t i b l e f o r some two-sided A-submodule of T
in T i f and only i f XY = Y X = A
I t i s c l e a r t h a t every i n v e r t i b l e two-sided A-submodule X of T i s an i n v e r t i b l e A-bimodule. Hence there e x i s t s a canonical oroup homomorphism : Inv(T,A)
, X
Pic(A)
+
[XI.
+
Let G be a group and o : G
for every
~ ( o =) Xo
G
If
CJ,T
C G. W e obtain the group homomorphism
CJ
'qn
9 Inv(T,A)
: Xo 60 0
3
7
XT
+
y C XT , define a f a c t o r s e t associated t o stronaly graded ring
= A c f,
#(p)
#(,p)
=
0
oCG
= can
+,
C.
>.
+.
Then we can define the
This ring i s called
the
'0.
Xo with the canonical multiplication.
2 ) Let A be a ring and
: G
'0
-+
A u t ( A ) a g r o u p homomorphism. We have
the group homomorphisms G 2
Aut(A)
We denote by f0,T : lAo
+
@
6
Pic(A), where
+
= h0 p
lAT
6(a) = [
lAm,
fo,T
(x@Y) =
I t i s e a s i l y seen t h a t f = ( f o , T ) o , T
to
+.
We denote by A
We have A
< 'D,
C.
lA,]
,
and we consider the A-bimodule isomorphism
+
< 'D, G >
> = 8
oFG
lAo,
0
X ~, fo,T ( x w y ) = xy, x c
generalized Rees rino of type G associated t o Hence
+
'0.
.
Pic(A)
C- then the maps f
F
Inv(T,A) a g r oup homomorphism. We p u t
+
~
X.O(Y).
i s a f a c t o r s e t associated
the strongly graded rin9s A
where the
<
f,@,C->.
multiplication i s defined by
x 0'
31
f. Graded Rings and Modules
the r u l e : x.Y = xo(y). where x Acl.,
c
y
lAT.
3) Consider a r i n g A, a group G and a Group e x t e n s i o n o f U(A) by G i . e . anexactsequence o f groups.
(&)
1
+
U(A)
5
H
+
G
1.
+
suppose t h a t t h e c o n j u g a t i o n a c t i o n o f H on U(A) d e f i n e d by ( c ) can be extended t o an a c t i o n
a
: H
+
Aut(A) ( .e. i f u
c
U(A) then
"(u) a = uau-l, a 6 A ) . For each u C G we choose a r e p r e s t e n t a t we ' D ~c n - l ( u ) and then we
Po
define t h e A-bimodule Exactness o f
( E )
= lAa(on)
-1
c
'om
e n t a i l s u = o';$~
U(A).
The previous r e s u l t s p r o v i d e us w i t h A-bimodule isomorphisms : lAa(9~)
f lAa(PT) -1
A
9
= lAa(pu;T)~lPa(~,)
a(q,)a(p,,)
where fo,T ( x ~ y =) x (a(p,)(y))
-1
and g,(z)
= ZU = z ( p 0 p p m ) .
L e t hLT,T be the A-bimodule isomorphism -1
x(a(p,)(y)
poPTcpm
,is
A [ €,a1
by A [ & , a ] ~
i.e. ho,T(x@y) =
+
: G
-L
Pic(A) d e f i n e d by @(u) =
a group homomorphism and t h e f a m i l y h = (hLT,T)u,T
i s a f a c t o r s e t associated denoted
fo,T
.
From the hypothesis i t f o l l o w s t h a t = [lAa(pu)l
o
= lAa(cu)
,
to
+.
The r i n g A < h,
+,
G > w i l l be
i t i s a s t r o n g l y qraded r i n o o f t y p e G w i t h :
and t h e p r o d u c t o f x F lAa(Du),
defined by x.y = x(u(P,)(Y))P,~,P,
-1
y C lAn(DT)
.
The s t r o n g l y graded r i n g A [ € , a ] i s c a l l e d t h e crossed p r o d u c t associated t o A, t h e e x a c t sequence
1 . 3 . 2 1 . Lemma.
(E)
and t h e rnorphism
I f u c U(R)nRo f o r some o C G, t h e n U - l
a.
C
R,-1.
is
A. Some General Techniques in the Theory of Graded Rings
32
Proof.
We may w r i t e u-'
Z vT w i t h vT c RT and a l l but a f i n i t e number
=
TFG
o f t h e elements vT a r e zero. From 1 =
Z
u vT i t
fGllOWS
t h a t vT = 0
TCG
unless
= a-' and i n t h a t case
T
u vs-l
Using t h e decree map de? : h(U(R))
--f
=
1 = V -1U i.e. u
-1
C
Ru-l.
U
G we o b t a i n a sequence o f group
homomorphisms : R >
<
E
1 + U(Re)
:
+
h(U(R)) d$O G
+
e
w h i c h i s e x a c t a t each p l a c e e x c e p t p o s s i b l y a t G. D e f i n e a g r o u p homomorphism for u 1.3.22.
c
aR :
h(U(R))
Aut(Re) by u R ( u ) ( y ) = uyu-'
--f
h(U{R)), y 6 Re. Lemma.
If
E
R
i s exact then :
I>
i
1. The r i n g R i s s t r o n g l y graded 2. I f 'u
uo c h(U(R)) n Ro t h e n t h e r e i s an isomorphism o f Re-bimodules : l(Re)aR(ua)
Ro,
+
g i v e n by
eu(A)
=
:
Xuu.
Proof. 1. Exactness o f E < R > y i e l d s t h e e x i s t e n c e o f a u n i t uU C G. S i n c e -1 u C Rdl, i t follows t h a t 1 6 R Ro f o r a l l 6 G. U
0-
1
2. Obvious.
Note.
L e t R be a s t r o n g l y Graded r i n o o f t y p e G such t h a t t h e sequence
e < R > is exact, then :
1. F o r e v e r y subgroup H o f G, t h e sequence
E
<
R(H) > i s e x a c t .
2. IfH i s a normal subgroup o f G t h e n t h e sequence
E
< R
(GIH) >
si
exact. The f o r e g o i n g r e s u l t s 1.3.23.
may now be combined i n :
Theorem (Crossed P r o d u c t ) L e t R be a graded r i n g o f t y p e G such
t h a t t h e sequence p r o d u c t Re [
E
<
E <
R >,
R
aR]
i s exact, then R i s isomnmhic t o t h e crossed
.
33
I. Graded Rings and Modules
n
R1 # 6 -1 ,ql t h e n R i s s t r o n g l y craded, t h e sequence c < R > i s e x a c t and R 2 RJ X,X 1.3.24.
C o r o l l a r y . I f R i s 2 - g r a d e d and i f we suppose t h a t h(U(R))
where q : Ro one
+
Ro i s an automorphism, X i s an i n d e t e r m i n a t e o f degree
and m u l t i p l i c a t i o n i s g i v e n by Xa = m(a)X.
An a r b i t r a r y r i n g S i s s a i d t o have t h e - l e f t I . B . N .
property (Invariant
B a s i s Number) i f f r e e l e f t Re-modules o f d i f f e r e n t r a n k c a n n o t be i s o m o r p h i c e.g.
commutative r i n g s , l o c a l r i n g s , l e + t ' i o e t h e r i a n r i n o s ,
e t c ... have t h e l e f t I.B.N.
P r o p e r t y . We r e f e r t o P. Cohn, [231, f o r
more p r o p e r t i e s o f t h e s e r i n g s .
We use t h e I.B.N.
property i n the
following : 1.3.25.
L e t R be a s t r o n g l y graded r i n q such t h a t Re has t h e l e f t
Lemma.
and such t h a t f i n i t e l y g e n e r a t e d p r o j e c t i v e l e f t Re-modules
IBN property
a r e f r e e . Then
Proof.
F
< R >
i s exact.
in
S i n c e R o i s a l e f t p r o j e c t i v e R -module we f i n d R o e RL') e
Re-mod f o r some s e t I.From R,-1
bo
i n Re-mod.
Re
R u % Re i t f o l l o w s t h e n t h a t ( R O - l ) ( I ) =
E u t R o - l = RLJ) i n Re-mod f o r some s e t J . Hence t h e I B N p r o p e r t y f o r Re e n t a i l s t h a t c a r d ( I x J ) Consequently R o such t h a t Ru
5
:
1.
Re i n Re-mod. By Lemma 1.3.14.2.
l(Re)a
as Re-bimodules.
Let
e
t h e r e e x i s t s a u C Aut(Re)
: l(Re)au
+
R o be a bimodule
U
isomorphism. P u t t i n g uo = e ( 1 ) y i e l d s Ro = Reuo = uURe and f r o m = RR,1-,
= R
Ro-l
Ru =
we may d e r i v e :
Re and u o - l R R u = Re. Hence t h e r e e x i s t X, + e ' e CJ c Re such t h a t 1 = u 1 u o - l , and 1 = u - 1 p uo. I t f o l l o w s t h a t each uu i s UaRe.Reuo-l
=
an i n v e r t i b l e e l e m e n t and t h e r e f o r e 1.3.26.
E iR >
i s e x a c t a t C. as w e l l .
C o r o l l a r y . L e t R be a s t r o n a l y araded r i n g such t h a t Re i s a
semi-local r i n g ,
Rz R~ r & <
R
i n t h e sense t h a t R/J(R) i s s e m i s i m p l e A r t i n i a n , t h e n
> ,aR
I.
Re
A. Some General Techniques in the Theory of Graded Rings
34
Proof. Apply Lemma 1.3.15.
1.3.27.
Corollary.
and 1.3.20.
-
I f R i s a s t r o n g l y graded r i n g such t h a t Re i s a l e f t
p r i n c i p a l i d e a l domain then R
, aR1
Re [ .; < R >
and R i s a l e f t
graded p r i n c i p a l i d e a l r i n g . Moreover i f G = Z then R
CI
-1, c p 1 where
Ro[X,X
X i s a v a r i a b l e of degree one and q i s an automorphism o f Ro.
~
Proof. The f i r s t statement
i s c l e a r from t h e lemma. Recall t h a t a
graded r i n g i s a l e f t graded p r i n c i p a l i d e a l i s p r i n c i p a l . Now i f
I.3.8.,
L
r i n g i f every l e f t i d e a l
i s a graded l e f t i d e a l o f R then by C o r o l l a r y
R o I e = Iu for a l l
0
c
G. But Ie = Rea
,
hence I = Ra.
The l a s t statement i s e v i d e n t from C o r o l l a r y 1.3.24. We now proceed t o study i n some d e t a i l t h e graded i d e a l s o f s t r o n a l y graded r i n g s . I f R i s a s t r o n p l y graded r i n g o f type G , then Mod(R1,R)
c
w i l l denote t h e f a m i l y o f a l l two-sided R1-submodules o f R. I f A and u c G then p u t Ao = R o - l A Ru.
Mod(R1,R)
I t i s c l e a r t h a t A 5 C Mod(R1,R);
we
say t h a t Ao i s the o-conjugate o f A. 1.3.28.
F i x i n n u c G , t h e map
Lemma.
: Mod(R1,R)
cp
+
Pod (R,R),
A
+
A"
i s b i j e c t i v e and i t preserves i n c l u s i o n s , sums, i n t e r s e c t i o n s and products i n R o f elements o f Hod(R1,R). P r o o f . That q, i s b i j e c t i v e i s c l e a r , a c t u a l l y t h e map A i n v e r s i v e . For
U,T
c G we g e t (A')T
n o t hard t o see t h a t A, B C MOd(R1,R) c (AU)"-l =
AO
n
B"
n .
Q"
then A
(B")"-l,
--f
A'
-1
f o r a l l A c Mod(R1,R).
= Am
is its
It i s
preserves sums and i n c l u s i o n s . Now consider
n
B = RoRG-l(A
hence A
n
n
B = Ro(Au
I n a s i m i l a r way one checks t h a t
Q,
B) RuRRa-l c R(,A'
n
B')Ru-l
n B')
and thus (A
preserves products.
Ro-l c
n B)'
=
35
1. Graded Rings and Modules
We say t h a t A
C
Mod (R1,
R) i s G-invariant
i f A'
= A f o r a l l u F G.
It
i s straightforward t o check t h a t any ideal of R i s a G-invariant element
of Mod(R1,R). l e t I be a
1.3.29. Corollary. Let R be strongly graded of type G , and graded ideal of R . Then Ie i s an ideal of R1,
I = RI e = IeR. The correspondence I
+
which i s G-invariant and
I e defines a bijection between the
s e t of graded ideals of R and the s e t of F-invariant ideals
Proof. 1;
c
For any
T
F
Ie
RT c I and thus R,-1
u F G , then :
I e . Pick any
I = I u-1 and I: e e
G, RT-l
Ie
=
u 0-1
(Ie)
c ' :1
of Re.
Ie RT c Ie or c I e , hence
= Ie.
I = RIe
By Corollary I . 3 . 8 . ,
=
IeR. Conversely, l e t J be any G i n v a r i a n t
ideal of Re then RJ = JR i s a graded ideal of R i s e a s i l y seen. The statement about the correspondence I 1.3.30. Lemma.
+
Ie
i s a l s o e a s i l y checked.
Let R be a strongly graded ring o f type G .
! . If M i s a simple l e f t Re-module then f o r every u
C G,
Ru
M is
60
Re
a simple l e f t Re-module. (Ra @ M ) then : I = JO. 2. Putting I = AnnR ( M ) , J = Ann e Re R~ Proof.
Let N c R u
60
M be a l e f t Re-submodule, then since R u - i i s a
Re Projective R -bimodule i t follows t h a t R,-l e
60
BY Proposition 1.3.6. i t follows t h a t R u - i 60 N
Re Or Ro-l
@
c R u - l 60 Ru 60 El.
N
Re c
M thus
Re Re R u - l 60 N = P Re
N = 0. The l a t t e r case obviously e n t a i l s t h a t N = 0 while the
Re f i r s t case y i e l d s N
=
Ra 60 M.
Re
A. Some General Techniques in the Theory of Graded Rings
36
2 . Consider a F Ru,
i c I , b F Ru-l,
then a i b F 1'-
1
and on t h e o t h e r
hand ( a i b) ( R a 60 M) = a 60 i b R u M . Re Now b R n M abi
6
c AnnR
M and r
(M),
1
A n n R ( R u 60 M ) . Thus Iu Re
Using t h e f a c t t h a t M = Ro-l
-1
60
thus i b R M = 0 , consequently c J and I c J u .
(RO
Re
Jc
60
R
M ) we a l s o o b t a i n J
c
I 0-1, hence
c I . Combination of both i n c l u s i o n s y i e l d s I = J'.
1.3.31. Lemma.
Let R be a s t r o n a l y
graded r i n g of type G.
I f P i s a prime i d e a l o f Re then Po i s a prime i d e a l of Re f o r a l l u C G . I f I and J a r e i d e a l s of Re such t h a t I J c P' -1 = 1o-l Ja - l (P')5-1 = P , hence I 0 c P say. Proof.
Then I c P' By
then ( I J ) '
-1
=
follows.
Prim ( - ) we denote t h e s u b s e t of Spec ( - ) c o n s i s t i n g of the p r i m i t i v e
i d e a l s of t h e r i n g under c o n s i d e r a t i o n . If R i s a s t r o n g l y graded r i n g o f type G then the
1.3.32. C o r o l l a r y .
Jacobson r a d i c a l of Re, J ( R e ) , and the n i l r a d i c a l o f Re,
rad(Re), a r e
both G - i n v a r i a n t . Proof.
Since J(Re)
=
n I P , P c Prim Re},
and rad(Re) =
n
{ P , P C Spec(Re)l
we only have t o apply Lemma 1.3.30. and Lemma 1.3.31. The following theorem i s a g e n e r a l i z a t i o n of t h e c l a s s i c a l " C l i f f o r d theorem' cf. [88]
theorem 2.16 pag. 2%.
t h e proof of theorem
1.3.33. Theorem.
The proof we give here c l o s e l y resembles
2.16. in loc. c i t .
Let R be a s t r o n g l y graded r i n g o f type G , where
i s a f i n i t e group. I f
G
i s simple i n R-mod then p2 c o n t a i n s a simple Re-
37
1. Graded Rings and Modules
submodule W such t h a t
e
M
=
z
Ro W. I n p a r t i c u l a r C i s semisimple i n
uFG
Re-mod. W 2 W } = { u F G , Ro W = Id } and l e t MW be the Re W . Then H i s a subcroup sum of a l l Re-submodules X of M such t h a t X
put H =
{U
F G , R O 69
of G and MW i s a simple R(H)-module such t h a t
M then
I”r 2 R
MW.
69
R(H)
proof.
I f x # 0 in
Ro,
G, i s a f i n i t e l y Generated Re-module i t follows t h a t
L c
o F
=
Rx
= ( C
aC G
Ro)x
=
C
oCG
R,x.
Since each is a
f i n i t e l y generated Re-module, hence there i s a maximal proper Re-submodule K of
E.
P u t KO
=
o6G
RoK.
R-submodule of
Clearly R,Ko
M and
sequence in Re-mod :
then KO
0
+
c KO f o r a l l o F
+ F1 implies
M
--f
0 UCG
G. Therefore, KO i s an
KO = 0. Thus we find a n exact
M / RuK.
-
B u t each R u i s an i n v e r t i b l e Re-bimodule, therefore each RUK i s also a maximal proper Re-submodule of
M.
semisimple Re-module and then a l s o
I n other words,
0 U6G
!/Ro
i s a semisimple Re-module. From
t h i s we deduce the existence of a simple Re-submodule of
M
=
RW e n t a i l s t h a t M = C
aCG
K is a
y,
W say. Clearly
Ro W. Again, RG being an i n v e r t i b l e Re-bi-
module, i t follows t h a t each ROW i s a simple Re-module. The canonical multiplication map Ra 69 W
+
ROW i s a n isomorphism because ROW i s
Re simple i n Re-mod a n d Ra i s i n v e r t i b l e . The second statement o f the theorem follows now immediately.
A, Some General Techniques in the Theory of Graded Rings
38
1.4. Graded Division R i n g s A graded r i n g of type 6 i s s a i d t o be a x a d e d d i v i s i o n r i n p i f every
nonzero homogeneous element i s i n v e r t i b l e . C l e a r l y i f R i s a g r a d e d d i v i s i o n r i n a then Re i s a s k e w f i e l d .
Let R be a graded r i n n of type G , then the followinc:
1.4.1. Lemma.
statements a r e e q u i v a l e n t : 1. R
has no graded l e f t i d e a l s d i f f e r e n t from
0,
R.
2 . R has no graded r i g h t i d e a l s d i f f e r e n t from
0,
R.
3. R i s a graded d i v i s i o n r i n g .
Proof.
Straightforward.
1.4.2. Theorem. Let R be a graded d i v i s i o n r i n g . 1. I f G' = { g
C
G , Rg # 0 } then 6 ' i s a subgroup of G and R i s a domain
i f G' i s ordered. 2 . The r i n g R ( G ' ) = 3. R(")
=
Re
[ E
c
Q
R
OF G'
i s a s t r o n g l y GI-graded r i n q .
R > , aR 1 where <
E
>
i s t h e following e x a c t sequence
of groups :
where R*,
=
Ro - { O } for a l l
u F
G , and
aR
i s t h e group morphism
Proof. 1. T h e f i r s t s t a t e m e n t i s obvious. I t i s a l s o c l e a r t h a t t h e r e a r e no
homogeneous z e r o d i v i s o r s and under t h e assumption of 1. we only have
39
1. Graded Rings and Modules
t o c o n s i d e r t h e s i t u a t i o n where ( 1 t a where 1 < u < a.
n
or bT
= 0
m
=
...
and 1 <
< o
<
T~
t..
u1
.+ au ) ( 1 t bT n
...
b7 ) = 0 m
t...t
1
zrn' Then a. b7 n m
=
0 entails
0. A f t e r r e d u c i n g t h e l e n g t h o f t h e d e c o m p o s i t i o n s a
c o n t r a d i c t i o n i s reached. 2 . , 3. a r e immediate consequences o f S e c t i o n A . I . 3 .
A s t r o n g l y G-graded r i n g R = Re R = Re[ Xu,
;T,O
a o r ~
0,T
c
G
[ F
<
R >,
aR]
I where t h e Xo
may be w r i t t e n as
,u C G, a r e u n i t s o f R and
m u l t i p l i c a t i o n i n R i s d e f i n e d by t h e r u l e s : 1" F o r a l l o F G , Xo a = a,(a)
Xu f o r a l l a E Re where au C Aut Re,
for
a l l u c G. 2" F o r a l l
0
, C ~G
, XoXT
=
c 0 9 7
XuTwhere t h e c
~ C , U ~(Re)
satisfy the following relation :
1.4.3.
Corollary.
I f R i s a graded d i v i s i o n r i n g o f t y p e
R = Ro, when t h e g r a d a t i o n i s t r i v i a l , o r e l s e R 'D
C Aut
2' R0
[ X,
Z then X-',
either
91
where
R o , and X i s an i n d e t e r m i n a t e such t h a t Xa = ,o(a)X f o r a l l a
A graded d i v i s i o n r i n g o f t y p e
C Ro.
Z i s a l e f t and r i g h t p r i n c i p a l i d e a l
domai n. Proof.
The f i r s t s t a t e m e n t i s o b v i o u s . The r i n g o f t w i s t e d p o l y n o m i a l s i s a l e f t and r i g h t p r i n c i o a l
S = Ro[X,,o1
i d e a l domain, t h e n R w i l l
have t h e same p r o p e r t y because i t i s o b t a i n e d f r o m S by l o c a l i z i n g
S at
Il,x,x
2
,... 1 .
1 . 4 . 4 . Remark.
The e q u i v a l e n t o f 1.4.3.
r i n g s o f t y p e G, f o r c e r t a i n G, e.p. i s n o t l e f t and r i g h t
i s f a l s e f o r graded d i v i s i o n
if G i s f i n i t e a b e l i a n t h e n k [ G I
p r i n c i o a l . I n t h e commutative case some of t h e s e
problems may be viewed i n t h e l i g h t o f t h e t h e o r y o f a r i t h m e t i c a l l y graded r i n g s , w h i c h we w i l l r e t u r n t o l a t e r .
A. Some General Techniques in the Theory of Graded Rings
40
1.4.5.
L e t R be a graded d i v i s i o n a l g e b r a o f t y p e G.
Lemma.
1. R i s s t r o n g l y graded i f and o n l y i f G
=
G'.
2. I f G = G ' t h e n t h e s e t H = { o C G, t h e r e e x i s t s x # 0 i n Z(R) fl Ru} i s a subgroup o f Z(G), t h e c e n t e r o f G. 3. I f G = G ' i s a b e l i a n t h e n Z(R) i s s t r o n g l y graded of t y p e H. 1 and 3 a r e o b v i o u s .
Proof. 2. I f
c
6 H, choose xT
T
Z(R)
n
RT such t h a t xT # 0.
F o r any u C G and e v e r y a C Ro we have xT a = a xT. S i n c e Ro
# 0 and s i n c e t h e r e a r e n o nonzero homoaeneous z e r o d i v i s o r s ,
i t follows that
UT
=
TO
i.e.
T
c Z(G).
T h a t H i s i n d e e d a subgroup i s r e a d i l y checked. 1.4.6.
Proposition.
L e t R be a s t r o n a l y graded d i v i s i o n r i n g o f t y p e G.
1. F o r o c H i t i s necessary t h a t u C Z(G) and .a 2. I f G i s a b e l i a n t h e n Z(R) = (Z(R)), d : H x H
+
U(Z(R),)
[Yo,
i s inner.
.
r e p r e s e n t s an e l e m e n t o f
t H
U,T
a
E
I , where
H2 (H,U(Z(R)e))
d e s c r i b i n g up t o isomorphism t h e s t r u c t u r e o f t h e c r o s s e d
product
Z(R). Proof. 1. I f o C H t h e n t h e r e i s a nonzero c ( ~ )Xo t Z(R) f o r some c ( ~ )c Re. F o r e v e r y a C Re we o b t a i n : C ( ~ ) Xa ~= c(")u,(a)
Xu = a c(O) Xu,
hence s i n c e a l l o c c u r i n g elements a r e homogeneous : .,(a)
=
(c("))"a
i . e . au i s i n n e r and i n d u c e d by ( c ( ~ ) ) - ' . 2. T h a t Z ( R ) i s o f t h a t f o r m f o l l o w s f r o m 1.4.5.3.
and
t h e crossed
p r e c e d i n g 1.4.3. i t f o l l o w s 2 i n d e e d d e t e r m i n e a n eleinent o f H (H, U ( ( Z ( R ) ) e ) ) . I f
p r o d u c t theorem. Because o f t h e remarks t h a t do,Tdoes
t h e Yo a r e r e p l a c e d by u ( ~ ) Yf o~ r c e r t a i n u ( ~ )C U((Z(R)),)
the
c ( ~ )
41
1. Graded Rings and Modules
dU,T
w i l l be r e p l a c e d by an e q u i v a l e n t 2 . c o c y c l e ,
therefore the structure
o f t h e c r o s s e d p r o d u c t Z(R) i s up t o isomorphism d e t e r m i n e d by {aU,ocG} and
1.4.7.
4. L e t R be a s t r o n a l y graded d i v i s i o n r i n g o f t y p e
Proposition.
-
G such t h a t Re i s commutative, t h e n :
1. R
=
Re
I Xo, c
;o T C 0
3
GI where c
r e p r e s e n t s an e l e m e n t 3T
7
c C H L ( G , U(Re)). 2. I f G i s a b e l i a n t h e n Z(R)
(Z(R)),[Yo,
do,T,
U,T
c HI where d0 , T
r e p r e s e n t s an e l e m e n t d o f H L ( H , U ( ( Z ( R ) ) e ) ) w h i c h i s e q u i v a l e n t t o resH c i n H[(H,U(R~)) Proof. __ 1. C o m m u t a t i v i t y o f Re a l l o w s
L4.8.
U,T
c
m y
= ao(CT,y)
co,Ty
f o r a l l U,T,Y
0,T
F G.
t o e s t a b l i s h t h e second s t a t e m e n t .
Remark.
These t e c h n i q u e s e n a b l e one t o c o n s t r u c t
homogeneous elements i n Z(R )
o f degree d i f f e r e n t from e p r o v i d e d one can f i n d a subgroup H of Z ( G )
42
A. Some General Techniques in the Theory of Graded Rings
such t h a t the ao, u c H , are inner and such t h a t resHc i s equivalent 2 in H ( H , U ( R e ) ) t o an H-invariant cocycle.
For example in the Z-graded case R = R e [ X ,
X-',
cp1 i t i s c l e a r t h a t ,
in order t o have a non-trivially Graded center, some power o f q i s inner i f Re i s a skewfield i . e .
cpn = 1
if R
e
i s a comutative f i e l d .
43
1. Graded Rings and Modules
1.5.
Graded Endomorphism Rings.
L e t R be a graded r i n g of t y p e G ,
M F R-pr. Then END (M) = HOMR(M,M) i s
R
a graded r i n g o f t y p e G. I n d e e d i t i s e a s i l y seen t h a t c o m p o s i t i o n o f graded endomorphisms o f M y i e l d s a m u l t i p l i c a t i o n making t h e araded a b e l i a n group HOMR(M,M)
i n t o a graded r i n g .
However, due t o t h e f a c t t h a t we w i l l work w i t h l e f t modules and l e f t o p e r a t i o n s , t h i s m u l t i p l i c a t i o n i s d e f i n e d as f o l l o w s : o.f = f C o n s i d e r M,
g.
N i n R-gr. We say t h a t N d i v i d e s M i n R-gr i f N i s i s o m o r p h i c
t o a graded d i r e c t summand o f M i . e .
i f and o n l y i f t h e r e e x i s t
c
F Horn (N,M) such t h a t f g = 1M' R-gr We say N weakly d i v i d e s M i n R-gr i f i t d i v i d e s t h e d i r e c t sum f E Hom (M,N), R-gr
dt) o f
a f i n i t e number o f t c o p i e s o f M. E v i d e n t l y N w e a k l y d i v i d e s M i n R-gr i f and o n l y i f t h e r e e x i s t : fl,
g1 ,..., gt F Hom (N,M) R- g r
...,
(F1,N) and f t F Horn R-ar
such t h a t lM = fl " g l
... +
+
ft
gt.
We say t h a t t h e graded modules M, N a r e weakly i s o m o r p h i c i n R - g r , ( n o t a t i o n M
-
N) i f each o f them weakly d i v i d e s t h e o t h e r i n R - o r . E q u i v a l a n t l y
M - N i f and o n l y i f t h e r e e x i s t n a t u r a l numbers n, m and graded modules Nm, N
M ' , N ' o f t y p e G such t h a t M 0 M ' 2
@
N'
1 M".
An M F R-gr i s s a i d t o be w e a k l y G - i n v a r i a n t i f M 1.5.1.
Theorem.
L e t M E R-gr. Then t h e
- P(o)
for all u
< 6.
graded r i n o ENDR(M) i s s t r o n a l y
graded o f t y p e G i f and o n l y i f M i s w e a k l y G - i n v a r i a n t .
Proof.
ENDR(M) i s s t r o n g l y graded i f and o n l y i f we have
lM F (ENDR(M))T (ENDR(M)&l g1 ,..., gn c ENDR(M)T and fl
for all
,..., fn 6
T
C G,
i f and o n l y i f t h e r e e x i s t
(ENDR(M))Tc-l such t h a t :
A. Some General Techniques in the Theory of Graded Rings
44
Now, (ENDR(M))T = HOM ( M ( o ) , M(Tu)), and R-gr (ENDR(M))T-l = HOM (M(Tu), R-gr
M(u)), f o r a l l
0
F G.
The r e l a t i o n (*) i s t h e r e f o r e e q u i v a l e n t t o t h e f a c t t h a t Mco) w e a k l y d i v i d e s M ( a ) f o r a l l u , F~ G. S i n c e G i s a grouo, t h i s l a s t c o n d i t i o n i s e q u i v a l e n t t o M b e i n g weakly G - i n v a r i a n t .
I . 5.2. C o r o l l a r y . L e t R be a s t r o n g l y graded r i n g o f t y p e G and l e t M be a l e f t Re-module. P u t Mu = R
@
M,
Re i f and o n l y i f M
Proof.
f o r a l l u C G . Then ENDR ( R 60 M) i s s t r o n g l y graded Re Ma i n t h e c a t e g o r y Re-mod, f o r a l l u C G.
-
By Theorem
i f and o n l y i f P
-
1.3.4.
i t f o l l o w s t h a t : R 60
M - (R
Re M0 i n Re-mod. T h i s h o l d s f o r e v e r y u
60
M) (u) i n R-gr
Re c G and hence
t h e Theorem p r o v e s t h e c o r o l 1 a r y . An M c R-gr i s s a i d t o be G - i n v a r i a n t i f f o r a l l u F G, 1.5.3.
M - M ( a ) i n R-or.
A non-zero M F R-gr i s G - i n v a r i a n t i f and o n l y i f t h e
Corollary.
sequence : eR <
M
: 1
+
U(End (M)) R-or
+
U(ENDR(M))
+
G
+
1
i s e x a c t . I n t h a t case we have t h a t ENDR(Y) i s s t r o n p l y graded and isomorphic t o t h e crossed product Proof.
End (M) [ R-or
ER
< M >, c M l
The s t a t e m e n t f o l l o w s i m e d i a t e l y f r o m Theorem 1.5.1.
. and t h e
t h e o r y expounded i n S e c t i o n 1.3. L e t us now f o c u s on m a t r i x r i n g s o v e r graded r i n g s and see how t h e s e r i n g s can be made i n t o graded r i n g s . I f R i s a y a d e d r i n a o f t y p e G, n F N , t h e n Mn(R) denotes t h e r i n g o f n by n m a t r i c e s w i t h e n t r i e s f r o m R . Fix
0
=
(u
un) c
subgroup o f Mn(R):
G".
To X c C. we a s s o c i a t e t h e f o l l o w i n g a d d i t i v e
45
1. Graded Rings and Modules
I t i s easy t o check t h a t Mn(R) equipped w i t h t h e g r a d a t i o n o f t y p e G
g i v e n by
(G),
M,(R)
(0)
Mn(R)X
Q
XfG
i s a araded r i n g , w h i c h we w i l l denote b y
c o r i t a i n i n g R a s a g r a d e d s u b r i n q . T a k i n g ul
(e)
y i e l d s Mn(R)
.....
o
n
.
e
w h i c h we denote s i m p l y by Fn(R) whenever t h e r e
cannot a r i s e any a m b i g u i t y w h e t h e r t h e l a t t e r r i n g i s c o n s i d e r e d as a graded o b j e c t o r n o t . 1.5.4.
Lemma.
say deg e . = 1
Proof.
L e t M f R-gr be g r - f r e e w i t h homogeneous b a s i s el, 0..Then
1
ENDR(P)
f c (ENDR(M))X, X F
If
c Po.,
Consequently f ( e i )
1
w i t h deg a . .
1J
Mn(R)X
2
=
o. X 1
J
Cn(R)
(0)
where
G, t h e n f(Mo)
i= c
.... en,
( ol,... ,on).
MuX f o r a l l o 6 G .
n f o r i = 1, ....n. So we may w r i t e f ( e i ) = C j = i aijej The m a t r i x ( a . .) a s s o c i a t e d t o f i s i n 1J
(0).
I . 5.5. C o r o l l a r i e s . 1. I f 9 i s a p e r m u t a t i o n o f t h e s e t
(0)
t h e n Mn(R)
2. W r i t i n g 0 . f~o r t h e s e t {ulT..... onTl we g e t
-
Mn(R)
(~(0)).
:
= Mn(R)(G).
Mn(R)(;..)
Proof.
1. Is obvious. 2. L e t M
c
R - o r be g r - f r e e w i t h homogeneous b a s i s el,
... ,en, say dep ei
=
46
=
A. Some General Techniques in the Theory of Graded Rings
5.
1
6 G.
Mn(R)(o) b a s i s el
From Lemma 1 . 5 . 4 .
ENDR(M). Consider M(T-’);
5
,..., en
where deg ei
and, u s i n g Lemma I . 5 . 4 . ,
=
0
i t i s g r - f r e e w i t h homogeneous
2 ENDR(M(-r-’)),
Cn(R)(&)
we have t h e s t a t e m e n t .
o f G. The graded r i n g Mn(R)(G) i s a s t r o n a l y graded r i n g o f t y p e G.
Proof.
Consider t h e g r - f r e e
M
R(al)
=
graded, Theorem 1 . 5 . 1 . e n t a i l s t h a t R
M(0)
= R ( o l ) ( ~ )@.*.@
of-,’
R(u) f o r a l l
CJ
C
~(oo,).
R i s strongly
G; c o n s e q u e n t l y : ~y t r a n s i t i v i t y
R ( c ~ ~ ) - R ( u o ~f )o r a l l u t G f o l l o w s .
-
i=1
C G.
-
Q...Q R(un). S’nce
R(a-01) Q . . . 8
R(on)(u) =
n Hence we o b t a i n t h a t : Ca R(ui)
u
2 ENDRM(-r-1)
= u i ~ , i = 1,..., n . S i n c e ENDR(M)
C o n s i d e r a s t r o n g l y graded r i n p R and a s u b s e t t o l,...unl =
Theorem.
1.5.6.
we o b t a i n :
n
R(aui) i=l 8
o r PI- M (u) f o r a l l
Again, because o f Theorem I.5.-1., weakly G - i n v a r i a n c e f o r M
e n t a i l s t h a t ENDR(M) i s a s t r o n g l y graded r i n g . Lemma 1 . 5 . 4 . f i n i s h e s t h e p r o o f o f t h e theorem.
1.5.7. Corollary.
L e t R be a graded r i n g o f t y p e G such t h a t
E
i s an e x a c t sequence (see S e c t i o n 1 . 3 . ) . F o r any n c N and any = to
an} c G,
t h e sequence
E
< Mn(R)(G) > i s e x a c t .
case, Mn(R)(G) i s t h e c r o s s e d p r o d u c t (Mn(R)(G))e Proof.
< R >
a
=
I n this
P n ( R ) ( n ) >, u I Vn(R)
[ E i
L e t M c R-gr be g r - f r e e and such t h a t :
Mn(R)(G) = ENDR(M), where
C 2 R(al) Exactness o f
@...a E
<
R
0
i s obtained from :
R(on). entails R
R(o) f o r a l l
5
c C,
hence M 2 M ( c )
f o r a l l o c G. Now C o r o l l a r y 1 . 5 . 3 . e n t a i l s t h e r e s u l t s t a t e d . A graded r i n g R o f t y p e G i s s a i d t o b e g r - s e m i s i m o l e i f R-gr i s a semisimple c a t e g o r y i . e . any o b j e c t o f R-gr i s semisimple ( i . e .
if
M
6 R-gr
47
1. Graded Rings and Modules
t h e n M i s a d i r e c t sum of
g r - s i m p l e modules). I t i s c l e a r t h a t R i s
i f and o n l y if : ( * ) R =
gr-semisimple
L 1 @...aLn, where t h e
L . , i = 1,..., n a r e m i n i m a l graded l e f t i d e a l s o f R . C l e a r l y , i f R i s g r - s e m i s i m p l e t h e n Re i s semisimple. A araded r i n g R o f t y p e G is
i f i t has a d e c o m o o s i t i o n ( * ) b u t w i t h
g r -simple
f o r e v e r y i , j = 1, ...,n, e q u i v a l e n t l y : t h e r e e x i s t s a
HOMR(Li,Lj)# oij
0
C G such
t h a t L . 2 Li ( u . . ) . A g r - s i m p l e graded r i n g R i s s a i d t o be ~ r J 1J u n i f o r m l y s i m p l e i f R i f R a d m i t s t h e d e c o m p o s i t i o n (*) b u t w i t h L . i n R-gr f o r any i,j.
L. 1
J
1.5.8.
Theorem. (The graded v e r s i o n of Weddenburn’s Theorem.)
L e t R be a r i n g o f t y p e G. Then t h e f o l l o w i n !
statements are equivalent :
1. R i s g r - s i m p l e ( r e s p . g r - u n i f o r m l y s i m p l e ) . 2. There e x i s t s a graded d i v r s i o n r i n g D and a such t h a t R Proof.
Let M
,..., T, c
t1
Mn(D)(o) ( r e s p . R
c
R-gr be g r - s i m p l e .
= ( o,...,on)
C G
n
z Vn(D)). Since R i s gr-simole there e x i s t
G such t h a t R ”=’ M ( T ~ )8 . . . @M(T,).
P u t D = ENDR(IU’), t h e n D i s a g r - d i v i s i o n r i n a .
z ENDR ( M ( T ~ ) @ . . . @ V ( T ~ ) ) . S i n c e ENDR(M(-ri)) z ENDR(En) = D f o r a l l i = 1, ...,n i t f o l l o w s f r o m = ( o ~..., , un) c Gn. The t h e f o r e g o i n g t h a t R z Enn(D)(E) f o r some
Then R T ENDR(R)
uniformly simple statement also follows.
1.5.9.
Remarks.
1. I f R i s g r - s i m p l e t h e n Re need n o t be a s i m p l e r i n g . As n o t e d b e f o r e
Re i s semisimple.
2. I f R i s g r - u n i f o r m l y s i m p l e t h e n Re i s a s i m p l e r i n g . 3 . If R i s g r - s i m p l e and suppose t h a t t h e r e e x i s t s a s i m p l e o b j e c t i n R-gr w h i c h i s G - i n v a r i a n t ,
then R i s gr-uniformly simple.
4a
A. Some General Techniques in the Theory of Graded Rings
In case G i s an ordered group we may d e r i v e some e x t r a r e s u l t s i n t h i s v e i n , we r e f e r t o S e c t i o n 11.1.
,
i n p a r t i c u l a r Lemma I I . 1 . 5 . ,
P r o p o s i t i o n 11.1.6. and t h e remarks i n 11.1.7.. For
more r e s u l t s
about gr-simple and gr-semisimple o b j e c t s o f R-qr i n case R i s
z
graded, we r e f e r t o S e c t i o n 11.6.
1. Graded Rings and Modules
49
1.6. Graded Rings o f F r a c t i o n s . R e c a l l t h a t i f R i s any r i n g and S i s a m u l t i p l i c a t i v e l y c l o s e d s u b s e t o f R such t h a t 1
c
S, 0 f S t h e n t h e l e f t r i n g o f f r a c t i o n s S - l R ,
with
r e s p e c t t o S, e x i s t s i f and o n l y i f R s a t i s f i e s t h e l e f t Ore c o n d i t i o n s w i t h respect t o S :
c
O1 : I f s E S, r 6 R a r e such t h a t r s = 0 t h e n t h e r e e x i s t s an s '
S
such t h a t s l r = 0.
O2 : F o r r C R, s F S t h e r e e x i s t r ' E R, s '
C S such t h a t s ' r = r ' s .
I f R i s a l e f t N o e t h e r i a n r i n g t h e n O1 always
h o l d s and o n l y has t o
v e r i f y 02. I f t h e Ore c o n d i t i o n s w i t h r e s p e c t t o S a r e b e i n g s a t i s f i e d t h e n : S
-1 a R = {
,
Y
5
2!=-.-tl.s
=
c
a x + b
x
S+T
a
.Y
s . t
c
R, s
,
R, s
c
S }, where o p e r a t i o n s a r e d e f i n e d by
where a, b 6 R a r e such t h a t u = as = b t 6
, where
c
tl
Recall also t h a t every W
S, x1
c
c
R a r e such t h a t tlx
=
xlt.
R-mod a l l o w s t h e c o n s t r u c t i o n o f a module o f
1 1 f r a c t i o n s S - l M which i s a l e f t S- R-module a c t u a l l y S- C 1.6.1.
Lemma.
S.
S - l R 66 M. R
L e t R be a graded r i n g o f t y p e G and l e t S be a m u l t i p l i -
c a t i v e l y c l o s e d s u b s e t o f R c o n t a i n e d i n h(R), i . e . c o n s i s t i n g o f homogeneous elements, t h e n R
s a t i s f i e s t h e l e f t Ore c o n d i t i o n s w i t h
r e s p e c t t o S i f and o n l y i f :
0; : If r
6 h(R),
s F S a r e such t h a t r s = 0 t h e n t h e r e i s an s ' 6 S
such t h a t s ' r = 0
0; : For any r
C h(R),
s C S, t h e r e e x i s t r '
c
h ( R ) , s ' 6 S such t h a t
s ' r = r's.
-
P r o o f . C l e a r l y O1 and O2 i m p l y 0: r 6 R,
where r = r j
+... t r n
1
n
and 0;. Conversely, c o n s i d e r s
w i t h ro.C h ( R ) o A , 1
1
ci
C
G. I f n
=
c
S,
1 then
50
A. Some General Techniques in the Theory of Graded Rings
the l e f t Ore c o n d i t i o n s f o r r , s a r e seen t o hold because 0'; and 0;
hold.
Now we proceed by i n d u c t i o n on n , supposing O1 and O2 may be v e r i f i e d f o r
R having a homogeneous decomposition of lenoth l e s s than n . By assumption 1 1 1 1 t h e r e e x i s t r F R , s c S such t h a t s ( r + . . . f r o ) = r s, and s2 C S , Ol n- 1 r2 C R such t h a t s 2 ru = r 2 s . By 0: and 0; t h e r e e x i s t u F S, v C h ( R ) i-6
such t h a t us1 = vs2 = n t ,and t c S i s non-zero. Then t r
2
(url + vr ) s ,
=
hence O2 holds a l s o i f r has a decomposition of lenpth n . Furthermore,
+...+ a. , a u F Ro., then a. s = 0 and ( a + . . . + a u )s u1 n n i n Ol n- 1 ... a = 0. By t h e i n d u c t i o n hypothesis we may pick a tl C S such t h a t t l ( a ) ul un-l
i f as = 0
with a = a
= 0 . Now from
t a s
t h a t t2tl a.
= 0. P u t t i n g s ' =
0 f S)
n
un
i t follows t h a t t h e r e e x i s t a t2 6 S such
0 and 0:
=
t2tl F S we s e e t h a t s ' a
=
0 ( s ' # 0 since
I f the graded r i n g R s a t i s f i e s t h e l e f t Ore c o n d i t i o n s with r e s p e c t t o some m u l t i p l i c a t i v e l y closed S c h ( R ) then we can d e f i n e a o r a d a t i o n on S-lR by p u t t i n o ( S - l R ) ,
=
I
:,
s C S, a F R such t h a t X = (dea s ) - l deg a ) .
1 . 6 . 2 . P r o p o s i t i o n . S-lR i s a qraded r i n g of tyDe G .
Proof.
If
+, f
C
(S'lR),
then X = (dep s ) - l d e g x
P u t t i n g u = a s = b t c S we g e t ax + by ) Hence : deg(--U
=
f +$
=
=
(deg t)-' deg y .
ax+bY , U
(deg u ) - l deg(ax)
(deg s ) - l (deg a ) - ' -1 = (deg s ) deg x =
(deg a ) (deg x )
Therefore (S-IR), i s an a d d i t i v e subgroup o f S-lR, f o r each X C 6 . In a s i m i l a r way one v e r i f i e s Obviously, S-'R=
Z
XFG
( s - ~ R )(,s - ~ RLL )c
(S-lR),.
( S - ~ R ) ~ ~ .
T h e common denominator theorem y i e l d s
t h a t the sum i s d i r e c t . 1.6.3. Corollary.
I f R i s a s t r o n g l y graded r i n g s a t i s f y i n g t h e l e f t
I. Graded Rings and Modules
Ore c o n d i t i o n s w i t h r e s p e c t t o S
c
51
h(R) then S - l R i s a s t r o n g l y graded
ring. Proof.
C l e a r l y (S-lR)u (S-lR)u-l
= R R - l = R UCJ
e'
= (S-lR)e
f o l l o w s from R u - l
Ra
=
52
A. Some General Techniques in the Theory o f Graded Rings
1.7. Jacobson R a d i c a l s o f Graded R i n g s . A l t h o u g h a theorem o f G . Bergman's s t a t e s t h a t t h e Jacobson r a d i c a l o f a Z - g r a d e d r i n g i s a g r a d e d i d e a l , many problems o f an i n t r i n s i c a l l y graded n a t u r e may be d e a l t w i t h by u s i n g a araded v e r s i o n o f t h e Jacobson r a d i c a l and, i n p a r t i c u l a r , a g r a d e d v e r s i o n o f Nakayama's Lemma. L e t G be a g r a d e d r i n g o f t y p e G. An S 6 R-gr i s s a i d t o be g r - s i m p l e i f o and S a r e i t s o n l y g r a d e d submodules. An M 6 R-gr i s o r - s e m i s i m p l e i f M i s d i r e c t sum o f q r - s i m p l e modules, see a l s o S e c t i o n I 5 . , i n p a r t i c u l a r
1.5.8..
1 . 7 . 1 . Lemma.
I f S i s g r - s i m p l e t h e n f o r each o
6
So
G,
= 0 or S
is
a s i m p l e Re-module. and x # 0 t h e n Rx = S , hence Rex =
If x F S
Proof.
Consequently, S
A graded
So.
i s a s i m p l e Re-module.
submodule N o f M 6 R-gr i s s a i d t o be pr-maximal i n M i f M/N
a g r - s i m p l e R-module.
f
I n o t h e r words, N i s gr-maximal i n M i f and o n l y
N i s p r o p e r and f o r a l l m F h(M), N
+
Rm=
M. N o t e t h a t a gr-maximal N n M
i s n o t n e c e s s a r i l y such t h a t N i s maximal i n
1 . 7 . 2 . Lemma.
S
p.
I f S i s gr-simple then there e x i s t s a praded l e f t i d e a l I
o f R w h i c h i s gr-maximal, and t h e r e i s an e l e m e n t u 6 G, such t h a t S '2 ( R / I ) ( r ) .
Proof.
Take x F h ( S ) T , x
# 0. The map a : R
--f
S(T) d e f i n e d by a ( r )
=
rx
i s g r a d e d o f degree e . S i n c e S ( T ) i s o r - s i m o l e t o o , a i s s u r j e c t i v e i . e .
S(T)
R/I
with
I
1.7.3. D e f i n i t i o n .
gr-maximal i s R. P u t t i n g Consider M
c
u =
T
-1 .
y i e l d s S 2 (%'I)(u).
R - g r . The Jacobson g r a d e d r a d i c a l o f M,
d e n o t e d by J 9 ( M ) , i s t h e i n t e r s e c t i o n o f a l l gr-maximal submodules o f M, t a k i n g Jg(M)=M i f
M possesses no p r o p e r gr-maximal subniodules.
53
1. Graded Rings and Modules
Lemma.
1.7.4.
L e t M F R-gr,
then :
1. I f M i s o f f i r , i t e t y p e t h e n Jg(M) 2 . Jg(M)
fl { Ker f, f c Horn (M, S ) , R-or = n { K e r f , f c HOW; (b!,S), =
3. If f C HOMR(t4,N) t h e n f
n
4. J g (RR) =
# M o r t4
=
0.
S pr-simole}
1
S gr-sirnDle
(J9(P)) c J g ( N ) .
{ AnnRS, S g r - s i m p l e i n R-gr
1
5. Jg(,R)
i s a graded i d e a l o f R.
6. Jg(,R)
i s t h e l a r a e s t p r o p e r graded i d e a l such t h a t any a C h ( R ) , f o r
w h i c h t h e image i n R/JP(RR) i s i n v e r t i b l e , i s i t s e l f i n v e r t i b l e .
Proof.
A l o n g t h e l i n e s o f p r o o f o f t h e ungraded e q u i v a l e n t s , l e t us
e s t a b l i s h o n l y 6. h e r e . Let
IT
: R
+
R/Jg(,R)
such t h a t .(a) l e f t ideal
M
be t h e c a n o n i c a l epimorphism. Suppose a C h(R) i s
i s i n v e r t i b l e . I f Ra # R t h e n t h e r e e x i s t s a ar-maximal of R such t h a t Ra c M. S i n c e Jg(,R)
i t f o l l o w s t h a t R = M, a c o n t r a d i c t i o n .
f o r some b c h ( R ) . From'
c
M and n(M)
=
n(R)
Hence Ra = R and t h e r e f o r e ba = 1
1 = n ( b ) n ( a ) i t f o l l o w s t h a t c b = 1 f o r some
c C h(R) and t h e r e f o r e b i s i n v e r t i b l e i n R. P o r e o v e r ba = 1 t h e n i m p l i e s t h a t a i s i n v e r t i b l e i n R . Now l e t I be a qraded p r o o e r i d e a l o f R such t h a t any a c h(R)
such t h a t n ( a ) i s i n v e r t i b l e i n nI(R)
(where nI:
R
--f
i s t h e c a n o n i c a l epimorphism) i s i n v e r t i b l e i n R . I f I # Jg(RR) t h e n t h e r e i s a gr-maximal l e f t i d e a l M o f R such t h a t I $ i t follows t h a t 1 = a
+
b with a C I, b
M.
From I + lul = R
M and i t i s c l e a r t h a t we may
assume t h a t a, b c Re. Because
m I ( b ) i s i n v e r t i b l e , b i s i n v e r t i b l e i n R,
b u t then M = R
Yields a contradiction. I n v i e w o f 5 and 7. i n t h e above lemma we w r i t e Ja(R) ( = Jg(RR)= Jg(RR))
R/I
A. Some General Techniques in the Theory of Graded Rings
54
f o r t h e Jacobson graded r a d i c a l o f R. From 6. i n t h e above lemma we r e t a i n t h a t Jg(R) i s t h e l a r g e s t p r o p e r g r a d e d i d e a l o f R such t h a t r C Re.
1
t
a r i s a u n i t f o r a l l a C Jg(R) fl Re,
I n o t h e r words, Jg(R) i s t h e l a r a e s t p r o p e r g r a d e d i d e a l o f R such
t h a t i t s i n t e r s e c t i o n w i t h Ro i s i n J ( R e ) . L
1.7.5. If
Lemma.
R - g r i s f i n i t e l y g e n e r a t e d t h e n Jg(R)M iM.
M c
From 1 . 7 . 4 . and s i m i l a r t o t h e ungraded case.
Proof.
1.7.6. Jg(R)
(Graded v e r s i o n o f Nakayaiaa's Leisma).
P r o p o s i t i o n . I f R i s a s t r o n g l y qraded r i n g o f t y p e G t h e n
n Re
=
J ( R e ) , where J(Re) i s t h e Jacobson r a d i c a l o f t h e r i n o Re.
P r o o f . I f I i s l e f t gr-maximal i n R t h e n Ie i s a maximal l e f t i d e a l o f Re,
( c f . C o r o l l a r y 1.3.8.
Since I
n
i n Section 1.3.).
Re = Ie and s i n c e e v e r y maximal l e f t i d e a l o f Re e x t e n d s t o
a p r o p e r graded l e f t i d e a l o f R, i t f o l l o w s t h a t Jg(R) An F1 F R-gr i s s a i d t o be l e f t g r . - N o e t h e r i a n , i f M s a t i s f i e s t h e ascending,
nRe
= J(Re).
resp. l e f t g r . - A r t i n i a n
r e s p . descendinp, c h a i n c o n d i t i o n f o r
graded l e f t R-submodules o f Y . C l e a r l y , a g r . - s i m p l e
bl i n R-gr i s b o t h
l e f t g r . - N o e t h e r i a n and l e f t g r . - A r t i n i a n .
1.7.7.
Theorem. (Graded v e r s i o n o f H o p k i n s ' theorem).
I f R i s l e f t g r - A r t i n i a n then R i s l e f t gr.-Noetherian. Proof.
F o r m a l l y s i m i l a r t o t h e p r o o f o f t h e unaraded e q u i v a l e n t . One
shows f i r s t t h a t R/Jg(R) i s a s e m i s i m p l e o b j e c t o f R - g r and t h a t (Jg(R))'=,C f o r some p o s i t i v e n C F o r each i, 1 5 i 5
Z. n-1,
hence an R/Jg(R)-module.
(Jg(R))i/
Jg(R))'"
i s a n n i h i l a t e d by J g ( R ) ,
I t f o l l o w s t h a t (Jg(R))'
/ (J'(R))'+'
is
s e m i s i m p l e i n R-gr and because i t i s a l s o l e f t g r - A r t i n i a n i t has t o be
55
1. Graded Rings and Modules
o f f i n i t e type, hence l e f t g r - N o e t h e r i a n . RR i s l e f t gr-Noether an i n R - o r ,
1.7.8.
Remark.
hence R i s l e f t g r - N o e t h e r i a n .
A l e f t gr.-Artinian
, where
F o r example k[X,X-'] i s gr.-Artinian
It i s then also c l e a r t h a t
r i n g R need n o t be l e f t A r t i n i a n .
k i s a f i e l d and X a v a r i a b l e o f degree 1,
and n o t A r t i n i a n .
N e x t we i n c l u d e a s h o r t d i s g r e s s i o n i n t h e unoraded t h e o r y w h i c h i s n e c e s s a r y i n o r d e r t o p r e s e n t some o f t h e deeper r e s u T t s c o n c e r n i n g t h e Jacobson r a d i c a l and t h e s y m p l e c t i c r a d i c a l o f p r a d e d r i n o s o f t y p e Z . 1.7.9.
Proposition.
L e t S be a r i n g c o n t a i n i n g a s u b r i n g R such t h a t
S i s as a l e f t R-module g e n e r a t e d by a f i n i t e s e t o f R - n o r m a l i z i n g
elements, al
,...,
an say ( i . e .
Ra. = a.R f o r i = 1,... n . ) . I f 1
1
M c
S-mod
i s a s i m p l e S-module t h e n RM i s semisimple and f i n i t e l y Generated. P r o o f . I f 0 # x C M t h e n M = Sx =
Cn Raix and t h u s RM has f i n i t e t y p e . i=l I f N c M i s a maximal p r o p e r R-submodule t h e n Ni = ( N : ai) = { x CM,aix c N }
i s c l e a r l y an R-submodule o f M. We c l a i m t h a t Ni R-submodule o f o r N + aiRx ai(y-u)
= z
y c (N:ai)
+
M
M-
t o o . Indeed, i f x
=
M.
If y 6
c
N
and y
-
M
t h e n aiy
=
u c(N : ai).
i s a maximal p r o p e r
Ni t h e n aix f N and
M
= NtRaix
z + aiu where z 6 N, u C R x, hence The l a t t e r means t h a t
Rx, o r i t f o l l o w s t h a t (N:ai)
i s a maximal R-submodule o f M.
Note t h a t n r j t a l l Ni a r e equal t o M s i n c e o t h e r w i s e N = M. n P u t L = 1-1 Ni. I f L # 0 t h e n SL # C. By d e f i n i t i o n o f L ( i t i s e a s i l y i =1 n n checked t h a t ) i t f o l l o w s t h a t SL c N, s i n c e S = 2 Rai, L = CI N . .
i=I
'
i=l
The s i m p l i c i t y o f SM y i e l d s SL = M hence N=M, a c o n t r a d i c t i o n . T h e r e f o r e L = 0 f o l l o w s . The nonzero M/Ni
RM
=
Q MJN.,
thus
a r e s i m p l e i n R-mod and we o b t a i n
M i s semisimple i n R-mod.
R
56
A. Some General Techniquesin the Theory of Graded Rings
I f J i s a primitive ideal of S then I = J n R i s a
1.7.10. Corollary.
f i n i t e i n t e r s e c t i o n of primitive i d e a l s of R . Let us just mention the following elaboration of the arauments i n the foregoing proof. 1.7.11. Lemma. Let S be a ring containina a subring R such t h a t
S is
generated as a l e f t R-module by a f i n i t e s e t of R-centralizinp elements of S , then : 1. I f M C S-mod i s simple then RM i s semisimple
o f f i n i t e lenath a n d
the simple R-submodules of RM are isomorphic t o one-another. 2 . If N
c ?-mod i s simple then S
i n R-mod.
n R
3. We have J ( S )
3
6h N
R
i s semisimple of f i n i t e length
J(R) and equality holds f o r example in each of
the following cases :
- does n o t a n n i h i l a t e a simple R-module R -1 b . R i s r a t i o n a l l y closed in S, i . e . i f x c R a n d x c S then x-'
a. S
Proof.
@
cf.
[ 8 ]
.
I n t h i s p r e p r i n t G . Bergman asks whether
equality
holds generally. For t h i s a n d r e l a t e d r e s u l t s we may r e f e r t o a . 0 . Robson's work on the theory of normalizing extensions, c f . [ 9 I ,[48 1. We introduce the followina notation; i f M c R-mod then JR(M), the Jdcobson radical
of M in R-mod, i s the i n t e r s e c t i o n of the proper
maximal R-submodules of M. 1.7.12. Lemma. Let S be a ring containing a subrina R such t h a t S i s free R-module generated by a f i n i t e s e t { a l , . . . ,a n } such t h a t each ai, i
=
1, ..., n, commutes with R in S. Then JR(M) = M n J s ( S 60 M ) ,
every M C R-mod. Proof.
Cf.
[ 8
1.
R
for
c R.
I. Graded Rings and Modules
1.7.13.
s
Proposition.
Let S
be a r i n g
57
c o n t a i n i n g a s u b r i n g R such t h a t
i s g e n e r a t e d as a l e f t R-module by a f i n i t e number o f R n o r m a l i z i n g
...,an
elements, al,
say. If J i s a maximal i d e a l o f S t h e n
J
il R i s a
f i n i t e i n t e r s e c t i o n o f maximal i d e a l s o f R.
proof.
1.7.9.
The p r o o f i s based upon t h e i d e a s o f p r o o f used
b u t adapted t o t h e bimodule s i t u a t i o n ; l e t us j u s t p r e s e n t a b l u e
p r i n t o f the proof.
I t i s n o t h a r d t o see t h a t J i s c o n t a i n e d i n a
p r o p e r maximal R-bimodule i n S ( u s i n g Z o r n ' s lemma). an R-bimodule i n S t h e n : (K:ai)l = {
s
i n Proposition
s C S, s ai c K } and (K : ai)r
=
C K } a r e a l s o maximal R-bimodules i n S . F u r t h e r m o r e any
ais
6 S,
= {
I f K i s maximal as
such K i n t e r s e c t s R i n a maximal i d e a l o f R as i s e a s i l y v e r i f i e d . Now n (K:ai)l and Lr = (K:ai)r, L = LrilL1. p u t L1 =
iil
icl
Since J i s an i d e a l o f S, L i J f o l l o w s , whereas on t h e o t h e r hand
K
S L c K, LS c
i s evident from the construction o f
L . Thus SLS i s a n
i d e a l o f S , c o n t a i n i n g J and c o n t a i n e d i n K. By m a x i m a l i t y o f J, SLS = J and L = J f o l l o w s . F i n a l l y , t h i s e n t a i l s R
n
( R fl (K:ai)r)l,
Ti
J = R Ti L =
ic;
c (Rn(K:ai)l)
which i s a f i n i t e i n t e r s e c t i o n o f maximal i d e a l s o f R .
We now a p p l y t h i s i n case R i s a graded r i n g o f t y p e
H.
The f o l l o w i n a
r e s u l t s s t e r n f r o m t h e a u t h o r ' s p a p e r [ 8 4 1 , b u t t h e b a s i c i d e a comes from G. Bergman's [ 8 1
.
1 . 7 . 1 4 . Theorem. L e t R be a graded r i n a o f tyDe Z/nZ f o r some n # 0 i n N, l e t M F R-gr. Then, i f x 6 R - g r . Then, i f x F J R ( V ) decomposes as X
0
+.. .+ x
Proof.
s
n-1'
Let
o
we have n x . F JR(M) f o r each j C Z/nZ. J be a p r i m i t i v e nth-
r o o t o f u n i t y i n II: and c o n s i d e r
= R bd Z [a]. We c o n s i d e r S as a graded r i n o
Z s j = Ri c% Z [ w I f o r a l l j C z / n z Z
If
o f type Z/nZ
M F R-gr t h e n on
by p u t t i n o
S c% 11 we R
n
A. Some General Techniques in the Theory of Graded Rings
58
consider the gradation defined by putting ( S S @ M c S-gr. Write R
=
60
R
Si s M k a n d then i+k=j
M) .= C
S 60 M. R
sj) = 2 ( Z d s . where jWnZ jtz/nZ s . F S j . Obviously a i s an automorphism. J If we define 0 : R -L R by putting ( 2 m . ) = C. mJm then j' J W ~ Z jcZ/nZ . . o(s.m.) = wliJ s . m . = o(s.)o(mi),where s F S . and mi f M i . J 1 J 1 J j~ Hence & i s a o-automorphism of R a n d i t i s c l e a r l y a l s o a araded morphism
Now we define u : S * S by putting
B
a
of degree o F Z/nZ. Consider x c JS(R) and l e t g :
R
+
T be an S-linear
map t o any simple
S-module T . We show t h a t g ( & ( x ) ) = 0. The automorphism a defines a " r e s t r i c t i o n of s c a l a r s " functor i n S-mod.
Consider
a'
=
g
0
o
=
U ,
: S-mod
b + f?
+
S-mod, and o,(T) i s simple
o,(T)
--f
as a set map ( u * ( T ) a n d
T ar? i d e n t i f i e d as s e t s ) .
From g ' ( s z )
=
g ( ~ ( s )& z ) )
~ ( s g) ' ( z ) with s
=
S, z C
f
P
i t follows
t h s t g ' i s a n S ' - l i n e a r map in the module s t r u c t u r e of o,(T).
Then
x F J s ( k ) e n t a i l s g ( x ) = 0 or & ( x ) C J s ( b ) foilcws. Thus o (J s (f? )) c c Js(R). Now applying Lemma 1 . 7 . 1 2 we obtain J,(M) = ! t I1 J s ( b ) . -i j -
I f we c a l c u l a t e t
i
=
c
w
jcZ/nz
c Z/nZ, then we find t
under the automorphisms
;J,
=
( c J ) ~ (xo
nxi. B u t t j F
+
f M
... +
x ~ - ~ f )o r, some fixed
n Js(W) since i t i s fixed
11,...n-1). Therefore n xi
Js(V) f o r
C
a l l i , follows.
1.7.15. Corollary.
Let R be a graded ring of type
z
and l e t
then JR(M) i s a graded submodule of M. I n p a r t i c u l a r , i f R obtain G . Berman's r e s u l t t h a t J(R) i s a graded ideal o f R.
=
M
F R-gr,
M , we
59
I. Graded Rings and Modules
Proof.
The Z-gradation induces a Z / n Z g r a d a t i o n f o r each n
a = a l +...+a h
JR(M) with 1
C
<
.. , a h .
By the theorem i t follows
t h a t n a i t JR(M) f o r a l l i . The same argument holds
+
0 . Consider
h . Take n > h-1, t h e n theZ /nZ-homogeneous
components of a w i 11 be e x a c t l y the a l , .
by n
>
i f n i s replaced
1 but then ( 1 + n)ai - n a i t J,.(P).
I f R i s Z-graded then j ( R )
1.7.16. Remark.
n
Indeed, i f a 6 J(R)
c
Jg(R).
Ro t h e n l + a i s a u n i t of R . Since J ( R ) i s graded,
t h e maximality property of J9(R) ( s e e I.7.4., 6j i m p l i e s J(R) c J g ( R ) . 1.7.17.
C o r o l l a r y . Let R be a graded r i n a of type Z and l e t M be a simDle
o b j e c t of R-gr then JR(M)
=
0.
Let A be an a r b i t r a r y r i n g and l e t S be a simple
1.7.18. C o r o l l a r y .
A-module, then J A L x 1( S [ X I ) = 0 , where X i s a v a r i a b l e commuting w i t h A and S
[XI
Proof.
The theorem y i e l d s t h a t M = JR(S [ X I ) i s a praded submodule of
S
[XI
=
S
8
A
(writing R
A [XI.
=
w i t h X Mi c M i + l .
A [ X I ) . We have t h a t M = Mo
B u t each Mi i s
IfM i s maximal l e f t i d e a l of R then M
maximal l e f t i d e a l s c o n t a i n i n g i t . Proof.
@
...a M . 8
...
... .
Let R be a graded r i n g o f type Z .
Corollary.
-
Pi
isomorphic t o an A-submodule of S
Consequently Mi = 0 f o r a l l i = 0 , 1, 1.7.19.
@
9
i s the i n t e r s e c t i o n of a i l
Then JR(X) i s graded. So t h e r e i s a graded g' l e f t i d e a l Y of R such t h a t M c Y c F! which maps t o JR(X) under t h e 9 canonical map R + X . Hence Y = Mg and JR(X) = 0 f o l l o w s . 1.7.20.
Ideal
P u t X = R/M
C o r o l l a r y . Let R a graded r i n g of t y p e z . If I i s a primitive of R then I
containing i t .
9
i s the i n t e r s e c t i o n o f the p r i m i t i v e i d e a l s of R
A. Some General Techniques in the Theory of Graded Rings
60
1.7.21.
Proposition.
w i t h A,
then J(.4
I
=
1. L e t A be an a r b i t r a r y r i n a , T a v a r i a b l e commut
[TI )
= I [ T l where
J ( A [ T l ) il A i s a n i l i d e a l o f A.
2. L e t A be an a l g e b r a o v e r t h e f i e l d k and l e t k ( T ) be t h e f i e l d o f r a t i o n a l f u n c t i o n s i n one v a r i a b l e Then J ( A
@
k
over k.
k ( T ) ) = I 69 k ( T ) where I = A k
n
J ( A 60 k ( T ) ) i s a n i l i d e a l o f k
A (S. Amitsur). Proof. 1. U s i n g t h e f a c t t h a t A [ T ] has an automorphism s e n d i n g T t o T p r o o f o f 1. becomes an
1, t h e
easy e x e r c i s e .
2. ( F o l l o w i n g G. Bergman, 1 8 1 ) . Conversely, any element o f J ( A where p ( T ) F k [ T I
+
69
k
That I @ k(T) c J(A
k ( T ) ) i s obvious k k ( T ) ) can be w r i t t e n as D ( T ) - I (amTm+.. @
i s nonzero and ai F A. Now k ( T ) i s n o t Z - g r a d e d b u t
i t can be Z/nZ-graded by p u t t i n o k(T)i
=
Tik(Tn)
for n
0, i F Z / n Z
L o o k i n g a t t h e i n d u c e d g r a d a t i o n o f A 60 k ( T ) and a D p l y i n g Theorem 1.7.14 . k we o b t a i n f o r a l l n > m : n ai T’ F J ( A 6n k ( T ) ) ; t h u s ai T’ f J ( A qh k ( k k and t h e r e f o r e ai F I . Pick
x c I, t h e n ( 1
+
xT)-’
t h e c o e f f i c i e n t s have t o
=
1 - xT
+
x2T2
-...
i n A 60 k ( ( T ) ) . However k
l y i n a f i n i t e d i m e n s i o n a l k-subspace o f A i . e
x i s a l g e b r a i c o v e r k . I f x were n o t n i l p o t e n t t h e n t h e p o l y n o m i a l e q u a t s a t i s f i e d by x w o u l d g i v e r i s e t o a nonzero i d e m p o t e n t e l e m e n t i n t h e Jacobson r a d i c a l , a c o n t r a d i c t i o n . I n view o f p r o p o s i t i o n 1 . 7 . 1 3 we may produce s t a t e m e n t s s i m i l a r t o I . 7 . F and 1.7.18.,
a two s i d e d
v e r s i o n o f 1 . 7 . 1 9 . and a analogon f o r 1.7.21.;
w i t h r e s p e c t t o t h e s y m p l e c t i c r a d i c a l J s ( R ) o f a graded r i n g . L e t us j u s t m e n t i o n t h e main r e s u l t and r e f e r t o [ 8 4 ]
f o r more d e t a i l .
I. Graded Rings and Modules
61
1.7.22. Proposi t i o n . Let R be a graded r i n g of type 2. T h e n : 1. J s ( R )
=
n tM,
M maxima
i d e a l of R} i s graded.
2. I f A i s any r i n g , X a v a r i a b l e commuting with A then : JS(A [ X I )= I [ X I
, JS(A
[X,
X-l])
= I"X,
X-l]
f o r i d e a l s I , I ' of A.
1.7.23. C o r o l l a r y .
Let R be a graded r i n g of type Z. I f M i s a maximal i d e a l o f R then M
9
i s t h e i n t e r s e c t i o n of t h e maximal i d e a l s c o n t a i n i n g Mg. I n p a r t i c u l a r any gr-maximal i d e a l i s an i n t e r s e c t i o n of maximal i d e a l s . 1.7.24. Example.
Let k be a f i e l d , R
=
k [XI
a l l maximal i d e a l s of R i s zero. Note t h a t M maximal i f M i s maximal; indeed M
=
(X-1)
t h e n the i n t e r s e c t i o n o f
9 i s n o t n e c e s s a r i l y gry i e l d s Mg = 0 c ( X ) .
A. Some General Techniques in the Theory of Graded Rings
62
1.8. Almost s t r o n g l y graded r i n g s 1.8.1. D e f i n i t i o n .
The r i n g R i s c a l l e d an almost
s t r o n g l y graded r i n g
over t h e group G i f t h e r e e x i s t s a decomposition sum ( n o t n e c e s s a r i l y d i r e c t sum) : (1) R =
I:
u6 G
Ro
such t h a t Ro RT
( a s a d d i t i v e groups) i n t o a d d i t i v e subgroups Ro, = Rm
for all
U,T
C
u C G,
G.
I f t h e decomposition sum (1) i s d i r e c t , then R
i s a s t r o n g l y graded
r i n g a5 i n 1.3. In 1 2 4 1 , Dade c a l l s an almost s t r o n g l y graded
r i n g : C l i f f o r d system.
1 . 8 . 2 . Remarks. 1) I f R =
Z Ro i s an almost s t r o n g l y graded r i n g , then R/I i s a l s o
0 6G
an almost s t r o n g l y graded r i n g , f o r a l l two-sided
i d e a l s I of R .
In p a r t i c u l a r i f R = 0 Ro i s a s t r o n g l y graded r i n g , then R/I i s an
d G
almost s t r o n g l y graded r i n g f o r a l l two-sided i d e a l s I of R . ( n o t necessarily 2) If R =
graded i d e a l s ! ) . 2 Ra i s an almost s t r o n g l y graded
d G
r i n g , then f o r a l l u
<
G,
Ro i s an i n v e r t i b l e Re-bimodule. In p a r t i c u l a r , we o b t a i n t h a t Ro i s a l e f t ( r i g h t ) f i n i t e l y generated Re-
module.
Moreover Ro i s a p r o j e c t i v e Re-module ( l e f t and r i g h t ) .
3 ) I f R i s an almost s t r o n g l y graded r i n g , then t h e r e e x i s t s a s t r o n g l y graded r i n g T and an i d e a l I
such t h a t R
Indeed we can d e f i n e t h e s t r o n g l y graded r i n p for all
CJ
c G. The m u l t i p l i c a t i o n of
T/I.
T =
8
where To = R o
T
aCc-
R given by maps
:
63
1. Graded Rings and Modules
Induces t h e m u l t i p l i c a t i o n of
To
Tz
+
T
g i v e n by maps,
TOT
The a p p l i c a t i o n
T 2 R,
q((xJucG)
=
z
uCG
xo i s c l e a r l y a s u r j e c t i v e r i n g
homomorphism.
I . 8.3. N o t a t i o n s .
c
1) If M
(M) we denote t h e l a t t i c e o f RRe submodules o f M( r e s p Re-submodules o f M ) . R-mod, t h e n by =CR(M) ( r e s p . d:
M) we denote t h e K r u l l d i m e n s i o n o f e o v e r t h e r i n g Re).
2) By K.dimRM ( r e s p . K.dimR t h e r i n g R (resp.
M
over
P) we denote t h e G a b r i e l d i m e n s i o n o f M Re over the r i n g R (resp. over the r i n g Re).
3) By G.dimR
M
( r e s p . G.dim
R = ;C
Throughout t h i s s e c t i o n
uCG
o v e r a f i n i t e group G. Let
M
Ru
i s an a l m o s t s t r o n g l y graded r i n g
M.
C R-mod and N be an Re-submodule o f
C l e a r l y N* i s t h e l a r g e s t R-submodule o f
-Remark -
Define
N**
c Nf*. that N -
1.8.4.
Lemma.
=
M
We denote by N*
=
I;
uC G
RON.
c o n t a i n e d i n N.
c RUN. T h i s i s t h e s m a l l e s t R-submodule o f F? such
u
If N i s an e s s e n t i a l Re-submodule o f
M,
t h e n so is,N*;
( t h e group G i s f i n i t e t h r o u g h o u t ) . Proof.
I f X c M i s Re-submodule t h e n R u - l
Hence N
n
= Ro N
n
RG-l X
# 0 and R,(N
fl R u - l
X e i t f o l l o w s t h a t Ru N
submodule of
n
X #
X ) # 0.
0 (because R o - l i s i n v e r t i b l e ) S i n c e Ro(N
n
Ru-i
X)
=
X # 0, t h u s RON i s an e s s e n t i a l Re-
M.
Since G i s f i n i t e , N* i s an e s s e n t i a l Re-submodule.
A. Some General Techniques in the Theory of Graded Rings
64
1.8.5. Lemma. i)
(M/RuL)
I
Re
i i ) d:
L e t M C R-mod and L
Re
d:
e
Re
be an Re-submodule o f M.
Then
(M/L)
(RUN) = LRe(N)
Proof. (M/L). C l e a r l y X/L + RoX/RoL i s an isomorphism of Re l a t t i c e s . The i n v e r s e isomorphism i s Y/RoL + Ru-l Y/L where i)
L e t X/L C
Y/Ro L t
d:
(M/RoL). Re The second statement i s c l e a r s i n c e Ro i s an i n v e r t i b l e Re-bimodule. 1.8.6.
L
M
Lemma.
r e s p e c t t o N* =
t R-mod. Then M contains an Re-submodule N maximal w i t h
o i F I } be a chain o f Re-submodules o f
L e t INi,
Proof.
# 0 then t h e r e i s an x c ( U Ni)*,
f o r each i C I . I f ( U Ni)* iCI i.e.
R x
c ( U
iC I
Ni)*,
c
iFI
U
iC I
Ni.
Hence
Ru x
Since R i s a l e f t f i n i t e l y generated Re-module,
u 6 G t h e r e e x i s t s an i o c I such t h a t Ro x Rx c Ni
#
and N :
c U N$ f o r a l l
iF I
= 0
N :
x # 0,
c
G.
i t follows t h a t f o r a l l
Ni
f o r a l l o c G. Thus 0
0, c o n t r a d i c t i o n .
0
0
Hence ( U iC I
P such t h a t
= 0 and
Ni)*
so Z o r n ' s lemma can be a p p l i e d t o y i e l d t h e
e x i s t e n c e of an N as desired. If M F R-mod, we l e t r a n k R M, rankReM denote t h e Goldie dimension o f
M
over these r i n g s . 1.8.7.
Lemma.
L e t o r d G = n and rankRM
= m.
maximal w i t h r e s p e c t t o N* = 0, then rank
Proof.
L e t A1,...,At
i s d i r e c t modulo N.
Re
I f N i s an Re-submodule
(M/N)
be Re-submodules o f M s t r i c t l y
5
m and rank
Re
M
5
mn.
c o n t a i n i n g N whose sum
65
1. Graded Rings and Modules
# 0 f o r each 1
Then A:
*
c
'j#i
*
n
A,)
A . Ti Ai
J
j#
.8.5.
By Lemma
n
0 = N* =
1.8.8.
Aj
j#i
and S O
UC G
i
.
t
5
If t > m
t h e n f o r some i :
ic 0.
z
Because
5
n
Ai)* 3
#
3 ( -
n
2 A ); j#i
i t follows that (
A:
N, a c o n t r a d i c t i o n . Thus r a n k
we have t h a t r a n k RON we have :
0
(M/RoN)
Re
+
M
+
5
8
oCF
Re
(M/N)
m for a l l u
M/RoN,
z
j#i
A . rl Air# J
5 in.
c
G. Because
and so r a n k
Re
M5m.n.
t h e n Re has
C o r o l l a r y . I f R has t h e f i n i t e G o l d i e dimension
f i n i t e G o l d i e dimension. Moreover, i f R = G o l d i e dimension ~
8
oCG
RU i s a s t r o n g l y graded r i n g , t h e n R has
i f and o n l y , i f
Re has
finite
finite
G o l d i e dimension.
Proof.
F o r t h e f i r s t s t a t e m e n t we a p l l y Lemma 1.8.7..
R =
RU i s a s t r o n g l y graded r i n g . S i n c e R o i s a p r o j e c t i v e f i n i t e l y
8
Now, suppose
that
UCG
generated Re-module f o r each dimension as an R -module. e dimension as an Re-module.
u C G,
t h e n Ro has f i n i t e
Because G i s f i n i t e ,
R has
Goldief i n i t e Goldie
Hence R has f i n i t e G o l d i e dimension as an
R-module. 1.8.9.
*
N
=
Lemma.
Let M
c R-mod and N 5 M be an Re-submodule o f
C such t h a t
0. Then :
i ) M i s a l e f t N o e t h e r i a n Re-module i f and o n l y i f M/N i s l e f t N o e t h e r i a n Re-module. i i ) K.dim
I4 = K.dim (P/N) i f e i t h e r s i d e e x i s t s . Re Re
iii) G. dimRe M = G.dimRe(M/N)
i f either side exists.
Note t h a t I4 embeds i n 8 M/RoN. uc G Lemma 1.8.5.
\
The r e s u l t now f o l l o w s f r o m
0.
66
A. Some General Techniques in the Theory of Graded Rings
1.8.10.
L e t M c R-mod. Then Y i s l e f t N o e t h e r i a n i f and o n l y
Theorem.
if
M i s l e f t N o e t h e r i a n . I n D a r t i c u l a r R i s l e f t N o e t h e r i a n i f and Re o n l y i f Re i s l e f t N o e t h e r i a n . Proof.
Suppose t h a t RM i s N o e t h e r i a n . By Lemma 1.8.9.
t o show t h a t M/N i s a l e f t N o e t h e r i a n R -module where N e
it i s sufficient
5P
i s a maximal
Re-submodule w i t h r e s p e c t t o N* = 0. L e t M1/N of M/N.
c M2/N
Then M1
. ..
c
j
be an ascending c h a i n o f non-zero R-submodules
N and hence 0
# MT
c
MI.
Since
M; i s R-submodule o f M
i t f o l l o w s by N o e t h e r i a n i n d u c t i o n t h a t M/N;
a l e f t N o e t h e r i a n Re-module.
Thus t h e c h a i n must t e r m i n a t e and hence M/N
i s a l e f t N o e t h e r i a n Re-
module. The converse i s c l e a r . 1.8.11.
L e t M c R-mod and supoose t h a t K.dim
Lemma.
K.dimRM = K. dim Proof. a
Re
M.
C. L e t K.dimRM = a. By i n d u c t i o n on Re 5 a. I t i s easy t o reduce t h e p r o b l e m t o t h e
C l e a r l y , K.dimR M 5 K.dim
we show t h a t K.dim
Y
Re case where M i s a - c r i t i c a l . L e t r e s p e c t t o N* = 0. I f X
so K.dim K.dim
Re
M e x i s t s . Then Re
Re
M/X*
<
3
f
N f M be a maximal Re-module w i t h
N, t h e n X* # 0 and hence K.dimRM/X*
a, by i n t r o d u c t i o n . Thus K.dim
M/N c a . T h e r e f o r e K.dim
Re
Re
M/X <
a
< a
and
and hence
M 5 a by Lemma 1.8.9.
L e t M 6 R-mod. Then K.dimR !-t = K.dimR C i f e i t h e r e s i d e e x i s t s . I n p a r t i c u l a r , K.dim R = K.dim R i f e i t h e r s i d e e x i s t s Re and R i s r i g h t A r t i n i a n i f and o n l y i f R i s r i a h t A r t i n i a n . e 1.8.12.
Theorem.
Proof.
F o l l o w s f r o m Lemma 1.8.11.
1.8.13.
C o r o l l a r y . I f RM i s a - c r i t i c a l ,
i t c o n t a i n s an e s s e n t i a l d i r e c t sum Re-modules.
then
C i s K-homogeneous and Re o f a t most n = o r d G, a - c r i t i c a l
1. Graded Rings and Modules
67
(Here M i s called K-homogeneous i f f o r a l l R -submodules N c M, N # 0 we have K.dim
Proof.
Re
N = K.dim
Re
V).
By Theorem 1.8.12., K.dim
M = a.
Re Let 0 # L c M be an Re-submodule of L . By Lemma I . 8 . 5 . , i i ) , we have
L__
K.dim
Re
L = K.dim
Hence K.dim K.dim
Re
Re
RoL f o r a l l u
Re
C
G.
( R L ) . Since M i s a - c r i t i c a l then K.dimR(RL) = a and hence
(RL) = a
(by Theorem 1.8.12.). Hence K.dim
L
Re
= a.
Thus M i s K-homogeneous over the ring Re.
Now l e t K
c
M be a a - c r i t i c a l Re-submodule of M. Because Ro i s an i n v e r t i b l e
Re-bimodule , RoK i s an a - c r i t i c a l Re-module f o r a l l
5
c G. T h u s R K
=
c
oCG
RuK
i s a sum of a t most n = o r d G , a - c r i t i c a l R e -modules. Since M i s a - c r i t i c a l
over the ring R , K.dimR V / R K < a and thus K.dimR ( M / R K ) < a . e I f RK i s n o t e s s e n t i a l i n V as an Re-module, then R K f? L = 0 f o r some
nonzero R -submodule 1 of V. Thus K.dim L c K.dim M/RK < a . Because e Re Re M i s K-homogeneous as an R -module we obtain a contradiction. e Thus R K i s e s s e n t i a l in M as an Re-module.
M i f e i t h e r side Re e x i s t s . I n p a r t i c u l a r G.dim R = G.dim Re i s e i t h e r s i d e e x i s t s . 1.8.14. Theorem.
M c R-mod; then G.dimR M
Let
Proof. The same proof
= G.dim
as f o r the theorem 2 . 2 . in
[ 10 I.
W e now turn t o the investigation of prime ideals in almost strongly graded rings. Let R
=
c Ro be an almost strongly graded r i n a . I f
OF G
i s a two-sided ideal of Re we define the 5-conjugate of A A" = R
0-1
ARu. Clearly A'
a Prime ideal in Re,
i s a two-sided ideal of R
then 'Q
e
i s a prime ideal in Re,
.
(5
A c Re
F G ) by :
Poreover, i f Q i s f o r any u F G.
A. Some General Techniques in the Theory of Graded Rings
68
c
I f Au = A f o r a l l o
t h a t the G-invariant ideal A o f R f o r G - i n v a r i a n t i d e a l s Bi
1.8.15.Lemma. L e t R
i d e a l . We say
G, A i s s a i d t o be a G - i n v a r i a n t
e
i s G-prime
i f and o n l y i f B1B2 c A
o f Re i m p l i e s t h a t B1 c A o r B2 c A.
1 R o be an a l m o s t s t r o n g l y oraded r i n c . I f P u
=
i s a prime i d e a l i n
R
Conversely, i f R =
0 oCG
R
i s a s t r o n a l y graded r i n g and Q i s a G-prime
t h e n t h e r e e x i s t s a t l e a s t one p r i m e i d e a l P o f R such
i d e a l i n Re,
t h a t P fl Re = Q. I f P i s an i d e a l i n R, t h e n Q = P
Proof.
o f R e . Suppose now t h a t B1B2 c Q, Because B1
where Bi
Re i s a G - i n v a r i a n t i d e a l
a r e G - i n v a r i a n t i d e a l s o f Re.
and B2 a r e G - i n v a r i a n t i d e a l s we have RBI
Now B1B2 c Q y i e l d s B1B2 c P hence RBIR
n
BR I
=
and RB2 = B 2 R .
a n d - t h e n R(B1B2)P. c P y i e l d s (RB1R)(RB2R) c P
c P o r RB2R cP. Thus B1 c Q o r B2 c Q.
Conversely, we suppose t h a t Q i s a C-prime i d e a l inR,.
S i n c e Q i s G-
i n v a r i a n t we have RQ = QR. Because R i s a s t r o n g l y graded r i n g , i t f o l l o w s t h a t RQ
n
Re = Q.
Hence, by Z o r n ' s Lemma, we may s e l e c t an i d e a l P o f respect t o P then I
n
n
Re and
Re = Q. If
J
I, J
a r e i d e a l s o f R p r o p e r l y l a r g e r t h e n P,
fi Re a r e G - i n v a r i a n t i d e a l s of Re p r o p e r l y l a r a e r t h e n
(I n RS)(J
Q. S i n c e Q i s G-prime, t h i s y i e l d s
IJ 4
R, maximal w i t h
n Re) 4 Q
and hence
P. Thus P i s p r i m e .
1.8.16. Theorem.
Let R =
Z Ro be an a l m o s t s t r o n g l y p r a d e d r i n g oCG
( G i s a f i n i t e group). If P i s a prime i d e a l o f R then there e x i s t s a p r i m e i d e a l Q o f Re such t h a t : P
n
Re =
n OCG
Moreover, Q i s m i n i m a l o v e r t h e i d e a l P fl Re.
Qu
69
I . Graded Rings and Modules S i n c e t h e r i n g R/P i s a l s o an a l m o s t s t r o n g l y v r a d e d r i n g , we
proof.
/
R
may assume t h a t P = 0. We v i e w
as an R -R-bimodule.
r e s p e c t t o Y*
subbimodule Y maximal w i t h
= 0. Hence
Choose an Re-Rf! Ro Y = 0. UC G
F i r s t we n o t e t h a t a Reb c Y f o r a C Re ' b c R i m p l i e s t h a t a 6 Y b
Y . Suppose t h a t a, b f Y . Because Y i s a maximal Re-R-subbimodule
w i t h r e s p e c t t o Y*
= 0 i t f o l l o w s t h a t (ReaR
P u t I = (Rea R + Y)*,
J =
+
# 0 and
Y)*
+
(Re b R
Y)*
# 0.
and R i s p r i m e we have IJ # 0.
+
On t h e o t h e r hand, IJ c (Re a R Re a (Re b R
+
C l e a r l y I and J a r e t w o - s i d e d
(Reb R + Y)*.
i d e a l s o f R. S i n c e I # 0, J # 0
c
or
Y) J
Y ) + Y c Re a Re b R
c
+
Hence 0 # IJ c Y . T h e r e f o r e 0 # IJ c Y*,
Re a RJ
+
Re a J
Y J c
+
Y c
Y c Y. contradiction.
P u t Q = Y f! Re. By t h e p r e c e d i n g s t a t e m e n t
we have t h a t Q i s a p r i m e
i d e a l i n Re.
n Qa
Hence
=
0.
OCG
L e t Q ' be a p r i m e i d e a l m i n i m a l o v e r t h e i d e a l P fl Re. i t follows that Q'
3
n Qa.
Hence t h e r e e x i s t s u
uCG
Q'
2
Qa
3
P
Clearly ( Q ' ) " 1.8.17.
n
Re.
-1
S i n c e Q" i s prime, Q ' = Q";
C
Since Q'
n 'Q
3
"CI:
I: such t h a t -1
hence Q =
)'Q(
"
=
(Q')"
i s a p r i m e i d e a l m i n i m a l o v e r t h e i d e a l P f! Re.
Corollary. Let R =
Ro be a s t r o n g l y graded r i n g ( G i s a
8
uCG
f i n i t e group) and Q c Re a p r i m e i d e a l o f Re. Then t h e r e e x i s t s a p r i m e ideal P of 1.8.18.
R
such t h a t P
Corollary.
n
Let R =
R
e
= Q i f and o n l y i f Q 8
ofG
is G-invariant.
Ru be a s t r o n g l y graded r i n g and Q
be a p r i m e i d e a l o f Re. Then t h e r e e x i s t s a
p r i m e i d e a l P o f R such
-1
70
A. Some General Techniques in the Theory of Graded Rings
t h a t Q i s one o f t h e minimal primes over P n Re. Proof.
P u t A = n Qu. C l e a r l y A i s G-invariant a n d a G-prime i d e a l i n OCG
Re. Now t h e r e exists a prime i d e a l P such i s minimal prime i d e a l over t h e i d e a l P
1.8.19. C o r o l l a r y .
Let R
=
2 oCG
t h a t P n Re = A. C l e a r l y Q
n Re.
Ra be an almost s t r o n g l y graded r i n g
( G i s f i n i t e group). Then rad(Re) = rad(R) fl Re.
71
I. Graded Rings and Modules
I . 9. E x e r c i s e s . Comments. References. 1. L e t A be a commutative r i n g and R an A - a l g e b r a graded o f t y p e G such t h a t Re i s an A - a l g e b r a . Then Ra
i s an A-bimodule f o r a l l o
c
F. I f S
i s any graded A - a l g e b r a o f t y p e H, show t h a t t h e A - a l g e b r a R R S
A
has
a natural G x H-gradation. 2 . W i t h n o t a t i o n s as i n 1. Show t h a t R 60 S
s
A
i s s t r o n g l y araded i f R and
a r e s t r o n g l y graded.
3 . L e t R be a s t r o n g l y Graded r i n g o f t y p e G such t h a t i ) Re c Z(R)
ii)G i s t h e d i r e c t p r o d u c t o f subgroups H and i i i ) F o r a l l u c H,
I(.
K and f o r a l l x 6 Ra, y
T C
RT we have t h a t
xy = y x . Show t h a t R 2' R(H) 8 R ( K ) . Re
4. L e t A be a commutative r i n a , G = H x K a d i r e c t p r o d u c t o f groups A [ H x K] . show t h a t A [ H l @ A [ [ ] A Can t h i s be g e n e r a l i z e d t o c r o s s e d p r o d u c t s ? 5. L e t R be a graded r i n g o f t y p e G and l e t H be a subgroup o f G. Consider a 6 R ( H ) . Show t h a t a i s i n v e r t i b l e i n R
i f i t i n v e r t i b l e i n R(H).
6. L e t R be a graded r i n g o f t y p e G and suppose t h a t t h e r e e x i s t s a family
{Hi,
i c I} of subgroups o f G such t h a t I; =
f u r t h e r m o r e t h a t t h e Hi, i n c l u s i o n . Show t h a t R =
i E
U
r,
U Hi and suppose iCI f o r m a d i r e c t e d system w i t h r e s p e c t t o
R( H i ) .
iCI
7 . L e t R be a g r a d e d r i n g of t y p e G where G i s an a b e l i a n t o r s i o n f r e e group. Show t h a t R 2' li+m Ta a Some n C N (depending on a ) .
where Ta i s a g r a d e d r i n a o f t y p e Z n f o r
A. Some General Techniques in the Theory of Graded Rings
72
8. Let R be a graded ring of type G and consider M c R-mod. I f M i s i n j e c t i v e then G C, i' s i' n j' e c t i v e in the cateaory R-gr.
9 . Let R be a graded ring of type G and consider M F R-gr. Show t h a t M i s a d i r e c t summand of
G
.
M i n R-gr.
10. I f G i s a f i n i t e Croup and R i s strongly graded of type 6 then Jg(R)
=
J(R).
Hint. Use C l i f f o r d ' s theorem. 11. A ring S i s called projective r e l a t i v e t o the subring R provided each exact sequence of S-modules :
0
--t
M'
+
M
--f
M"
+
0,
which s F l i t s when considered as a sequence in R-mod; i s s p l i t i n S-mod. __ Lemma (The crossed product version of Maschke's lemma).
Let R
be a strongly graded ring of type G where 6 i s a f i n i t e aroup
of order n say. Suppose t h a t n a u n i t in R and U ( R ) n Ro # 6 f o r every 0
~
F G . Then R i s projective
be such t h a t N i s a n Re-direct s u m a n d of
Proof. Let N c M in R-mod Pick uo c U ( R ) n
R0 f o r a
Define an R-linear f : M
r e l a t i v e t o the subring Re.
--f
1 o 6 G .
M.
Re
N as follows :
where g i s an R - l i n e a r map s p l i t t i n g 0 e
-+
N
+
M.
Check t h a t f i s R-linear a n d f IN i s the i d e n t i t y of N. 12. Let R a n d G be as in the lemna in 11.Show t h a t , i f Re i s a semisimple
Artinian r i n g , then R i s semisimDle Artinian too. 13. Let R be strongly graded of type G , where G i s a f i n i t e group, and
suppose t h a t U ( R ) fl Ro # $
for all o
6 6.
Let H be a subgroup of 6
such t h a t i t s order i s a unit of R . Show t h a t , i f Fn i s an i r r e d u c i b l e
73
I. Graded Rings and Modules
R-module.
t h e n t h e r e s t r i c t e d R(H)-module R ( ~ ) Mi s c o m p l e t e l y r e d u c i b l e
of f i n i t e l e n a t h o v e r t h e r i n a R ( H ) . H i n t : A p p l y C l i f f o r d ' s theorem and Maschke's lemma. be as i n 13 and suppose moreover t h a t f o r each uo
14. L e t R and G
t h e automorphism o f Re g i v e n by X
+
u-'
X u
c
U(R)
i s an o u t e r automorphism.
Show t h a t R i s s i m p l e i f Re i s s i m p l e . 15. L e t R and G be as i n 13. P i c k an uo t U(R) F o r any
IY1
C R-mod,
: HornR
Q
e
by Q ( f )
n
Ro f o r reach a
c
G.
define :
(M, Re)
HornR (Y,R)
+
= ? where ?(m)
=
z
u? G Prove t h e f o l l o w i n g s t a t e m e n t s
uo f ( u i ? n )
i ) q i s a Z-isomorphism ii) R
Hom (R, Re) as l e f t R-modules. Re iii)I f Re i s l e f t s e l f - i n j e c t i v e t h e n R i s l e f t s e l f - i n j e c t i v e . 16. L e t R be s t r o n g l y graded by a f i n i t e group G. If R
e
i s a q u a s i F r o b e n i u s r i n g t h e n so i s R.
17. Check whether i n a l e f t g r - A r t i n i a n r i n g t h e a r e gr-maximal
graded p r i m e i d e a l s
.
18. i ) L e t A be a commutative graded r i n g o f t y p e
Z.
I f A i s g r - A r t i n i a n then A i s isomorphic t o a (unique) f i n i t e d i r e c t product o f gr- A r t i n i a n
gr-local rings.
ii)Check i)i n case A i s graded o f t y o e 6 , G a r b i t r a r y .
iii)
T r y t o e x t e n d i )i i ) t o t h e non-commutative case.
19. L e t R be a s t r o n g l y araded r i n g of t y p e G .
If E
i s an i n j e c t i v e o b j e c t i n R-gr t h e n E
for a l l u
c
G.
I f Re i s a l e f t
i n j e c t i v e module.
i s an R - i n j e c t i v e module e
N o e t h e r i a n r i n g t h a n E i s an Re-
n Ra
A. Some General Techniques in the Theory of Graded Rings
74
20. L e t R = Put :
C
Ru be an a l m o s t s t r o n g l y graded r i n o o v e r t h e group G.
oCG
= {r
D = C(Re,R)
c R / r a = a r f o r a l l a c Re} ( i t i s c a l l e d the
c e n t r a l i z e r o f Re i n R ) . L e t u C G; s i n c e R R -1 = Re t h e r e e x i s t o u xi
c R
U'
a ) If z c
c Ru-l
yi
D
c) q,(z).x
: D
D,
+
2
xi yi = 1. Prove t h a t :
i=1
z yi E D. cp,(z)
=
n
xi z yi i s a r i n g automorphism. i=l C
f o r a l l x c Ro and z : D
= xy
d) The map 'a: G e ) By d)
n C xi i=l
then
b) The map cp,
such t h a t
+
Aut(D), e(u) =
e d e f i n e s on a c t i o n o f
G where D = { z E
D 1
,p
,(z)
p ',
i s a group homomorphism.
I; on t h e r i n g
D.
Prove t h a t DG = Z(Re)
= z f o r a l l u C G }.
f ) I f we suppose t h a t R = Re.D t h e n p r o v e t h a t t h e r e e x i s t s a group homomorphi sm a : G
Aut(R), ) . ( a
+
( a z ) = a cpo(z)
f o r a l l a E Re and z E D ( i . e . a d e f i n e s an a c t i o n of G on R ) . g) I f A i s a commutative r i n g and Re i s an Azumaya A-algebra p r o v e t h a t R = Re.D
.
Moreover i f R = 0
uCG
Ru i s a s t r o n g l y graded r i n g t h e n D i s
a s t r o n g l y graded r i n g , where Do = Ru
21. L e t R =
0
uCG
R
a graded R-module.
n
D f o r a l l u E G.
be a s t r o n g l y graded r i n g , M
c
R-mod and N = 0 Nu oCG
I f cr E C. we d e f i n e t h e oroup homomorphisms:
Hom (Nu, M), g i v e n by a ( f ) = f i o where f C HornR (N,M) Re and where iu: N o + N i s t h e c a n o n i c a l i n c l u s i o n homomorphism;
a :
HornR( N, F1)
p : HomR(M,N)
+
Hom (M, Nu), Re i s the canoncial p r o j e c t i o n . +
g i v e n by p ( g ) =
nu
9 where no : N
a. Prove t h a t a and p a r e i n j e c t i v e . b. I f G i s a f i n i t e group t h e n p r o v e t h a t a and p a r e b i j e c t i v e .
+
Nu
75
I. Graded Rings and Modules
Hint.
cc
i t follows that 1 =
z xf y: icI(T)
From R T R T - l = Re f o r a l l
T
T
and 1 ( 7 ) i s a f i n i t e s e t . Ifa ( f ) = 0
T
where xi,
t h e n f(n,) -1
R -1 r e s p .
C ,R,
yi
a l l n u C Nu.
= 0 for
C G
But f(nT) =
x u- 1 , f (yyl,n
C
=
i€J(u-%)
nT 6 Nu y i e l d = 0. C o n s i d e r fu C Hom (No, Re
and y;
)
M).
Check t h a t f i s R - l i n e a r and a l s o check t h a t a ( f ) = fu.
22. L e t R be a s t r o n g l y graded r i n g o f t y p e G such t h a t Re i s an A-algebra. R~ = R B A
Prove : a. RO,(R,)~
proj.dim
= R~
Re
(Re)
e (R ) e
c proj.dim
Re
( R ) where
.
b. I f R i s a s e p a r a b l e A-algebra t h e n R i s a s e p a r a b l e A-algebra. e 23. L e t R be a s t r o n g l y graded r i n g o f t y p e G and l e t 0 be End
(R). Re Show t h a t D i s a s t r o n g l y graded r i n g o f t y p e G w i t h g r a d a t i o n g i v e n by : Du =
I f c D, f ( R T )
c
Rm
for all
T
F G }, f o r a l l
0
€ G.
34. L e t R be a graded r i n g o f t y p e G and c o n s i d e r M C R-Fr. X =
M(u), Y =
8
u€G
n
UC G
Put
P(u).
Show t h a t X and Y a r e G - i n v a r i a n t o b j e c t s o f R-gr. 25. I f M
c
R-gr i s G - i n v a r i a n t .
of M i n R-gr,
show t h a t Eg(M),
the injective h u l l
i s G-invariant.
26. I f R i s a graded r i n g o f t y p e G and l e f t g r - A r t i n i a n t h e n J g ( R ) c J ( R ) .
If G = 27. L e t
z then
Jg(R) = J ( R ) .
R be a graded r i n g o f t y p e G and P C R-gr a f i n i t e l y g e n e r a t e d
P r o j e c t i v e module. The t r a c e i d e a l ideal.
o f P, T ( P ) =
z
Imf
f (HornR( P ,R)
,
i s a graded
o
76
A. Some General Techniques in the Theory of Graded Rings
28. G i s a u . p . group i f f o r any two nonempty f i n i t e s u b s e t s E , F o f G , element i n E.F
t h e r e e x i s t s a t l e a s t one
i n t h e f o r m xy w i t h x 6 E, y c
F.
which i s u n i q u e l y p r e s e n t e d
A l l o r d e r e d groups a r e
b u t no+ a l l u . p . groups can be o r d e r e d , c f . [ 8 8 1 a. Show
U.P.
aroups
.
t h a t a m i n i m a l p r i m e i d e a l o f a G-graded r i n p R i s homogeneous
i f G i s a u.p. group. b. L e t A be a r i n g w i t h u n i t , H a normal subgroup o f a group G such t h a t G/H i s a u.p.
group show t h a t r a d (A [ G I ) = ( r a d ( A [ G I ) !l A [HI ) A [GI.
c . L e t R be graded by t h e
U.D.
group G
.
L e t I be an i d e a l o f R and
a F I an e l e m e n t such t h a t i t s d e c o m p o s i t i o n a = a has m i n i m a l l e n g t h . I f b a then b a Oi
“j
c = 0 f o r some 1
5
01
+
j 5 n, b
.. . + c
Ro,
a
On
# 0
c C RT,
c = 0 f o r a l l 1 c i 5 n.
d. The l o c a l l y n i l p o t e n t r a d i c a l o f a r i n g R, i s a graded i d e a l , c f .
graded by a u.p. group G
[561.
e. L e t A be a r i n a w i t h u n i t y and H a c e n t r a l subaroup o f a arouo G such t h a t G/H i s a u.p.
group.
Then L(R [ H I ) . R [ G I = L ( R [ GI), where L denotes t h e l o c a l l y n i l p o t e n t radical . f.
L e t G be a u.p.
group. L e t A be an U-semisimple r i n a where U denotes
t h e upper n i l r a d i c a l . Then e v e r y c r o s s e d p r o d u c t o f A by G i s U-semisimple. 29. If G i s a u.p.
group t h e n i n e v e r y E . F
where E and
F
a r e nonempty
f i n i t e s u b s e t s o f G one o f which i s n o t a s i n g l e t o n , t h e r e a r e a t l e a s t two elements u n i q u e l y p r e s e n t e d i n a f o r m x . y w i t h x C E, y C
F.
cf.
A. S t r o j n o w s k i , A n o t e on u.p. groups, Comm. i n A l g e b r a 8(3), 1980, 231-234.
77
1. Graded Rings and Modules
Chapter I may be c o n s i d e r e d as an e x t e n s i v e w o r k - o u t o f p a r t o f t h e c o n t e n t s of [831. C o n s i d e r i n g a r b i t r a r y g r a d a t i o n s o f t y p e G we p r e s e n t a more g e n e r a l t h e o r y h e r e . L e t us r e f e r t o some b a s i c papers by E . C . by t h e a u t h o r s , [ 821
Dade, [ 241
, 1 251
and a p a p e r
.
More r e s u l t s a b o u t c r o s s e d p r o d u c t s , t w i s t e d group r i n g s and group r i n g s have appeared i n papers d e a l i n g w i t h t h e t h e o r y o f group r i n g s . L e t us m e n t i o n : D. Passman, [ 881 Related
, [89 I , M.
L o r e n z [ 671
r e s u l t may be f o u n d i n J. R o s e b l a d e ' s , [ 95
and a l s o 1 7 0 1
1
.
The r e p r e s e n t a t i o n t h e o r y o f A r t i n i a n a l g e b r a s , c f . paoers by and o t h e r s , has a graded e q u i v a l e n t i . e .
.
P.
Auslander
representations o f l e f t gr.
A r t i n i a n a l g e b r a s . The l a t t e r has been bought t o an a t t e n t i o n by R. Gordon
[ 4 2 1 ; we d i d n o t go i n t o t h e s e m a t t e r s h e r e . I n t h e s e c t i o n o f Jacobson r a d i c a l s t h e i m p o r t a n c e o f f i n i t e n o r m a l i z i n g e x t e n s i o n s i s e v i d e n t . The r e s u l t s on a l m o s t s t r o n o l y graded r i n g s we i n c l u d e d i n S e c t i o n 1.8. g e n e r a l i z e s i m i l a r r e s u l t s c o n o e r n i n g n o r m a l i z i n g e x t e n s i o n s , f o r t h e l a t t e r we r e f e r t o J.C. Robson and A. H e i n i c k e [48 1 ,
and 3. B i t - D a v i d
9
I,
J.C. Robson
1 101.
Some of t h e e x e r c i s e s have been e x t r a c t e d f r o m a o a p e r by E . Jespers, J.
Krempa, E . Puczylowski,[ 5 6 1 .
II: Some General Techniques (in the Theory of Graded Rings)
11.1. G r a d a t i o n by Ordered Groups. L e t us r e c a l l some d e f i n i t i o n s and b a s i c p r o p e r t i e s o f o r d e r e d g r o u p s .
An o r d e r e d group, o r an 0 - g r o w , i s a group G t o g e t h e r w i t h a s u b s e t S ( t h e s e t o f p o s i t i v e e l e m e n t s ) o f G such t h a t t h e f o l l o w i n g c o n d i t i o n s are being f u l f i l l e d : OG1
. e f S
OG2
. . .
OG3 OG4
I f a C G t h e n e i t h e r a F S, a = e, o r a-'
C S.
I f a, b F S t h e n ab F S. F o r any a C G, a S a - l c S. A c t u a l l y i t i s e a s i l y
verified that
t h i s i s e q u i v a l e n t t o as a - l = S f o r a 6 G. W r i t e b < a i f b-'
a E S. By OG4
t h i s i s e q u i v a l e n t t o ab-'
c
S. From
OGg i t f o l l o w s t h a t < i s a t r a n s i t i v e r e l a t i o n OG, and O G p i m o l y t h a t
f o r any aib
C G we have e i t h e r b < a,
c F G t h e n ( c b ) - l ( c a ) = b-'a bc
C S,
b = a, o r a < b. I f b < a and
hence c b < ca and i n a s i m i l a r way
a c f o l l o w s . Combining t h i s w i t h t h e t r a n s i t i v i t y o f < we f i n d t h a t
i
b < a and a < c e n t a i l s b d < ac. I f b < a t h e n a b - l c S hence (a-')-' and t h u s a-'
b-'CS
< b-'.
Conversely, i f t h e elements o f G a r e l i n e a r y o r d e r e d w i t h r e s p e c t t o a relation <
such t h a t b < a i m p l i e s bc < a c
f o r a l l c C G, t h e n
S = { x C G, e < x } s a t i s f i e s OG1-4.
11.1.1. Examples 1. Any t o r s i o n - f r e e n i l p o t e n t group i s o r d e r e d . 2. Every f r e e group i s an o r d e r e d group. F o r t h e s e examples c f . [88
I , Lemma 1.6. and C o r o l l a r y 2.8.
I n t h e sequel of t h i s s e c t i o n we suppose
t h r o u g h o u t t h a t G i s an 0-group.
II. Some General Techniques
79
Let R be a graded r i n g of type G . An M 6 R-qr i s s a i d t o be l e f t - l i m i t e d j r e s p . r i g h t l i m i t e d ) i f t h e r e i s a o0 C G such t h a t Po ( r e s p . u > u0 ) . I f Mu graded
=
0 for all
u < o0
0 f o r each u < e then M i s s a i d t o be D o s i t i v e l y
=
and i f M,, = 0 f o r a l l u
P
e then M i s n e g a t i v e l y graded. I t i s
c l e a r t h a t a s t r o n g l y graded r i n g of type G cannot be l e f t ( o r r i g h t ) l i m i t e d ( n o t e t h a t f i n i t e groups and groups c o n t a i n i n g t o r s i o n elements cannot be o r d e r e d ) .
.
11.1.2. Lemma
Let R be a l e f t l i m i t e d graded r i n g of type G and c o n s i d e r
M E R-gr. The following p r o p e r t i e s hold : 1. I f M i s f i n i t e l y generated then P i s l e f t l i m i t e d . 2 . I f M i s l e f t l i m i t e d then t h e r e e x i s t s a g r - f r e e R-module F which i s
and such t h a t t h e r e i s an e x a c t sequence F
l e f t limited
Y
+
+
0.
i n R-gr.
C
3. M i s f i n t e l y presented i n R-gr i f and only i f
i s f i n i t e l y presented
i n R-mod Proof. Since M i s f i n i t e l y generated t h e r e i s a f i n i t e l y generated g r - f r e e F F R-gr such t h a t F
+
M
+
0 i s an e x a c t sequence i n R-gr. Since R i s
l e f t l i m i t e d and F being f i n i t e l y g e n e r a t e d , i t follows t h a t F i s l e f t l i m i t e d and so i s M. 2 . Let u0 F
MT
=
G be such t h a t R u = 0 f o r
0 for all
T
< -c0.
u <
uo and l e t
I f x # 0 i s i n h ( M ) then deg x
corresponds a morphisms o f degree e , q X : R(ux -1 ) where ux = deg x . For any u hand zo
5 u
X
-1 y i e l d s ox
Therefore R u u - l X
= 0,
<
U
we ~ have mo -1
~
T
5 T ~ hence ,
thus R ( O ; ' ) ~
+
=
u u-' X
5
0 for all
-co F
? T ~ To .
Y , aiven by
< u 0
o T-' < u 0
G be such t h a t
x there (1) = X ,
'p
X
and on t h e o t h e r 0'
u < u T 0 0 '
80
A. Some General Techniques in the Theory of Graded Rings
Now, p u t t i n g 0
xch(M)
=
-1 ) R(ux
0
xfh(M) M
+
+
obviously a g r - f r e e
y i e l d s an e x a c t sequence i n R-gr :
(px
and the f i r s t term i n t h e sequence i s
0
and l e f t l i m i t e d graded module.
3. Obvious. 11.1.3. Lemma.
M
C
Let R
=
0
u6 G
Ro be a p o s i t i v e l y araded r i n g and l e t
R-gr be a l e f t l i m i t e d module. Then, i f we w r i t e R+ f o r the i d e a l
8 Ro of R , R+M = M i f and only i f M = 0. me Proof. I f M were non-zero then we could pick u C G such t h a t M
b u t Mu = 0 f o r a l l u
C l e a r l y F.l
00
n R+M
=
0
00
# 0
< o0.
0 , t h u s R+M # M.
Let R I I . 1 . 4 . .
be graded o f type G where G i s an 0-group.
Then : 1. A graded i d e a l P o f R i s prime i f and only i f f o r a , b f h(R) such t h a t
a R b c P i t follows t h a t a o r b i s i n P. 2 . R i s a doirain i f and only i f R has no homogeneous z e r o d i v i s o r s .
Proof. __ 1. I f P i s a prime i d e a l then c l e a r l y t h e property mentioned holds. Conversely
suppose a R b c P implies a o r b i n P i n case a , b f h ( R ) and c o n s i d e r
x R y c P for certain x, y
f R. We may decompose
x
=
x
+
...
+ x
un ' Then x R y c P + ... + yTm w i t h u1 < ... < u and T < . . _ c T T1 n 1 m' y i e l d s xun R yTm c P , thus xan o r yTm i s i n P . Repeatina t h i s argument
y
=
01
y
leads t o t h e s t a t e m e n t 1. 2 . One i m p l i c a t i o n i s obvious and t h e o t h e r follows
ordered decompositions.
again by i n t r o d u c i n g
II. Some General Techniques
Consider a graded d i v i s i o n r i n g p o s s i b l e , f o r o r d e r e d G,
D
o f type G
e
5
5
__ Proof.
...
= ( oI,...,~,,)
c Gn as d e s c r i b e d
c Tn
< o
Proposition.
D
C Gn
such t h a t
Mn ( D ) ( G ) as graded r i n g s .
such t h a t Mn(D)(G)
Let
e l e m e n t o f G f o r which
t h e r e corresponds a
C Gn
( 2 ) and use t h e o r d e r i n g on G .
Apply C o r o l l a r y 1.5.5.
11.1.6.
0
be t h e s m a l l e s t p o s i t i v e
# 0. Then t o any 0
Do
I t i s now
With these n o t a t i o n s :
Lemma. L e t u
11.1.5.
as i n S e c t i o n 1.4.
t o g i v e a b e t t e r d e s c r i p t i o n o f t h e p o s s i b l e G-
g r a d a t i o n s on Mn(D) o f t h e t y p e Mn(D)(G), i n Section 1.5..
81
be a graded d i v i s i o n r i n g o f t y p e Z and l e t
1 = o . The number o f graded isomorahism c l a s s e s o f t h e Mn(D)(d),
4
c Zn
is finite.
~
be t h e s e t { d < Zn, 0 5 dl 5 n,l introduce the f o lowing equivalence r e l a t i o n : Let X
Proof.
...
e x i s t t, ql,
d.
1
be
d
+ t
=
lq.
+
qn C
d 3(i),
Z and
u a p e r m u t a t i o n on
...,
i = 1,
...
-
5
d
c 1 }. On Xn
,
3 ,
a a'
i f and o n l y i f t h e r e
...,
t 1,
n . L e t Cn(D) be Xn,l/-
n } such t h a t and l e t cn(D)
t h e c a r d i n a l i t y o f Cn(D), w h i c h i s f i n i t e s i n c e 1 i s f i n i t e . C l e a r l y
- d'
y i e l d s Mn(D)(d) 2 M n ( D ) ( a ' ) . Conversely i f t h e isomorphism h o l d s
c o n s i d e r V c D-gr, V = D(-dl) Then ENDDV = ENDOW.
8
. . . 8 D(-dn),W
By C o r o l l a r y I . 4 . 3 . ,
D
W = D ( - d i ) 8 . . . 8 D(-dA).
6 D-ar,
i s a p r i n c i p a l i d e a l domain
and t h i s e n t a i l s t h e e x i s t e n c e o f an isomorphism o f degree z e r o : a : D and an a-isomorphism o f degree z e r o : f : V(-p) I t follows t h a t
D(-dl-p) Choose
8
. ..
+
W f o r some i n t e a e r p.
t h e r e e x i s t s an a-isomorphism: 8 D(-d,
- p) 2 D(-di) 8
c Zn such t h a t F p
=
'i;T +
.. . 8 D(-d;).
dTand
w i t h dy
1 f o r c = 1,
i
...
n.
Then f o r any m C Z t h e r e i s a D-isomorphism o f degree z e r o : D 2 D ( - l m ) . The e x i s t e n c e o f f t h e n e n t a i l s t h e e x i s t e n c e
o f an a-isomorphism o f
+
0,
82
A. Some General Techniques in the Theory of Graded Rings
degree zero : D(-dy)
@. ..@
Because
D1 = D2=
d!’ = d ’
u(i)
1
11.1.7. Let
@...aD(-dA).
‘Z D ( - d i )
D(-d;)
. ..
=
D1-l
=
0, t h e r e e x i s t s a
u F Sn such t h a t
f o r a l l i, 1 5 i c n.
Remark
D be a graded d i v i s i o n r i n g o f t y p e Z such t h a t 1
= 1, then Cn(D) = 1.
I f 1 = 2 t h e n Cn(D) = 1 + [ ! !1 .
2
11.1.8. P r o p o s i t i o n .
L e t R be a Graded r i n g o f t y p e G where G i s an 0-group,
then Z(R) i s graded o f type G. Proof.
l e t z F Z(R) and decompose z as z
For x E
Ro we have zx
...
u u and u o1 c n
Now
U ~ U
z
C Z(R) f o l l o w s .
z
On ai
= xz hence
z
. .. <
x 01
u1
+ ... + z
+ ... + z
Cm
“n
...
w i t h u1 <
x = x z
01
t
...
t
< un ’
x z
un
t h e r e f o r e z x = x z . Consequently ou n’ % an
Repeatina t h i s argument
we f i n a l l y end up showing t h a t
i s i n Z ( R ) f o r a l l i = 1,...,n , i . e . Z(R) i s graded o f t y p e G.
11.1.9. Remark.
I f G i s a b e l i a n then t h e r e s u l t o f 11.1.8.
i f G i s n o t ordered !
.
i s true
even
83
II. Some General Techniques
1 1 . 2 . Internal Homogenization.
Let G be an 0-group t h r o u g h o u t the section; R i s a araded ring of type G . Consider M c R-gr and l e t X c written in a unique way as x
,
f", resp. , X
be an R-submodule. An x C X may be
+ . . . + x on
01 f o r the submodule of
r
a l l x c X, i.e.
for
F
with u1
.. . <
<
u
n'
Write
, resp. x 01'
generated by x
i s generated by the homogeneous components
of highest degree appearing i n elements of X. With these notations : 11.2.1. Lemma. 1.
r and ,X
are graded submodules of M.
2 . We have t h a t X = 3 . If X c Y
c
-M
f" i f and only i f X i s a araded submodule of M .
then
r
and ,X
c
,Y
c
.
4. We have X = 0 i f and only i f f"
=
5 . I f L i s a l e f t ideal of R and
a n R-submodule of M then L"N,
a n d L-N-
c (LN)_.
0, i f and only i f X,
=
0.
Therefore i f L i s a n ideal then L" a n d L,
c (LN),
are
ideals. ___ Proof. The a s s e r t i o n s 1-4 are easy
5. Consider %
F
h(L-),
enough t o prove.
'j; c h ( r ) a n d l e t a F L , x F N be such t h a t
--,
a, x
are the components of highest degree in a , x resp.
If % ? #
0 then i t i s the homopeneous component of highest
degree of a x c L N hence
If M
C
R-gr and X c
'j;
c LN hence % 'i( c (L N),.
i s an R-submodule
the submodule of M generated by X
then ( X )
g
will denote
n h ( M ) . One e a s i l y v e r i f i e s t h a t
( X ) g i s maximal amongst submodules of o f M. Let us
.
X
which a r e graded submodules
write Spec,(-) f o r the subset of Spec(-),
of graded prime ideals of the ring under consideration.
consisting
A. Some General Techniques in the Theory of Graded Rings
84
Lemma. L e t G be an 0-group and l e t R be a graded r i n g o f t y p e
11.2.2.
G. Then t h e f o l l o w i n g p r o p e r t i e s h o l d :
1. I f I i s an i d e a l o f R t h e n ( I )
i s a graded i d e a l o f R.
9
2 . I f P i s a graded i d e a l t h e n P F Spec (R) i f and o n l y i f f o r a, b F h(R) 9 such t h a t a R b c P, a F P o r b 6 P f o l l o w s
3. I f P C Spec(R) t h e n ( P f
6 Spec R. g 9 4. I f P F Spec(R) i s m i n i m a l t h e n (P),
5 . I f I i s a graded i d e a l then r a d
I
0 o r P F Spec ( R ) . 9 = rl { P, P F Soec(R), P =
7
I 1 is
a g a i n a graded i d e a l . Proof. 1. Obvious. 2. P r o p o s i t i o n I I . 1 4 . , 1. 3, 4 , 5. F o l l o w i m m e d i a t e l y f r o m 2. F o r M C R-gr we i n t r o d u c e M+ =
0
o?e
Mo,
M- =
8 Mo.
oce
I t i s c l e a r t h a t R+ and R- a r e graded s u b r i n a s o f R and t h a t
i n R+-gr,
(M/N)'
have : 11.2.3.
If N,
M- i s i n R--gr. = M+/N+
M
6 R-gr a r e such t h a t
'P i s
N c M t h e n we
= M-/N-.
and (M/N)-
Proposition.
L e t R be a graded r i n g o f t y p e Z
.
Consider R-submodules X c Y
c M
f o r some M F R-gr. The f o l l o w i n g s t a t e m e n t s a r e e q u i v a l e n t : 1.
x
= Y.
2. X"
=
and X
n M-
3. ,X
= Y-
and X
n
Proof. o1 c
1
...
u
=
Mt =
Y fl M-
Y
P M'.
2 . Take y c Y and decompose i t as y = y
. Ifom 5 e t h e n y c M- and t h u s y
F X.
01
+ . ..
t
yOmw i t h
I f om > e t h e n
II. Some General Techniques
f'=X
- y i e l d s t h a t t h e r e exists x c X such t h a t x
with
<
... <
T
n-l
85 =
x
Tl
t
... + x
C l e a r l y , a f t e r a f i n i t e number (x' +
... +
of
s t e p s we f i n d xl,...,xk
x k ) c Y n M-
=
C
urn.
X
X n M- and t h e r e f o r e y F X .
Equivalence of 1 and 3 i s e s t a b l i s h e d i n a s i m i l a r way. 11.2.4.
X
=
C o r o l l a r y . I f M i s l e f t l i m i t e d ( r e s o . r i g h t l i m i t e d ) then
Y i f and only i f
f'
=
t y
I t follows t h a t y - x c Y has a homogeneous
c urn.
decomposition i n which t h e l a r g e s t degree aDpearing i s l e s s than
such t h a t y -
Tn-l
Y" ( n o t e : we a l r e a d y assumed X
c Y).
86
A. Some General Techniques in the Theory of Graded Rings
11.3. Chain C o n d i t i o n s f o r Graded Modules. L e t R be a graded r i n g o f t y p e G; M F R-gr i s s a i d t o be l e f t g r N o e t h e r i a n , r e s p . g r - A r t i n i a n (see s e c t i o n 1.7.), ascending,
i f M satisfies the
r e s p . descending, c h a i n c o n d i t i o n f o r graded submodules o f
M. I t i s s t r a i g h t f o r w a r d t o v e r i f y t h a t M i s l e f t g r - N o e t h e r i a n i f and o n l y i f each non-empty f a m i l y o f graded submodules o f C has a maximal e l e m e n t . D u a l l y , F1 i s l e f t g r - A r t i n i a n i f and o n l y i f each non-empty
M
f a m i l y o f graded submodules o f
has a m i n i m a l element, o r i f
and o n l y
i f each i n t e r s e c t i o n o f graded submodules may be reduced t o a f i n i t e intersection.
I I . 3 . 1 . . Let R be l e f t l i m i t e d . Then M i s and o n l y if
be a graded r i n g o f t y p e Z. L e t M F R-gr l e f t gr-Noetherian,
resp. g r - A r t i n i a n ,
if
i s l e f t N o e t h e r i a n , r e s p . l e f t A r t i n i a n , i n R-mod.
Proof.
A p p l y C o r o l l a r y 11.2.4..
11.3.2.
Lemma. L e t R be a graded r i n g o f t y p e G. I f M F R-gr i s l e f t
g r - N o e t h e r i a n ( g r - A r t i n i a n ) t h e n Mo i s l e f t N o e t h e r i a n ( A r t i n i a n ) i n Re-mod, f o r a l l u F G. Proof.
I f N c Mu
i s an R -submodule t h e n N = RN e
a s c e n d i n g c h a i n N1 c N2 c
...
c Nn c
corresponds an a s c e n d i n g c h a i n RN1 c
..., ...
n Mu.
of submodules o f Mo t h e r e c RNn c
...,
o f graded
submodules o f M. S i n c e M i s l e f t g r - N o e t h e r i a n RNn = RNn+l= some n, f o l l o w s . Thus Nn = Nn+l The p r o o f 11.3.3.
=
...
for
... .
i n t h e A r t i n i a n case i s s i m i l a r .
Corollary.
L e t R be graded by a f i n i t e group G. I f M C R-gr
i s l e f t gr-Noetherian ( A r t i n i a n ) then i n R-mod.
To an
i s l e f t Noetherian, ( A r t i n i a n ) ,
87
11. Some General Techniques
proof. 0
By t h e lemma, Ma i s a Noetherian Re-module. Since
u
i s finite,
M i s Noetherian i n Re-mod, hence i n R-mod.
oCG
CJ
The same proof works i n the A r t i n i a n case. 1 1 . 3 . 4 . P r o p o s i t i o n . Let R be a graded r i n g of t y o e Z . If M C R-gr i s
l e f t gr-Noetherian, then : 1. Mt i s l e f t Noetherian i n Rt-mod. 2. M- i s l e f t Noetherian i n R--mod.
Conversely, i f M+, M- a r e l e f t Noetherian i n Rt-mod, r e s p . R--mod, then M i s l e f t gr.-Noetherian. Proof. 1. By P o r p o s i t i o n 11.3.1. i t i s s u f f i c i e n t t o show t h a t M+ i s l e f t
gr-Noetherian i n R+-gr. Let N be a graded Rt-submodule of
RN i s f i n i t e l y generated a s R N = Rx
01
t
... +
P u t o = max.
{ of,
Rx
a; graded
with x
“k
ui
T ?
0.
Since
R-module we assume t h a t
c h(N), i
i = 1,...k } and l e t y,
= 1,
1,... , k .
c h(N)Tbe such t h a t
There e x i s t X1 ,..., Xk C h ( R ) such t h a t y, Since
M’.
=
X x
1 01
+...+
Xk x
f o r a l l i i t follows t h a t Xi F R+ f o r i = 1,
T
3
9.
ak’
...k.
Apply Lemma 1 1 . 3 . 2 . and d e r i v e t h a t 8 M i s f i n i t e l y generated ezyzo Y i n Re-mod, say by t y l , . . ,y,}.
.
Since N = (
8
i t is clear that I x
x , Y1. ..Y,} Yl” ’ uk t h e r e f o r e M+ i s a l e f t gr-Noetherian R-module.
N ) 8 ( 8 N,)
ezyzo g e n e r a t e s N over R’;
e 0
”
2. S i m i l a r t o 1. Now assume M’,
M- t o be l e f t Noetherian i n R+-mod, r e s p . R--rnod and
consider and ascending chain, N1
i
N2 c
... c
N
c
...,
P R-submodules of M. From t h i s we g e t ascending chains :
o f graded
A. Some General Techniques in the Theory of Graded Rings
88
N;
N;
c
...
c
c
NP
...
c
c
The assumptions amount t o NL = t h e n by P r o p o s i t i o n I I . 2 . 3 . , 11.3.5.
M-
Ni+l=
Nk = Nk+l
...
-
...
=
-
and Nk = Nk+l
=
...
f o r some k ;
follows
Theorem.
L e t R be a graded r i n g o f t y p e Z, l e t M are equivalent
c
R-gr.
The f o l l o w i n q a s s e r t i o n s
:
1. M i s l e f t g r . - N o e t h e r i a n 2.
Proof.
i s a l e f t N o e t h e r i a n R-module. Since 2
3
1 i s o b v i o u s l e t us show 1 = 2 .
Consider an a s c e n d i n g c h a i n say. By p r o p o s i t i o n 11.3.4. =
M- n
Xi+l
=
II.2.3.(2.) 11.3.6.
...
<
and
=
o f R-submodules o f t h e r e . i s an no
= Xi+l
=
...
i
>
for i
X1 c . . . c
Xp c...
such t h a t M-
n Xi
?
M
C
1. Me i s a l e f t N o e t h e r i a n Re-module 2 . M i s l e f t gr-Noetherian P r o o f . Immediate f r o m Theorem 1.3.4. ___ Lemma. L e t R be a s t r o n g l y graded r i n g o f t y p e Z .
I f Ro i s a l e f t N o e t h e r i a n r i n g t h e n R i s a l e f t N o e t h e r i a n r i n o . Proof.
From Lemma 11.3.6.
From Theorem 11.3.5.
i t follows t h a t R i s l e f t pr.-Noetherian
i t follows t h a t R i s a l e f t Noetherian r i n g .
A y o u p L: i s p o l y c y c l i c - b y - f i n i t e i f t h e r e e x i s t s a f i n i t e s e r i e s { e } = L,:
c
GI
c
... c
Gn = G o f subproups o f
=
no.
then the f o l l o w i n g p r o p e r t i e s a r e e q u i v a l e n t :
11.3.7.
,
no. By p r o p o s i t i o n
be a s t r o n g l y graded r i n g o f t y p e G and
Let R
Lemma.
c N
q+l= ... f o r a l l
i t f o l l o w s t h a t Xi
M,
F such t h a t each Gi-l
R-gr,
89
II. Some General Techniques
i s normal i n Gi 11.3.8.
and Gi/Gi-l
Theorem.
i s e i t h e r f i n i t e o f c y c l i c f o r each i.
L e t R be a s t r o n g l y graded r i n g o f t y n e G where G i s
a p o l y c y c l i c - b y - f i n i t e group. If Re i s a l e f t N o e t h e r i a n r i n g t h e n R i s a l e f t Noetherian r i n g . proof.
C o n s i d e r t h e normal s e r i e s { e ) = Go
c
5l c...
i
= G.
G
If n = 1 t h e n G = G1 i s f i n i t e o r c y c l i c and f r o m C o r o l l a r y 11.3.3.
of Lemma
11.3.7.
we deduce t h a t R i s l e f t N o e t h e r i a n . We p r o c e e d by
i n d u c t i o n on n, assuming t h a t t h e s t a t e m e n t h o l d s f o r normal s e r i e s of l e n g t h l e s s t h a n n. P u t H = Gn-l,
6' =
G/H and l e t
: G
IT
+
G ' be
t h e c a n o n i c a l homomorphism. Consider R ( G , ) = S (see i n t r o d u c t i o n of 1.1.). I f e ' i s the n e u t r a l element o f G ' then S e t = 8
R
=
R( H ) .
oCH
and R(H) a r e s t r o n g l y graded r i n g s and as R ( H ) i s l e f t (G') N o e t h e r i a n ( i n d u c t i o n ) , whereas G ' i s f i n i t e o r c y c l i c , i t f o l l o w s t h a t Both R
S i s a l e f t Noetherian ring.
11.3.9.
L e t A be a l e f t N o e t h r i a n r i n g ,
Corollary.
automorphism. Then A [ X, P r o o f . P u t S = A [ X, X-',
'D]
ip]
and A [ X , X - l , , p )
.
11.3.10.
:
P
-L
A
a ring
a r e l e f t Noetherian.
C l e a r l y S i s s t r o n g l y graded o f t y p e Z ,
thus t h e theorem a p p l i e s . S i n c e A f X , 91 = S e n t a i l s t h a t A[ X,,P]
Q
t
,
P r o p o s i t i o n 11.3.4.
i s l e f t Noetherian.
Application.
The group r i n g A[ GI o f a p o l y c y c l i c - b y - f i n i t e
group o v e r a l e f t N o e t h e r i a n r i n g A i s a l e f t N o e t h e r i a n r i n g . A s i m i l a r r e s u l t holds f o r crossed products.
11.3.11. Example. L e t R and S be any two r i n g s and l e t t h e r e be g i v e n bimodules M c R-mod-S, consider the r i n g
T
= (
N F S-mod-R.
R V )
= { ( r
m ) , r 6 R, m F M, n C N, s C S },
90
A. Some General Techniques in the Theory of Graded Rings
with t h e obvious a d d i t i o n b u t with m u l t i p l i c a t i o n defined by
r m
( n
1
r' ( n'
m'
rr'
1
s'
= ( nr' + sn'
rm' + ms' ss'
Define a Z-gradation on T by p u t t i n o T-l = (
T1 = (
O
M ) and Tn = 0
). ) , To = ( R o ) ,
i f n # -1, 0 , 1
L e f t graded i d e a l s of T a r e of the form : ( NI ,
'J"'). where I and J a r e
l e f t i d e a l s i n R and S r e s p e c t i v e l y , while M' i s an R-submodule o f M, such t h a t MJ c M', NI c N ' . W e then may formulate
N' an S-submodule o f N ,
the following s t a t e m e n t : T i s l e f t Noetherian ( A r t i n i a n ) i f and only i f R and S a r e l e f t Noetherian
( A r t i n i a n ) and M i s f i n i t e l y generated
i n R-mod, N i s f i n i t e l y generated i n S-mod. Let G be an a r b i t r a r y group, H a subgrouo of G
and t u i , i F I } a s e t
of r e p r e s e n t a t i v e s f o r t h e r i g h t H-cosets of G. r i n g of type G , l e t M F R-or. For each i t Z
Let R be a graded
we p u t M(OiYH)=
0
h FH
Mhoi
I t i s c l e a r t h a t M('i>H) i s a graded R(H)-module with g r a d a t i o n given by ( M ( ' i s H ) ) h
=
Mho,
f o r a l l h F H. 1
11.3.12. Lemma. RP
Let P be a graded R(H)-submodule of M('iyH) then
n ~ ( ~ i =y P .~ ) If y F h(RP
Proof.
xk F h ( P ) , k = l . . . n .
.
n
z n Xk xk with X~ c h ( R ) , k=l = dea x k , k= 1,..., n . Then xk c ( M ( u i y H))T
Mtui'H))then y
Put
T~
=
I f we w r i t e deg y = T i n M ( u i y H )then y F M T u , . I t follows k i 1 from a l l t h i s t h a t : -cui = (deg X k ) . ~ k ~ fi o r a l l k = 1, ..., n . =
MT
-1 On the o t h e r hand deg(X ) = -c-ck f o r a l l k = 1, ...,n . k Hence Xk F R ( H ) f o r a l l k = l , . . . , n and t h e r e f o r e y RP
n M(0i.H)
C
P
i.e.
c P.
11.3.13. C o r o l l a r y . I f M i s a l e f t gr-Noetherian R-module then f o r every i c I , M ( O i Y H ) i s a l e f t gr.-Noetherian R(H)-module.
k
=
ll. Some General Techniques
91
11.3.14. C o r o l l a r z . IfH i s of f i n i t e i n d e x i n G t h e n t h e f o l l o w i n s s t a t e m e n t s are equivalent : 1. M i s a l e f t g r . - N o e t h e r i a n R-module
2. ~
(
proof.
~ i s il e f~t g r~- N o )e t h e r i a n i n R(H)-mod f o r e v e r y i c 1 . 1
-
2 i s j u s t 11.3.13.
2 = 1. S i n c e I i s f i n i t e and s i n c e MZ @.M('iyH) i 61
a l e f t g r . - N o e t h e r i a n R(H)-rnodule b u t t h e n
P
i t follows that M i s
i s c e r t a i n l y l e f t gr.-
N o e t h e r i a n i n R-mod.
11.3.15. C o r o l l a r y . R(H)
I f R i s a l e f t gr-Noetherian r i n g then the r i n g
i s l e f t gr.-Noetherian.
A. Some General Techniques in the Theory of Graded Rings
92
11.4. I n v e r t i b l e I d e a l s . The Rees R i n o and t h e G e n e r a l i z e d Rees Rina. L e t R be any r i n g , I an i d e a l o f R. The Rees r i n o t h e r i n g R(1) = R
+
+ ...
IX
t
In Xn
+ ...
associated t o I i s
c R [ X I which obviously i s
2 a graded s u b r i n g o f R [ X I i s o m o r p h i c t o R 8 I @ I @ . . . @ A r i n g T i s an o v e r r i n g o f R
In Q
...
i f R c T and lT = lR. An i d e a l I o f R i s
an i n v e r t i b l e i d e a l i f t h e r e i s an o v e r r i n g T o f R c o n t a i n i n g an R-bimodule J such t h a t J I = I J =
R.
11.4.1.
R
Example.
Let
be a l e f t N o e t h e r i a n p r i m e r i n g and l e t x E R
be a n o r m a l i z i n g e l e m e n t ( i . e . x R = R x ) . By G o l d i e ' s theorems t h e i d e a l I = Rx
xR i s a n i n v e r t i b l e
idea
:
o f R w i t h inverse x-lR = R x
-1
i n t h e r i n g o f q u o t i e n t s o f R. I f I i s an i n v e r t i b l e i d e a l o f R ( i n t h e o v e r r i n ? T) t h e n we can d e f i n e t h e g e n e r a l i z e d Rees r i n g o f t y p e Z, graded s u b r i n g o f T[ X,X-.'].
t o be # ( I ) = 8 In Xn w h i c h i s a ncZ
Note t h a t , w i t h n o t a t i o n s as b e f o r e , # ( I ) + = R ( 1 ) . 11.4.2.
Proposition.
If R
i s l e f t N o e t h e r i a n and t h e i d e a l I o f R i s
i n v e r t i b l e t h e n # ( I ) and R ( 1 ) a r e l e f t N o e t h e r i a n r i n a s . Proof.
C l e a r l y # ( I ) i s s t r o n g l y graded o f t y p e Z
# ( I ) i s l e f t N o e t h e r i a n and by P r o p o s i t i o n 11.3.4. L e t I be an i d e a l o f R. On an
(*): M =
M0
3
M
1
2
. . _3
Mn
2
M c
...
,
t h u s by Lemma I I . 3 . 7 . ,
R ( I ) i s l e f t Noetherian.
R-mod we c o n s i d e r t h e I - a d i c f i l t r a t i o n ; (More a b o u t f i l t r a t i o n s i s i n t h e
f i n a l appendix i n t h i s book). Note t h a t t h i s f i l t r a t i o n has t h e p r o p e r t y
IMn c Mn+l,
f o r a l l n . The module R(M) = Po @
... Q Mn
Q.. i s called
t h e Rees module a s s o c i a t e d t o P and t h e f i l t r a t i o n ( * ) . I t i s c l e a r t h a t R(M) i s a graded R(1)-module.
The f i l t r a t i o n (*) i s s a i d t o be I - f i t t i n g i f t h e r e i s a
D 6
N
such t h a t
II. Some General Techniques
M I,
93
f o r a l l n 2 p. As an example one may consider t h e f i l t r a t i o n
= Mn+l
...
...
n Mn = I M f o r a l l n 3 0. I f M = M0
J
f i l t r a t i o n then f o r any submodule
N o f M we may d e f i n e an I - f i t t i n p
MI
f i l t r a t i o n on M/N by p u t t i n g (M/N),
3
3Mn
3
i s an I - f i t t i n g
Mn+N/N.
=
Lemna. L e t R be a l e f t Noetherian r i n g and l e t I be an i d e a l o f
11.4.3.
R. Consider a l e f t Noetherian M 6 R-mod and equip i t w i t h the I - a d i c f i l t r a t i o n (*) M = Mo
2
MI
...
3
3
Mn
... .
3
The f o l l o w i n g statements a r e e q u i v a l e n t : 1. The f i l t r a t i o n (*) i s I - f i t t i n g .
2. R(M) i s an R(1)-module o f f i n i t e Proof. 1 = 2. I f I C n = Mn+l
type.
f o r a l l n 3 p f o r a certain p P
c l e a r l y a f i n i t e l y generated’R-module,
whence t h e statement f o l l o w s .
2 = 1. L e t R(M) be generated by elements xl,.
11.4.4.
i=l,
l e f t Noetherian. L e t
M c
Mo = M
2
Hl
. . ,xn
o f degree dl,.
} and check t h a t M+,l
=
. . ydn
IMP.
L e t R be a r i n g and I an i d e a l o f R such t h a t R ( I ) i s
Lemma.
2
...n
0 then
M as an R(1)-module. The l a t t e r i s
R(M) i s generated by Mo 8 V1 8 . . . @
resp. Put p = max { di,
?
Mn
3...3
R-mod be f i l t e r e d by the I - a d i c f i l t r a t i o n (*)
... .
If M i s f i n i t e l y generated and (*) i s I - f i t t i n g then f o r every R-sub-
module
N o f M, t h e induced f i l t r a t i o n
(*)N : N =
-
N nM
0
2
N
n
M~
3
...
3
:
N r l Cn
2
...
i s also I - f i t t i n g .
Proof. By the f o r e g o i n g lemna R(M) i s a f i n i t e l y generated R(1)-module hence a l e f t Noetherian R(1)-module.
Now R(N) i s an R(1)-submodule of
R(M) and t h e r e f o r e R(N) i s f i n i t e l y generated as an R(1)-module. Lemma 11.4.3 implies t h a t
(*)N i s I - f i t t i n g .
94
A. Some General Techniques in the Theory of Graded Rings
11.4.5. C o r o l l a r y .
(Strong Artin-Rees p r o p e r t y )
L e t R be a
ring,
I an i d e a l o f R, such t h a t R(1) i s l e f t Noetherian. L e t M be a f i n i t e l y generated R-module c o n t a i n i n g t h e R-submodules K and L , then t h e r e i s a p
c
JY
I"-P
such t h a t f o r a l l n 2 p :
n L)
(IPK
Proof. Put Mi
=
= I ~ Kn L .
i I M. From Lemma 11.4.3. we d e r i v e t h a t R(M) i s a l e f t ( I n K n L ) i s a f i n i t e l y generated @ n?O By Lemma 1 1 . 4 . 4 . t h e f i l t r a t i o n
Noetherian R(1)-module. Thus R(1)-submodule
R(M).
L, n 3 0 } i s I - f i t t i n g . The r e s u l t f o l l o w s .
{ In K fl
11.4.6.
of
Corollary.
L e t R be a l e f t Noetherian r i n g . I f I i s an
i n v e r t i b l e i d e a l o f R then I has the s t r o n g Artin-Rees p r o p e r t y .
11.4.7. Remark.
L e t R be a l e f t Noetherian r i n g and l e t I be an i d e a l
generated be c e n t r a l elements. We c l a i m t h a t R ( I ) i s a l e f t Noetherian r i n g . Indeed, suppose al,
...,
canonical homomorphism
: R [ X
Obviously
Q
cp
a
n
c Z(R) generate I , then we have a Xn]
+
R(1) d e f i n e d by o(Xi)= aiX.
i s s u r j e c t i v e , thus R ( I ) i s l e f t Noetherian and by C o r o l l a r y
11.4.5. i t f o l l o w s t h a t I has the s t r o n g Artin-Rees p r o p e r t y .
95
II. Some General Techniaues
11.5. Krull Dimension of Graded Rings. The Krull dimension of ordered s e t s has been defined by P. Gabriel and Rentschler, i n [ 3 3 1 , f o r f i n i t e ordinal numbers,
and G.Krause 59 1 . Let us
generalized the notion t o other ordinal numbers c f . recall some d e f i n i t i o n s and elementary f a c t s . Let ( E , {
5)
x t E, a
be an ordered s e t . For a , b E E we write [ a , b ] 5
x c b
} and we
p u t r(E) =
f o r the s e t
( a , b ) , a 5 b }.
{
By t r a n s f i n i t e recurrence we define on T ( E ) the following f i l t r a t i o n : = t ( a , b ) , a=b}
T-l(E)
r o ( E ) = { (a,b) t
supposing T,(E)
rP ( E )
T(E), [ a , b ] i s Artinian }
has been defined f o r a l l p
= { (a,b) t T ( E ) ,
W b 2 bl
n F N such t h a t
?
[ bi+l
ia,
... 5
bn ?
bi] t
r 0( E )
We obtain an ascending chain r-l(E)
c
ro(E)
There e x i s t s an ordinal 5 such t h a t
r
E)
I f there e x i s t s an ordinal u such t h a t have Krull dimension.
r,(E)
then
5
c
... 2
... c
= T(E)E+l
r(E)
=
a
for all i r,(E) =
r(E),
c
there i s an ?
n 1.
... .
... .
then E i s said t o
The smallest ordinal with the property t h a t
= r ( E ) will be c a l l e d the Krull dimension of
E a n d we denote
i t by K dim E . 11.5.1.
Lemma.
Let E , F be ordered s e t s and l e t :
E
+
F be a s t r i c t l y
increasing map. I f F has Krull dimension then E has Krull dimension and K dim E 5 Kdim F. ( c f . 1 43 1 ) .
1 1 . 5 . 2 . Lemma.
E x
I f E, F a r e ordered sets with Krull dimension then
F has Krull dimension and Kdim ExF = sup (Kdim E, Kdim F )
(Note t h a t E x F has the product ordering).
96
If
A. Some General Techniques in the Theory of Graded Rings
A
i s an a r b i t r a r y a b e l i a n category and M i s an o b j e c t o f
A
then we
M inA ordered by i n c l u s i o n .
consider the s e t EM o f a l l subobjects o f
If EM has K r u l l dimension then M i s s a i d t o have K r u l l dimension and
we denote
i t by KdimA M o r simply Kdim M i f no ambiguity can a r i s e . -
I n t h i s case we may reformulate t h e d e f i n i t i o n o f K r u l l dimension as f o l lows : I f M = 0; Kdim = -1; i f
a
i s an o r d i n a l and Kdim M
provided t h e r e i s no i n f i n i t e descending c h a i n : M subobjects Mi o f M i n A such t h a t f o r i = 1, An o b j e c t M o f A having Kdim M =
a
...,
4: a 3
then Kdim M =
Mo
3
M1
3
Kdim (Mi-l/Mi)
...
4:
a
of a.
i s s a i d t o be a - c r i t i c a l i f K d i m ( M / M ' ) i a
f o r every non-zero subobject M ' o f M. For example, M i s o - c r i t i c a l i f and o n l y i f M i s a simple o b j e c t o f Also, i t i s obvious t h a t any nonzero subobject
A.
o f an a - c r i t i c a l o b j e c t
i s again a - c r i t i c a l . 11.5.3.
Lemma.
I f M i s an o b j e c t o f
Kdim M = sup { Kdim(M/N),
A
Kdim N } and
and N i s a subobject o f M then e q u a l i t y holds p r o v i d e d e i t h e r
side exists. c f [43 1 .
Proof.
Using Lemna I I . 5 . 2 . ,
11.5.4.
Lemma.
Proof.
See P r o p o s i t i o n 1.3. [ 4 3 I .
11.5.5.
Lemna.
Every Noetherian o b j e c t o f
Suppose t h a t
A
d i r e c t sums. I f an o b j e c t M o f
A
has
K r u l l dimension.
i s an a b e l i a n category a l l o w i n g i n f i n i t e
A
has K r u l l dimension then M cannot
c o n t a i n an i n f i n i t e d i r e c t sum o f subobjects. I n p a r t i c u l a r M has f i n i t e Goldie dimension. 11.5.6.
Lemna. Suppose t h a t A i s an a b e l i a n category
d i r e c t sums and suppose the o b j e c t M o f
A
allowing i n f i n i t e
has K r u l l dimension.
97
II. Some General Techniques
p u t u = sup I 1 a
t.
subobject of M}. Then
N a n essential
KdimA (M/N); -
KdimM.
proof.
For I I . 5 . 5 . , 11.5.6. we r ef er t o
[
43
Let us now come back down t o earth and p u t
A
_ L
]
= R-mod.
The Krull dimension
of M i n R-mod i s c al l ed the l e f t Krull dimension of M and we denote i t The Krull dimension of R considered as a l e f t R-module i s by Kdim M. R called the l e f t dimension of R and we denote i t by Kdim R . If R i s a graded r i n g , putting
A
for t h a t p a r t i c u l ar M
C
R-gr). The purpose for M
Kdim ( M ) and Kdim (M) R R-gr 11.5.7. Lemma.
R-gr we define Kdim M
=
C
R-qr
07
C
t h i s paragraph i s t o compare
R-gr.
Let R be a graded ring of type
(or r i g h t ) limited M
as before ( i f i t e x i s t s
z and
consider a l e f t
R-gr, t h e n :
1. M has Krull dimension i f and only i f !v! has Krull dimension and in
t h i s case we have : Kdim M = Kdim M . R-gr R 7 . If M i s a-crit'ical then i s a-critical. Proof. 1. Follows from Corollary 11.2.4. 2. If
X#
Kdim R
(F/X)5
0 i s any submodule of M then Kdim ( M / f " ) < a . R-gr BY proposition 11.2.3. and Corollary 11.2.4. and Lemma 11.5.1. we have
11.5.8.
Kdim ( M / f " ) . R-gr
Consequently
,
is a-critical.
Proposition.
Let R be a p r o s i t i v el y graded ring of type Z . For M F gr the following Properties hold : 1. M has Krull dimension i f and only i f M has Krull dimension and i f t h i s
i s the case then Kdim M = Kdim M. R-gr R
A. Some General Techniques in the Theory of Graded Rings
98
2. I f M i s a - c r i t i c a l then Proof.
i s a-critical.
For u E S, the p o s i t i v e elements o f G, we d e f i n e uM,
=
8 .M, T>U
i s a graded submodule o f M and M,/,
I f i s c l e a r t h a t, ,M
is right
l i m i t e d . From t h e f o r e q o i n q lemma we r e t a i n : Kdim M R-qr u' and a l s o Kdim (M/M>u) R-gr
= Kdim
R
= Kdim M (,),
R -
(M/M > u). --
A p p l i c a t i o n o f Lemma 11.5.3. y i e l d s the statement. 2. Consider a nonzero submodule
Combining t h e f a c t t h a t
M
X o f E . For
# 0.
some u E G,X fl, ,M
i s a - c r i t i c a l w i t h t h e r e s u l t of t h e foregoing
fl Mu) < a. Now t h e r e e x i s t s a s t r i c t l y lemma we deduce t h a t Kdim X M (/,, R-gr i n c r e a s i n g map from t h e l a t t i c e o f submodules o f i n t o t h e set-product
y/X
o f the l a t t i c e s o f submodules o f JM , X fl, ,M -~ By Lemma I I . 5 . 2 . ,
and because Kdim (M/M,u R --
Kdim (M/Z) < a; hence
R -
11.5.9.
Lemma.
and o f MM / u,
-
, g i v e n by
) < a, i t f o l l o w s t h a t
i s a-critical.
L e t R be a l e f t Noetherian graded r i n g o f type Z , and
l e t M 6 R-gr be a u n i f o r m o b j e c t . Suppose t h a t e i t h e r M i s l i m i t e d ( l e f t o r r i g h t ) o r t h a t P i s p o s i t i v e l y graded. Then
i s uniform i n
R-mod. Proof.
Theorem 2. 7. o f [ 431 s t a t e s t h a t an o b j e c t from any a b e l i a n
category, having K r u l l dimension a, c o n t a i n s a nonzero subobject which i s a-critical.
Since R i s l e f t Noetherian, we may apply Lemma 11.5.4.
and t h e f o r e g o i n g remark t o deduce t h a t M c o n t a i n s a graded Rsubmodule
N which i s a - c r i t i c a l . U n i f o r m i t y o f M e n t a i l s t h a t M i s an e s s e n t i a l extension o f N i n F g r . then
i s an e s s e n t i a l extension o f
Lemma 11.5. 7. and P r o p o s i t i o n
11.5.8.
yield that
(see 1.2.8.)
i s anacritical
99
II. Some General Techniques
i s u n i f o r m too.
R-module, hence i t i s u n i f o r m and t h e r e f o r e 11.5.10.
Theorem. L e t R be
R-gr then
uniform o b j e c t o f
proof.
M -x c -
M, -u
For u E G l e t
such t h a t f o r some
n M,IJ)
Indeed i f x E h(Xwith x = y
and
a graded r i n g o f type Z . I f M E P-gr i s a
u1 <
M
i s a u n i f o r m R-module.
stand f o r IJ
... urn. Since
that X
n , ,M- #
0
T 3
so we may use Lemma 11.5.9.
R-module. Consequently
1! n uM,-
= 0. Then X- n , ,M = 0. then t h e r e i s a y E X such t h a t y = y, +...+yo
6 G we have
u1 hence y = 0 thus (Xw)T = 0 f o r a l l
i n R-gr,
# 0 and , X
u. Now , X
and deduce t h a t , X
-
+
R -module. I n view o f Lemma
i t w i l l be s u f f i c i e n t t o e s t a b l i s h
Rx
Uniformity o f M y i e l d s R x
0.
n
Ry
n M, # -u
for L p E
n
u n i f o r m i t y o f M+ i n R+-gr.
+ R -gr.
IJ,
n
Ry
M+
11.5.11.
+ o y i e l d -x n -Y n Corollary.
T
n MzIJ) and w r i t e z
Thus z E R+x
n
and
M+# 0 ; hence -
5n1
I f R i s as i n 11.5.10.
=
kx = py
Rty and t h e r e f o r e M+
Finally, i f X, Y a r e nonzero submodules of M then
-Y n
deg y =
R y # 0 and by o u r assumptions
0. P i c k z # 0 i n h(Rx
h(R); n o t e X,p E R'.
i s uniform i n
i s a uniform
0 f o r any nonzero submodule X o f M and f o r a l l u E G .
P i c k y, x E h(M ) , nonzero elements w i t h deg x = 5
i s uniform
i s a u n i f o r m R-module. L e t us suppose now
t
7
1 y E M,>,-
5 u1 i t f o l l o w s t h a t
We proceed t o show t h a t Mt i s a u n i f o r m 11.5.9.
Consider a nonzero submodule
.M,
Q
T? U
X -
n Ft #
0 and
p 0.
i s moreover l e f t Noetherian
then the G o l d i e dimension o f M i n R-gr equals t h e G o l d i e dimension of
-M
i n R-mod.
Proof. 11.5.12.
D i r e c t f r o m t h e theorem and Lemna 1.2.8. Lemma.If R i s as i n 11.5.10,
N
and
module t h a t N has a composition s e r i e s N
3
E R-gr i s a l e f t Noetherian
N1
2
...
3
Nn = 0,
m
100
A. Some General Techniques in the Theory of Graded Rings
where N i - l / N i
i s a c r i t i c a l module
Proof.
I
As i n t h e ungraded c a s e , c f . [43
11.5.13. Lemma.
5
i
5
n
.
Let R be a s i n 11.5.10. and l e t M 6 R-gr have Krull
dimension a . Then Kdim M Re Proof.
f o r each 1
c
f o r every o 6 G .
a
For any submodule N o of Mo we have t h a t No
=
R N O n Mo;
then
apply Lemma 11.5.1. 11.5.14. Let ~ _ _Lemma. _ _
R be a y a d e d r i n q of t y p e G . Consider an a - c r i t i c a l
M 6 R-gr then Kdim M+ 5 a + 1 and Kdim P- 5 a + 1. R+ RProof. Pick x # 0 in h ( M + ) . W e i n t e n d t o show by t r a n s f i n i t e
induction
on a t h a t Kdim (M+/R+x) 5 a . If a = 0 then M i s a simple o b j e c t R-gr R+ and i n t h i s case t h e a s s e r t i o n follows from t h e s t r u c t u r e of simple o b j e c t s (proof i n s e c t i o n 1 1 . 6 . ) . I f a # 0 then Rx # 0 y i e l d s Kdim (M/rx) 6 a and the i n d u c t i o n hypothesis, combined w i t h Lemma R-gr + I I . 5 . 3 . , Lemma I I . 5 . 1 2 . , implies t h a t Kdim(M/Rx) 5 a .
R+
Since (M/Rx)+ = M+/(Rx)+ and R+x c (Rx)'
o
+
(RX)+/R+X
If degx
= T
II.5.13.,
+
M+/R+~ + M+/(R~)+
then (Rx)+/R+x
(
0
yc[T-l,e]
i t follows t h a t Kdim ( R x ) + / R + x R+
we f i n d t h e e x a c t sequence :
+
o
.
R x)/Rex a s Re-modules. By Lemma 5 a.
Then Lema 11.5.6. a p p l i e d t o
the above e x a c t sequence y i e l d s : Kdim M+ c a + 1. R+ Kdim M- 5 a + 1 f o l l o w s . R 11.5.15. C o r o l l a r y .
In a s i m i l a r way,
I f a graded R-module N i s l e f t Noetherian and such
a + 1, Kdim M- 5 a + 1. t h a t Kdim M = a , then : Kdiy M+ R-gr R RProof. D i r e c t l y form the lemma, Lemma 11.5.12. and Lemma 11.5.3.
11.5.16. Theorem.
Let R be a s before and l e t M be l e f t
R-gr with Kdim M = a , t h e n a c Kdim R-gr R
C a
+ 1.
Noetherian i n
101
11. Some General Techniques
proof. Clearly
a 5
we o b t a i n :
Kdim R
F.
By Lemma 11.5.2.
and
Proposition 11.2.3.
sup( Kdim M, Kdic F1-). R-gr R Then the above c o r o l 1a r y f i n i s h e s the p r o o f . KdimR
11.5.17.
5
and a - c r i t i c a l .
1. M i s an
(a
c
L e t R be as b e f o r e and l e t M
Corollary.
Suppose t h a t Kdim M = R -
a
+
R - c j r be l e f t Noetherian
1 then :
+ 1 ) - c r i t i c a l R-module.
2. M+(M-) i s an ( a
+
l ) - c r i t i c a l R+-(R--)-module.
proof. L e t X # 0 be a submodule o f M. Obviously Kdim Mt = Kdim M- = R+ RIfX
n
u 6 G,
MZa
=
n M5u
0 then
we may assume t h a t X
Then Kdim R- g r
M/T
< a
+
1.
0. Considering t h a t Kdim M(u) =
M-
and Kdim MR-
By p r o p o s i t i o n 11.2.3.
M i s (a
=
n
+
a
R-gr
# 0. 5 a
a
f o r every
(see the p r o o f o f Lemma 11.5.14.).
and C o r o l l a r y 1 1 . 2 . 4 . we o b t a i n Kdim M c R -
a;
hence
1 ) - c r i t i c a l because o f t h e f o r e g o i n g r e s u l t s
IfE? C R-gr i s l e f t Noetherian and a - c r i t i c a l then one expects t h a t
Kdim M = R
a
i s a-critical.
implies t h a t
I n the commutative case t h i s i s indeed t h e case : 11.5.18.
Proposition.
L e t R be a commutative graded r i n g o f t y p e
l e t M C R-or be a - c r i t i c a l .
Then
and
i s an a - c r i t i c a l o r an ( a + l ) - c r i t i c a l
R-module. proof.
M i s a-critical
i n R-gr i f and o n l y i f t h e r e e x i s t s a graded
i d e a l I o f R such t h a t f o r any x
c h(M), x # 0, we have Ann x
.
R
= I
and
Y-dim R / I = U . L e t y c M, y # 0, then y = y +. .+ yon w i t h uc l . . . < un ' R-gr 01 with I t i s c l e a r t h a t I c Ann y. I f X F Ann y, say X = X + .. . + X R Tm
102
T~
A. Some General Techniques in the Theory of Graded Rings
<
...<
X?1 yo2 --
then X y
T ~ ,
... -- A T l
=
0 yields X
yun = 0 hence
= 0 i.e. X F I . Then 71 y,l T1 c Ann y . X R
,. . . , X,
c I and Ann y = I . Since R / I i s m R a domain, R/I i s a - c r i t i c a l o r ( a + l ) - c r i t i c a l a s an R-module i . e . is Furthermore, we o b t a i n A,
2
a - c r i t i c a l o r ( a t 1 ) - c r i t i c a l i n R-mod. Let us now focus on Krull dimension f o r s t r o n g l y graded r i n g s . We use n o t a t i o n s of 11.3.12. and subsequent r e s u l t s . 11.5.19. P r o p o s i t i o n .
Let R be a graded rino of type G , where G i s
a r b i t r a r y . I f M F R-gr has Krull dimension then f o r each
i C 7 , M(Oi 'H)
Kdim M. I f H R-yM(OiYH)}. has f i n i t e index i n G then Kdim M = s u p { Kdim R-gr ic7 R(H)-gr
has Krull dimension i n R(")-pr
Proof.
and
Kdi
RTH)-w
M(ui'H)
5
The f i r s t s t a t e m e n t follows from Lemma 11.3.12. and i t s c o r o l l a r i e s
M ( u i y H ) } 5 Kdim M . t h a t s u p {Kdi ic7 RTH)-gr R-gr On the o t h e r hand, t h e l a t t i c e of graded submodules of M maps i n t o t h e
For t h e second s t a t e m e n t , note f i r s t
product of l a t t i c e s of graded submodules of M ( ' i y H ) , N
+
ic1, a s follows :
( N ( " i y H ) ) i c l . This map i s s t r i c t l y i n c r e a s i n g and we may apply
L e n a 11.5.1. 11.5.20. Corollary.
I f the graded rino R has Krull dimension i n R-gr
then, f o r every subgroup H of f i n i t e index i n G , the r i n n R ( H ) has Krull
R = Kd'm R ( H I dimension and Kdim R-gr RIH)-gr 11.5.21. C o r o l l a r y .
I f R i s y-aded by a f i n i t e group G
i f M has Krull dimension i n R-gr then M h a s Krull dimension i n R-mod and :
Kdim R
=
Kdim M = s u p { KdimRe Mo } R-gr UE G
103
II. Some General Techniques
Proof.
_c
t o H = { e } t h e n Kdim M = R-gr
Apply ng P r o p o s i t i o n 11.5.19.
= sup { Kdim
Mo }. S i n c e R-submodules o f M a r e a l s o Re-submodules,
Re
d G
F i n i t e n e s s o f G e n t a i l s : Kdim V = sup {Kdim Flu}. Re oCCHence KdimR M 5 Kdim M. R-gr On t h e o t h e r hand i t i s c l e a r t h a t Kdim M 5 Kdim R-Gr R
M.
Lemma. L e t R be a s t r o n g l y graded r i n g o f t y p e G and l e t
11.5.22.
M
t R-gr
M = Kdim Me be l e f t N o e t h e r i a n . Then Kdim R-gr Re P r o o f . I m m e d i a t e l y f r o m Theorem 1.3.4. L e m a . L e t R be a s t r o n g l y graded r i n g o f t y p e Z
11.5.23.
N o e t h e r i a n and has K r u l l dimension, Proof.
a
I f Re i s l e f t
say, t h e n Kdim R 5 a + 1.
Because o f Lemma 1 1 . 5 . 2 2 . and Theorem 11.5.16.
11.5.24.
Theorem. L e t R be a s t r o n g l y graded r i n g o f t y p e G where G
i s a p o i y c y c l i c - b y - f i n i t e group w i t h
n o r m a l i s i n g s e r i e s {e}=Goc G1c.. .cGs=G.
Suppose Re is l e f t N o e t h e r i a n and has K r u l l d i m e n s i o n a t h e n a
<
Kdim R 5
-
P r o o f . I f s = 1 t h e n G i s f i n i t e o r c y c l i c . Then by C o r o l l a r y 11.5.21.
o r Lemma 11.5.23.
the statement follows, i.e.
Kdim R 5
a
+ 1. We proceed
by i n d u c t i o n , assuming t h a t f o r s s m a l l e r t h a n n t h e s t a t e m e n t h o l d s . Put H = Gn-l,
G ' = G/H and l e t
TT
: G
--f
G' be t h e c a n o n i c a l epimorphism.
Consider R ( G , ) and, i f we l e t e ' be t h e n e u t r a l element o f G ' t h e n ( R ( G 8 ) ) e I =R ( H ) . Kdim R(H) 11.5.26.
5 a
+
A p p l y i n g t h e i n d u c t i o n h y p o t h e s i s t o R(H) y i e l d s t h a t : s - 1, hence Kdim R 5 a
Corollary.
+
s - 1
+
s = n
+
s, f o l l o w s .
L e t A be a l e f t N o e t h e r i a n r i n g w i t h Kdim A = a .
L e t G be p o l y c y c l i c - b y - f i n i t e ,
t h e n t h e aroup r i n g A I G I ( s i m i l a r f o r
a
+
S.
A. Same General Techniquesin the Theory of Graded Rings
104
any crossed p r o d u c t o f G over A) has K r u l l dimension and u 5 Kdim A [ G I
5
a
+
where s i s as i n t h e theorem. A c t u a l l y i t i s e a s i l y seen t h a t we may i n f a c t take s = h(G) ( t h e H i r s c h number, i . e .
t h e number o f c y c l i c n o n - f i n i t e
i n t h e normal s e r i e s ) .
Gi/Gi-l
IfA i s a r i n g and cp i s an automorphism o f A then we consider t h e skew
I f F1 c R-mod then M [ X,U,]
polynomial r i n g s A I X s c p l and A [ X, X-',q].
M i n R [ X,vl - mod and s i m i l a r l y f o r M t X,X-l,cpI
for R [ X , q l
@
Put M [ X,cpli
= {
is
J.
graded
11.5.27.
R
X
i
@m, m C M } f o r i
0, then M [ X,ql
R [ X,G 1 -module. S i m i l a r l y , G[
=
8
icN
. M [ X,cpli
i s a CradedN X,X-l,,$]-modul
X.X-',,q]
Theorem. I f M i s l e f t Noetherian i n A-mod and i f KdimAM = M [ X,
Kdim A[ x,X-l,~l
x-',(~I
5 a
Theorem 11.5.16.
a
then
+ 1.
is s t r o n g l y graded o f type Z we may apply
Since A [ X, X-l,cp]
Proof.
?
stands
and Lemna 11.5.22.
Theorem. L e t M c A-mod.
11.5.28.
has K r u l l dimension if and o n l y i f M i s l e f t Noetherian and
1. M [ X,cp]
then Kdim A [ X,ql
M [ X,q]
= Kdim M
R
+
1.
2. I f M i s l e f t Noetherian and a - c r i t i c a l then M [
X,(pl
i s (at1)-critical.
Proof. N = M [ X,U,].
1. Put S = A [ X , q l
I f M i s l e f t Noetherian then N i s l e f t
Noetherian i n S-mod and 1. f o l l o w s . Conversely, assume t h a t N has K r u l l dimension i n S-mod and t h a t M i s n o t l e f t Noetherian, i . e . t h e r e e x i s t s an i n f i n i t e ascending chain Mo
c
MI c
...
c Nn c
...
Put : K
=
l @M1 + X t% M2
+... +
X
L
=
1@ M + X @MI
+...+
X
0
k
@
Mktl
k baMk
+
...
+ ...
:
105
II. Some General Techniques
ObviouSlY L c K a r e graded submodules of N and X K c L. L e t ni be t h e c a n o n i c a l map Mi Note t h a t (ao
M ~ / H f~o r- ~a 1
a, X + ...) ni(m)
t
Define 8 : K/L
+
.
-1, i=1,2.. .
Mi/!'
->
i = 1,2
= ni(q
1-i
Q M2/M1
M1/Mo
,...,
i s an S-module b y :
(ao)m), f o r e v e r y in C Mi, @
. . .,
as f o l l o w s :
i f f = 1 @ m m , t X @ m 2 +... C K , p u t
e ( f mod L ) = (nl(ml), n2(m,) ,...). I t i s s t r a i g h t f o r w a r d t o check t h a t
e
i s an S-isomorphism. S i n c e N
has K r u l l dimension, so does K/L and t h u s t h e l a t t e r has f i n i t e G o l d i e dimension ( s e e Lemma 11.5.5.)
but this contradicts the f a c t that 8 i s
an isomorphism. 2. Assume t h a t M i s a - c r i t i c a l .
S i n c e N i s a Graded S-module,
to establish that N i s (atl)-&itical
we o n l y have
i n S-gr and t h e n b o t h t h e e q u a l i t i e s
i n 1 and 2 w i l l f o l l o w f r o m t h i s . Consider t h e i n f i n i t e s t r i c t l y d e c r e a s i n o sequence : N
3
X N
where XiN/XitlN
3
X
2
N
3
...
2
Xi
N
3
...
2 N/ XN 2 M.
+
O b v i o u s l y Kdim N 2 a S
1. I f z F h(N), z F h ( N ) , z = X
k
@
m # 0.
has degree k t h e n we c o n s i d e r t h e f o l l o w i n g sequence i n S-gr :
k
(*) N 1 XN 3 ... 3 X N where KdimS aiN/Xi+lN
M
2
=
Sz, a and KdirnS X k N/Sz i s Kdim((M/Rm) [ X , q I ) .
i s a - c r i t i c a l i t f o l l o w s t h a t Kdim((M/Rm)[
X,Q]
5 a.
Since
Lemma 11.5.3.
a p l l i e d t o t h e sequence ( * ) e n t a i l s now : Kdim(N/Sz) 5 a and t h e r e f o r e N i s ( a -+ l ) - c r i t i c a I
106
A. Some General Techniques in the Theory of Graded Rings
11.6. The s t r u c t u r e of Simple Objects i n R-gr.
I n t h i s section the graded rings considered a r e exclusively graded of type 2 . Recall t h a t an S c R-gr i s said t o be gr-simple or simple in
2-gr i f 0 and S a r e i t s only 11.6.1. Lemna.
graded submodules.
If S i s or-simple then f o r each i F Z , Si = 0 or Si i s
a simple Ro-module. Proof.
If Si # 0 f o r c e r t a i n i C
z, say
x # 0 in S i , then Rx
=
S and
thus R x = S i . Consequently S . i s a simple Ro-module. 1
0
11.6.2.
Remarks.
1. I f R i s positively graded and S c R-gr i s gr-simple then S = S . f o r J some j 6 2 . Indeed, i f both Si a n d S . are nonzero, say x i # 0 i n Si and J x . # 0 in S . , then x. = rx and x . = - s x i f o r c e r t a i n r. s c h ( R ) . J J 1 j J Depending whether i > j or j > i , one of these r e l a t i o n s i s impossible
whence i = j follows.
5
2 . If S i s l e f t - ( o r r i g h t ) limited and or-simple then
i s simple in
R-mod. 11.6.3. Lemma.
Let S be gr-simple. Then 0 = ENDR(S) i s a or-division
ring and i f Do # D Proof.
then
5
i s 1 - c r i t i c a l in R-mod.
Consider a nonzero graded morphism of degree n , f : S
f may be represented by a morphism of degree zero f : S
+
--f
S, t h e n
S(n).
B u t both S a n d S(n) are simple i n R-gr, t h u s 7 i s an isomorphism i n R-gr i . e . i n v e r t i b l e . I n order t o prove the second statement we f i r s t e s t a b l i s h t h a t _S / -X has f i n i t e length, f o r every nonzero R-submodule
-X of 2.
Since X # 0
, ,X # 0
hence ,X
= S. I f
5 0 5'
=
0 then , X
n S+
e n t a i l s S+ = 0 b u t then S i s right-limited and Remark 11.6.2.2. then yields that
5
i s simple.
=
0
107
IJ. Some General Techniques
SO
we may assume t h a t
X fl '2
# 0. Now l e t J1 J
descending chain of submodules o f
5
J
X be a
containin! J. We then o b t a i n
descending chains o f R+-submodules o f
5'
:
... An n -S+ J ... 3 -X fi -S', and n 5+)" = (J, n ?+)- ... 3 (-nx n s+)- = ... J(xn - s+)" ,
2, n 2' (J,
X2 = .. .
J
J2 0
5'
2
J
J
the o n l y
where the o p e r a t i o n ( )- i s c a r r i e d o u t i n S+. We c l a i m t h a t
graded submodules o f S+ a r e the S, w i t h p 5 0 ; t h e p r o o f o f t h i s -P c l a i m i s p a r t 1 o f t h e f o l l o w i n 9 Theorem 11.6.4..
(X n ?+)- =
Write
(X, n 2)-
S,p' I t f o l l o w s t h a t f o r some j,p
= pj+l=
-pk
where p1 c
. . . . , hence
pp
(X. -J
c .. . :p k c .
n S')"
=
-
.. c p. (zj+l n z+ )- =.
i t f o l l o w s t h a t X . = Jj+l=. . . -J i s l e f t A r t i n i a n whereas by Theorem I I . 3 . 5 . , 5 i s l e f t
By P r o p o s i t i o n 11.2.3.
Therefore
j
= S,
S/J
Noetherian, thus
and C o r o l l a r y 11.2.4.
z/J has
then D Z Do[ X, X-',Q]
f i n i t e ; l e n g t h . I f D has n o n t r i v i a l gradation, by C o r o l l a r y 1.4.3.
The endomorphism o f
5
represented by I - X w i l l be a n o n - i n v e r t i b l e i n j e c t i v e morphism and thus we o b t a i n t h e s t r i c t l y descending chain o f R-submodules i n
s=(l-X)J
'(1-x)
Consequently
2 is
that
5
11.6.4.
2
5
2
:
J....
n o t l e f t Artinian,
KdimR _S = 1 and t h e above i m p l i e s
i s 1-critical. Theorem.
L e t S be gr-simple, then :
1 . Every graded R+-submodule o f S+ ( s i m i l a r l y f o r &submodules f o r some p ? 0. -P 2 . The R+-submodules of S+ and t h e R--submodules o f
of S - )
i s o f t h e form S,
5- a r e
principal
submodules.
3. Every R-submodule o f 4.
Sis
5
i s principal.
e i t h e r simple i n R-mod o r a 1 - c r i t i c a l R-module.
5. The i n t e r s e c t i o n o f a l l maximal R-submodules of
5
i s zero.
108
A. Some General Techniques in the Theory of Graded Rings
Proof.
L e t M 8 Fli be a graded R+-submodule o f S'. L e t p be m i n i m a l i n i30 N such t h a t M # 0. By Lemma I I . 6 . 1 . , Fl = S and t h e n R+S = S, f o l l o w s P P P P -P f o l l o w s , t h u s M = S, . f r o m RS = S. T h e r e f o r e , M 3 R+Mp = pS, P -P 2. Easy enough.
3. C o n s i d e r a nonzero R-submodule -_ X c S. I f
n S+
b u t a l s o ,X
i s a nonzero y
c
J such t h a t y
=
S i n c e deg x 2 0 i t f o l l o w s t h a t y
c
g r . s i m p l i c i t y o f S. Now assume pS,
f o r some p
There e x i s t s y c T h i s means t h a t
(J n 2')"
=
(&)-
c N
(where
5n' 2 such (X n S+)"
3
-
- = S = (X)-. Ry # 0, (Ry)-
only i f there
i s nonzero
2'
2 '
# 0. Then we o b t a i n
t h a t y = y. 11
+ ... +
(a)--3
(R'y)-
n _S+ 2
X St,
# S but t h i s contradicts
i s d e f i n e d i n S').
5n' 2
(5 n >+)_=O
w i t h deg yi = i and m j j hence y c hence y F n
11
S+
and t h e r e f o r e J
On t h e o t h e r hand we have :
= 0 then
+...+ yi
y.
i s t h u s c l e a r t h a t X-
contradiction. It
=
n S')
= 0. Indeed x C h(X_
1!n -S+
=
2
(X fl 2')"
P i c k a nonzero x yim = x
P
= S,p,
Rxp
hence
5 f' ' 2
=
F S
P' and il< ...-4m hence
Q.
a,
P
=
Ry n
2+ .
Since
Now t h e r e s u l t s o f S e c t i o n 11.2. y i e l d t h a t
x = Ry. 4. and 5 . F o r t h e p r o o f o f t h e s e s t a t e m e n t s we need some p r e l i m i n a r y f a c t s , p r e s e n t e d h e r e as sublemma's. Any M c R-gr may be viewed as a graded l e f t module o v e r T = ENDR(M) and t h e graded r i n g Bg(M) = ENDT(M) i s c a l l e d t h e graded bi-endomorphism r i n g M. There i s a c a n o n i c a l graded r i n g morphism q : R Sublemma 1.
+
E9(M), d e f i n e d by q ( r ) ( x ) = r x f o r a l l x C M.
(The graded v e r s i o n o f t h e d e n s i t y theorem)
I f M i s g r - s e m i s i m p l e ( i . e . a d i r e c t sum o f g r - s i m p l e o b j e c t s ) and if xl, "(xi)
..., xn
C h(M),
a
c
h(Bg(M)), t h e n t h e r e i s an r
= r x . f o r i = 1,..., n.
c
h(R) such t h a t
109
II. Some General Techniques
Sublemma 2.
L e t L be minimal as a graded l e f t i d e a l o f R. Then e i t h e r
L~ = 0 o r L = Re where e i s an idempotent i n h(R). Voreover, t h e r i n g eRe
i s a gr-division ring.
Conversely, i f R i s a semiprime r i n g and e an idempotent i n h(R) such i s a g r - d i v i s i o n r i n g , then L = Re i s minimal as a graded
t h a t eRe
l e f t i d e a l o f R. I f eRe
i s a f i e l d then C i s minimal as a l e f t i d e a l .
S i m i l a r t o t h e ungraded e q u i v a l e n t s .
Proof.
L e t us now r e t u r n t o t h e p r o o f o f t h e theorem.
4. Since
5
i s f i n i t e l y generated i n R-mod, D = ENDRS = EndRS i s a g r -
division ring. If
0is
n o t a f i e l d then Lemma 11.6.3.
i s 1 - c r i t i c a l . I f D = Do i s a f i e l d , l e t P
implies t h a t
be t h e a n n i h i l a t o r o f
5
S i n R.
I t i s c l e a r t h a t P i s a graded prime i d e a l and because _S i s an R/P-module
we reduce t h e p r o o f t o t h e case P = 0. According t o sublemma 1, R i s dense in
Bg(S) = T. But S i s simple i n T-gr and isomorphic t o a graded l e f t
i d e a l o f T, L say. Since L2 = L and ENDT(L) L hence second sublemma t o deduce t h a t -
5
N
ENDT(S) = D, we may use t h e
i s a simple T-module.
Choose
+...t xi , deg xi = ij,il < ... in 1 n j i c i t y of S i n T-mod e n t a i l s y = ax f o r c e r t a i n a c T, say a = a . +...+a c
S and x
...
# 0; w i t h x
= xi
JI
< jm. By t h e f i r s t sublennna we o b t a i n r .
Jk
<
h(R) such t h a t
x . = r . x . f o r k = 1,...m, 1 = l . . . n . Thus a . x = r . x and hence Jk ' 1 J k '1 Jk Jk Y = ax = ( zm r . )x, p r o v i n g t h a t 5 i s simple i n R-mod. k = l Jk a.
5. By 4. i t w i l l s u f f i c e t o consider the case where D = ENDR(S) i s a g r - d i v i s i o n r i n g w i t h n o n t r i v i a l g r a d a t i o n . By sublemma 1, a maximal T-submodule o f S i s a l s o a maximal R-submodule. Therefore i t i s a c t u a l l y s u f f i c i e n t t o show t h a t t h e Jacobson r a d i c a l J(_S) o f But
5
5
i s a p r o j e c t i v e T-module o f f i n i t e type, thus :
over T i s zero.
jm
,
110
A. Some General Techniques in the Theory of Graded Rings
ENDR(S/J(S))
ENDT(S)/J(ENDT(S)) D/J(D).
For a g r - d i v i s i o n r i n g one e a s i l y checks t h a t J(D)
=
0 ( s e e a l s o Theorem
1 . 7 . 9 . ) . So i f J ( S ) were nonzero then S / J ( S ) would be a T-module of f i n i t e l e n g t h , b u t then ENDT(_S/J(S)) 2' D e n t a i l s t h a t D i s a f i e l d and t h e r e f o r e t r i v i a1 l y graded, a c o n t r a d i c t i o n . 11.6.5. C o r o l l a r i e s . 1. I f R i s graded of type 2 then J ( R ) c J a ( R ) . 2 . I f R i s p o s i t i v e l y graded then J9(R) = J(Ro) + R,.
Proof. 1. See Remark 1.7.16. o f Theorem I I . 6 . 4 . , 5 . 2 . I f S C R-gr i s gr-simple then R,S
follows. By Remark I I . 6 . 2 . , l . ,
S
=
# S , hence R,
0 s i n c e R,S
we have t h a t S
=
c Jg(R)
f o r some no, where
S
i s a simple Ro module. T h u s J(Ro)Sn = 0 o r J(Ro)
+ R,
c J g ( R ) . Now
0
J9(R)
=
Jo
+ R,.
with the t r i v i a l g r a d a t i o n . C l e a r l y S i s gr-simple and
S o equipped
JoSo = JoS
I f S o i s a simple Ro-module, then we t a k e S c R-gr t o be
=
0 f o l l o w s , hence J o
c
J(Ro).
11.6.6. Lemm. I f R i s a s t r o n g l y graded r i n g of type 2 and S i s gr-simple then So i s a simple Ro-module and S 2 R Proof.
See Theorem 1.3.4.
66
Ro
S
0'
We apply this t o modules of polynomials over simple o b j e c t s : 11.6.7. Theorem.
Let R be an a r b i t r a r y r i n g and l e t S be a simple R-
module. Consider S I X ] c R[ X I -mod and S [ X , X - l ]
over R [ X.X-']
t h e following
p r o p e r t i e s hold : 1. S [ X , X - ' ] 2 . Every R [
i s a I - c r i t i c a l R [X,X-'] -module. 1 X,X1- submodule of S [ X , X - l ] i s p r i n c i p a l .
3. The maximal R [ X,X-']
-
submodules of S [ X , X - l ]
i n t e r s e c t i n 0.
111
II. Some General Techniaues
4. SJXJ
i s 1 - c r i t i c a l i n R [ X I -mod.
5. E v e r y
R I XI-submodule o f SJXJ
-
6. The maximal R [ X I
i s principal.
submodules o f
i n t e r s e c t in' 0.
Proof. i s s t r o n g l y graded o f t y p e Z, S [ X,X-l]
Since R [ X,X-l]
Thus 1,2.,3.
i n R [ X,X-l]-gr.
4. Since S [ X I = S [ X , X - l ] that
f o l l o w f r o m Theorem 11.6.4. we may a p p l y C o r o l l a r y 11.5.17.
+
i s 1-critical i n
i s gr-simple
R
t o obtain
[ X ] -mod.
5. D i r e c t f r o m Theorem I I . 6 . 4 . , 3 . 6. P u t D = ENDR(S), Bg(S) = END,,(S).
Note t h a t Bg(S) i s Von Neumann r e g u l a r .
As a Bg(S)-module, S i s i s o m o r p h i c t o a m i n i m a l The c a n o n i c a l
P
Evidently, S [ X I
:
R
l e f t i d e a l L o f Bg(S).
Bg(S), e x t e n d s t o a morphism R [ X I
--f
L [XI
i s isomorphic t o
.
i s a d i r e c t summand i n Bg(S) [ X I r e s u l t o f Amitsur y i e l d s t h a t
+
Bg(S) [ X I .
i n Bg(S) [ XI-mod and t h e l a t t e r a classical
S i n c e Bg(S) i s r e g u l a r ,
J(Bg(S) [ X I )
= 0 ( a c t u a l l y we may use g r -
r e g u l a r i t y o f Bg(S) and Theorem 1.7.9.!) Tbe p r o o f may now be f i n i s h e d by showing t h a t a maximal Bg(S) [XI-submodule
-M of
S [ X I i s a l s o maximal as
Pick s(x) = s Then
+
0
+
slx
Bg(S)s(X) =
t h e r e e x i s t a.
+...+
R
s . XJ 3
[ X I - submodules.
S. Given
,...,am c
c SJXJ bo
+
R such t h a t aosi
0 5 i 5 j . Thus Bg(S) s(X) = R [ X I s(X) 11.6.8.
- l. bl X =
... + bm Xm bosi,.. ., amsi +
and hence
C o r o l l a r y . L e t R a n a r b i t r a r y r i n g . If F?
3+
M 1 X,X-
proof.
Bq(S) [ X I ,
=
bmsi
f o r each
R [ X I s(X) =
c R-mcd
S.
has f i n i t e l e n 9 t h
n t h e n e v e r y R [ X I - submodule o f M[XI and e v e r y R [ X,X-'] of
c
-
submodule
1
I may be g e n e r a t e d by l e s s t h a t n e l e m e n t s .
I f n = 1 then
i s s i m p l e and t h e s t a t e m e n t t h e n f o l l o w s f r o m
t h e f o r e g o i n g theorem. We now proceed by i n d u c t i o n o n n, assumin? t h e
A. Some General Techniques in the Theory of Graded Rings
112
statement to be t ru e f o r modules of length a t most n-1. Consider a maximal l e f t submodule 0
+
J1
M/M1
+
+
M, of
y.
The exact sequence in R-mod : 0,
+
gives r i s e t o a n exact sequence in R [ X ] -mod :
0 +MI [ X I If
y
i s an R
[
+
NM1)[XI
+
X I - submodule of
n - 1 elements , whereas
le s s t h a n
and i t may thus be be
M[Xl
+
then
lj/Y
0 .
Y n MI
P,.l[xl
[
X] i s generated by
i s a submodule of (W/M1)
generated by a s i n g l e element. Consequently
IX I
may
generated by ( l e s s than) n elements.
The second statement follows in formally the same way. I f D i s a gr-division ring, l e t d(D) be the minimal positive integer f o r which there e x i s t nonzero homogeneous elements in D of t h a t degree. Putting d
=
d ( D ) we have :
11.6.9. Proposition.
Let R be a graded ring of type Z and l e t S be a
gr-simple, then : 1.
5
i s a semisimple Ro-module.
2 . If D = ENDR(S) then the number of isotopic components of the Ro-module
S cannot exceed d .
Proof. 1. Easy (see Lemma 11.6.1.)
2 . I f a i s nonzero i n Dd then a : S
+
S i s an R-automorphism of degree d .
Thus, f o r any n c Z , the mapping Sn
+
Sd+n defined by x
+
a ( x ) i s an
R -isomorphism. I t i s cl ear t h a t statement 2.isa d i r e c t consequence 0
of t h i s . 11.6.10. Example. Let K be a ( t r i v i a l 1 y graded)
f i e l d and l e t V =
@ \In be a graded Kn>O K ( n ) , summation over ( n + 1)-terms.
vector space where V n = K(n)
@... 8
II. Some General Techniques
P u t R = ENDK(V) and i d e n t i f y K w i t h a s u b r i n g o f Ro.
113
I t i s n o t much o f
a p r o b l e m t o v e r i f y t h a t V i s g r - s i m p l e i n R-gr. B u t as V i s l e f t l i m i t e d as a graded R-module we o b t a i n t h a t ! t h e n dimK Vn #
dimK Vn,
i s a s i m p l e R-module. I f n # n '
; a f o r t i o r i Vn
7
V,
i n f i n i t e l y may i s o t o p i c components i n Ro-mod.
i n R -mod. T h e r e f o r e , V has
A. Some General Techniques in the Theory of Graded Rings
114
11.7.
P r i m a r y Decomposition.
I n t h i s s e c t i o n R w i l l always be a graded r i n g o f t y p e G where G i s an i s a nonzero R-module t h e n a p r i m e i d e a l P o f R i s
o r d e r e d group. I f
s a i d t o be a s s o c i a t e d t o M i f t h e r e e x i s t s a nonzero submodule such t h a t P = Ann f o r the set r i n g then P
=
Ann
M" f o r e v e r y submodule
R-
c
!'I
of
!'.
of
W r i t e Ass
I f R i s a commutative
i f and o n l y i f P = AnnRx f o r
Spec(R) i s a s s o c i a t e d t o
!.
Lemma.
Suppose M # 0 i n R-gr. I f R i s l e f t g r - N o e t h e r i a n t h e n
# 6.
Ass Proof.
F o r any graded submodule N c M, AnnRN i s a graded l e f t i d e a l o f R .
# 0 a graded submodule o f P
Consider t h e s e t P = {AnnRN, N
P #
e.
o f prime i d e a l s o f R associated t o
some x # 0 i n 11.7.1.
M'
R-
E'
+.
L e t P = Ann N be a maximal e l e m e n t o f P. R o
a F R, b 6 R-P t h e n RbNo # 0 i n No.
11.7.2.
Remark.
c P with
It i s e q u a l l y o b v i o u s t o see t h a t
AnnR(No) c AnnR(RbNo) and hence P = AnnR(No) i m p l i e s t h a t a C AnnR(RbNo) and
If a R b
}. O b v i o u s l y
=
AnnR(PbNo).Now a RbN,
t h e r e f o r e a 6 P. Thus P
c
= 0
AssM. -
W i t h o u t f u r t h e r assumptions on R and 6 i t i s n o t t r u e
t h a t R i s l e f t N o e t h e r i a n i f i t i s l e f t g r - N o e t h e r i a n . F o r example G = Z ( J ) where J i s i n f i n i t e , i s an o r d e r e d group. L e t k be a f i e l d . The group r i n g R = k [ G I i s a s t r o n g l y graded r i n g o f t y p e G and s i n c e k i s a f i e l d , hence N o e t h e r i a n , R i s l e f t g r - N o e t h e r i a n . On t h e o t h e r hand R i s n o t a l e f t N o e t h e r i a n r i n g i f G does n o t s a t i s f y t h e a s c e n d i n g c h a i n c o n d i t i o n
on subgroups, (see Lemma 2.2. p . 420, [ 8 8 11.7.3.
Theorem.
Let 0 # M
c
] ).
R-gr and suppose P
c
Ass
M, t h e n
_.
1. P i s a graded p r i m e i d e a l o f R .
2. There e x i s t s an x F h(M), x # 0, such t h a t P
=
AnnR(Rx).
:
115
II. Some General Techniques
Proof. 1. P = AnnR(M') f o r some R-submodule 0 # M ' c M.
Take X c P, x c M ' , say A x = x
+...+
xT
Tl
with
n
e h ( R ) . From
Choose a
= X
<
T~
+...t
01
... <
w i t h u1 < . . . < u and am f o r ul, urn, T ~ ... , 7 , c G. X
...
T
h a c P we p e t t h a t
A a x
0 hence
=
AD a x-
n bxT
Ln
=
0.
t Xu = 0 if F o r any b c h(R) we have h b x = 0 i . e . lo b xT m n-1 m-1 n am y T ~ = - Xu ~ yrn w i t h y = deg b, o r e l s e = 0. P- 1 'Tn-1
'
I n t h e s e e q u a l i t i e s we may r e p l a c e b by b ha a : i f hU b xT t Xa b xT = 0 m n-1 m-1 n m then
ho
m
b Xu
a xT
m
n-1
= 0, hence ( h o R ) L xT
m
n-1
= 0; on t h e o t h e r hand,
= 0 a l s o e n t a i l s t h a t : (Xu R)' t h e e q u a l i t y Xu b xT m n-1 m
I t f o l l o w s t h a t (Au R ) p xT = 0 f o r a c e r t a i n p F m i for all
T~
p i c k one p c
N (dependinq on T ~ ) ,
N f o r a l l xT . Consequently, (Au R)'Rx i
and deduce la c P f o r i = 1,
i
3'.Then
2 . Take 0 # x c
P u t Ji = AnnR(R x
'i
= 0, t h u s ha R c P.
m
m
c P and we may r e p e a t t h i s a r g u m e n t a t i o n M
..., m.
So, P i s graded.
P = AnnR(Rx), w i t h x = xT
1
t . ..t x
=
AnnR(Rxi).
W i t h n o t a t i o n s as i n t r o d u c e d e a r l i e r we have : Theorem.
AssA [ X 1 M [ X I
=
L e t A be any r i n g , M 6 A-mod, (P [ X I ,
P c Ass M
then :
1 , and,
~
"'A
[ X,X-l]
Tn
, T ~ < . . .
) i = 1, ...,x . One e a s i l y v e r i f i e s t h a t P = J1 fl...
and t h u s P = Ji f o r some i .
11.7.4.
= 0.
n- 1
and s i n c e t h e number 'of components o f x i s f i n i t e we may
I t now f o l l o w s t h a t X - Xa
Therefore P
xT
M [X,X-l]
= { P [X,X-lI
,
P c Ass
M 1.
"
Jn
n'
A. Some General Techniques in the Theory of Graded Rings
116
Proof.
I f P C Ass M t h e n P
AnnAM' f o r some nonzero
M' [ X I . Let
t h a t P [ X I = AnnA [ S i n c e we have t h a t P hence P [ X I C Ass
=
=
A [ X l
m ' X V be a nonzero e l e m e n t o f h(M' [ X I ) .
(M [ X I . ) __ ~
xl(m).
There i s a nonzero homooeneous e l e m e n t mX" then Q
c M. I t i s c l e a r
AnnAAm' i t f o l l o w s t h a t P [ X I = Ann A 1 x 1 (A [ X I m ' x " ) ,
Conversely, c o n s i d e r Q F AssA
I f P = AnnA(&)
M'
=
P [XI,
such t h a t
Q = AnnA [ X I ( A [ X I mx")
P i s p r i m e and P C AssAM.
The second s t a t e m e n t o f t h e theorem f o l l o w s i n a s i m i l a r way. Let
Ec
i s a p r i m a r y submodule o f M i f Ass(M/N)
i n R-mod. We say t h a t
i s a P - p r i m a r y submodule o f J! i f Ass(M/N)
i s a s i n g l e t o n , and we say t h a t =
IP
o
} . R e c a l l f r o m [ 1 2 9 1 , ch. 4, t h a t , i f R i s a commutative N o e t h e r i a n
r i n g , and
i s an R-module o f f i n i t e t y p e t h e n
and o n l y i f
fl i s
1 1 . 7 . 5 . Lemma.
i s i n primary i n
if
a c l a s s i c a l primary-submodule. L e t R be araded o f t y p e G, G an 0-Group,
R i s l e f t g r - N o e t h e r i a n . Consider !I
and suppose t h a t
c Spec(R). The f o l l o w i n o s t a t e m e n t s
are equivalent.
1. 0 i s a P - p r i m a r y submodule o f M. 2. The s e t ( 0 : P ) M = { m F M, Pm = 0 } i s an e s s e n t i a l submodule o f
M
and P c o n t a i n s any graded i d e a l w h i c h a n n i h i l a t e s a nonzero oraded submodule o f M. Proof.
1
11.7.1.
: Ass(?)
3
2. I f X =
# 0 i s a graded submodule o f M t h e n by 1 and Lemma {P}. Hence X
n
(0 : P)M
# 0.
I f I i s a graded i d e a l o f
R a n n i h i l a t i n g a nonzero qraded submodule M ' c M t h e n f r o m Ass
Q
i t f o l l o w s t h a t t h e r e i s an M" c M ' , 11"
However IM" = 0 y i e l d s I M"
=
=
tP1
0, and P = AnnR(M").
0 , hence I c P.
2 = 1. Because R i s l e f t g r - N o e t h e r i a n , we may a p p l y Lemma 1 1 . 7 . 1 . t o
f i n d a Q F Ass M. By t h e assumptions Q
c
P f o l l o w s . Now c o n s i d e r a nonzero
=
II. Some General Technioues
submodule M ' c M
such t h a t
Q
=
AnnR(M'). S i n c e M '
117
n
(O:P)b, # 0,
P c AnnR(M' n (O:P)M) f o l l o w s , t h u s P c Q and hence P = Q. 11.7.6.
Lemma.
L e t M c R-gr and l e t !
i s essential i n M/(N)g In particular, i f
Xis
then f / ( N ) g ,
c
5c
I f J/!
be R-submodules.
r e s p . X-(N)
9
i s e s s e n t i a l i n N/(N)g.
r and ,X
a l e f t e s s e n t i a l submodule o f M t h e n
a r e l e f t e s s e n t i a l i n M.
x
and l e t be i t s imaoe i n PI/(N) . S i n c e 9 9 x f N, i = x mod N i s nonzero i n y/N. The assumptions a s s u r e t h e e x i s t e n c e Proof.
Take x c h(M)-(N)
of a A c R such t h a t X 2 F
L/N,
X R # 0.
t.. .+ Xo , ol< 01 n t h e n we may r e p l a c e X by + = X-Xo
Write X = h
Then Xx = Xo x + ... + Xo x . I f Xu x C 1 n n w h i l e X i = p i , s o we may assume t h a t Xo x f n
1
x/N
c
hand, X i
t h e r e f o r e Lo
n
yields that X x c
x 6 r/(N)o
X
N,
Xo x
n
hence Xo x 6 n
i s nonzero. Lemma 1.2.8.
f o r t h i s case. The s t a t e m e n t a b o u t Xw/(Fi)q
# 0. On t h e o t h e r
f". B u t
Xo x f (N)g,
n
f i n i s h e s t h e proof
may be e s t a b l i s h e d i n a
s i m i 1a r way. 11.7.7.
Theorem. L e t R be a l e f t N o e t h e r i a n qraded r i n o o f t y p e
M F R-gr and suppose t h a t N.c
1i s
a P - p r i m a r y submodule o f
M.
G. L e t Then
submodule o f M.
( N ) g i s a (P) - p r i m a r y 9
X by X/lj
(O:P)M/N; c l e a r l y t h e l a t t e r i s a l e f t -_ (Lemma 1 1 . 7 . 5 . ) . The f o r e g o i n g lemma e s s e n t i a l R-submodule o f M/! Proof.
Define
=
e n t a i l s t h a t f ' / ( N ) g i s l e f t e s s e n t i a l i n M/(N)q. From deduce t h a t t h a t (P)
9
f
c (N)q, hence f / ( N )
9
P
Xc
c (O:P)M/(N)g
N-, we
and
t h e r e f o r e t h e l a t t e r i s l e f t e s s e n t i a l i n M(N)
I of R w h i c h a n n i h i l a t e s a nonzero Y/(N), Then I T
c
Consider a araded i d e a l q' i n M/(N)g.
(N)g and s i n c e Y # ( N ) g we have ( N ) o # \r i . e .
T / ( N ) g # 0.
T h e r e f o r e we may assume t h a t I a n n i h i l a t e s a nonzero araded submodule
< cn
n
1i a
A. Some General Techniques in the Theory of Graded Rings
As b e f o r e we t h e n deduce t h a t IT c N, so we a r r i v e g' a t t h e c o n c l u s i o n t h a t I ( T + N/N) = 0 w i t h \r+ N/N # 0. The hypotheses T ( N ) g o f M/(N)
imply I
c
P, t h u s I
c
(P)
9
f o l l o w s . Now we have reduced t h e p r o o f t o
an a p p l i c a t i o n o f Lemma 1 1 . 7 . 5 .
1 1 . 7 . 8 . C o r o l l a r y . L e t A be any l e f t N o e t h e r i a n rim, M a nonzero l e f t A-module. P
[XI -
o f M i f aqd o n l y i f 0 i s
Then 0 i s a P - p r i m a r y submodule primary i n M
[XI,
if
and o n l y i f 0 i s P [ X,X-']
-primary i n
P r o o f . A consequence o f 1 1 . 7 . 7 . and 1 1 . 7 . 4 . o f p r i m a r y submodules o f PI i s a
We say t h a t a f i n i t e f a m i l y (Ni)i
p r i m a r y d e c o m p o s i t i o n f o r t h e submodule N o f M i f
N
=
n
iCJ
Ni.
The
d e c o m p o s i t i o n i s s a i d t o be reduced i-f t h e f o l l o w i n a c o n d i t i o n s a r e met :
RD1.
,R. Ni
14
Ni
for all i E J.
.J#1
RD2.
I f Ass(M/Ni)
= Pi
t h e n t h e i d e a l s Pi a r e two-by-two d i s j o i n t .
1 1 . 7 . 9 . Lemma. L e t R be a qraded r i n g of t y p e G , C an 0-group, and l e t M
c
R-gr be f i n i t e l y generated. I f R i s l e f t g r - N o e t h e r i a n t h e n each
graded submodule N o f M
a d m i t s a reduced p r i m a r y d e c o m p o s i t i o n .
P r o o f . S i n c e M/N has f i n i t e t y p e , i t i s a l e f t g r - N o e t h e r i a n module. There e x i s t araded submodules N1,. and such t h a t t h e M/Ni
a r e or-uniform,
M such t h a t N = N1 0 Np 17.. .nNk
f o r a l l i = 1, ..., k . U n i f o r m i t y
i n R-gr e n t a i l s , because o f Lemma I I . 7 . 5 . ,
o f M/Ni i = I,
. . ,Nkc
..., k .
t h a t Ass(M/Ni)=
{Pi},
Since t h e obtained decomposition i s f i n i t e i t i s n o t hard
t o v e r i f y t h a t we may v e r y w e l l reduce i t t o a reduced d e c o m p o s i t i o n .
11.7.10.
Remark.
The l i n e s o f p r o v e used i n t h e above lemma f o l l o w
c l o s e l y the p r o o f o f the corresponding
ungraded s t a t e m e n t (where R
i s l e f t N o e t h e r i a n ) . More d e t a i l a b o u t p r i m a r y d e c o m p o s i t i o n may be
f o u n d i n [ 129 ] ch. 4 .
II. Some General Techniques
11.7.11.
Corollary.
119
L e t R be a l e f t N o e t h e r i a n araded r i n o , and suppose
M c R-gr i s f i n i t e l y generated. L e t N be a araded submodule o f M such that N =
n
i
{Ni.
J } i s a p r i m a r y d e c o m p o s i t i o n o f N i n M. Then t h e
f
following properties also hold :
i c J } i s a p r i m a r y d e c o m p o s i t i o n o f N i n M. g' 2. I f t h e p r i m a r y d e c o m p o s i t i o n N = n { Ni, i c J } i s reduced t h e n 1. N = f' { ( N . ) 1
N =
n
' g'
i c J } i s r e d u c e d and f o r any i 6 J , Ass (M/Ni)
{ N.)
=
= ASS ( M / ( N - ) ) .
1 g
Proof. 1. F o l l o w s f r o m Theorem 11.7.7. 2 . We have Ass(M/Ni)= Supose t h a t
n
(N.)
j#i
J
Ass(F1/(N) ) f o r a l l i f J . ' g c (Ni)g, t h e n N = n (Nj)q. j#i
There e x i s t s an i n j e c t i v e homomorDhism,
Q I
: M/N
f
We c o n c l u d e t h a t : Ass(M/N) c Ass (
8 M/(NJq) j # i__ .
=
=
U Ass(M/(Njla) j # i____
U Ass(M/Nj)
j#i
,
y i e l d i n g a c o n t r a d i c t i o n . Therefore t h e d e c o m p o s i t i o n N =
n
{(N.) 1
q'
i
f
J}
i s reduced. By a c l a s s i c a l P - p r i m a r y submodule o f M C R-mod we mean a submodule N o f M
w h i c h i s P - p r i m a r y and such t h a t Pn M c N f o r some n f
N. Theorem 11.7.7.
i m p l i e s t h a t f o r a c l a s s i c a l P - p r i m a r y submodule N o f M,
(N)
g
i s a classical
( P ) g - p r i m a r y submodule o f M. C o r o l l a r y 11.7.11.
s t i l l h o l d s i f " p r i m a r y " i s r e p l a c e d by " c l a s s i c a l
p r i m a r y " . T h a t , f o r a l e f t N o e t h e r i a n r i n g R and an R-module M o f f i n i t e type, 0 need n o t have a c l a s s i c a l p r i m a r y d e c o m p o s i t i o n i n PI, may be j u s t l y termed " o l d h a t " .
120
A. Some General Techniques in the Theory of Graded Rings
11.8. E x t e r n a l Homoqenization. H o m o l o g i c a l Dimension f o r Graded R i n o s . I n t h i s s e c t i o n , R i s a qraded r i n g o f t y p e 2. The r i n g
R
[ T I may be o r a d e d as f o l l o w s : den T = 1,
, I - ri TJ, ri c Ri
= {
c
M [TI
R
c
}. S t a r t i n g f r o m M
i+j =n
[ T I -qr. A c t u a l l y M [ T I = R [ T I
R [TIn
=
R - q r we may c o n s t r u c t
M
6a
R
where t h e t e n s o r
p r o d u c t i s graded i n t h e u s u a l way. If x
t o i t , x* c M [ T I
* x
= x
... +
M i s o f t h e f o r m x - ~+
c
Tm+n
-m
+ x
, 1-m
xo
+ ... +
Given by : Tmtn-l
+...+
x
0
Tnt
... t
Then u* =
x
n'
o f x. I f
We say t h a t x* i s t h e ( e x t e r n a l ) homoqenized say u = u-
xn t h e n we a s s o c i a t e
u
c h(M [ T I ) ,
Tk+J ... + u T j + . . . + u . , w i t h ui c 11, i = k ,..., j . k 0 J u - ~ + . .. + uo + . . . + u . c M i s s a i d t o be t h e dehomogenized
J o f u. The f o l l o w i n g p r o p e r t i e s a r e e a s i l y v e r i f i e d .
i)
I f f c h ( R [TI ) , u c h(M [ T I ) t h e n (fu),
ii)
If u, v c h(M [ T I ) have t h e same deoree t h e n ( u + v ) * = u* + v*
i i i ) If x
c
M, (x*),
If u
c
h(M I T ] , t h e n (u,)*
iv)
c
For D
= f,
ut.
= x.
.
Tk = u, where k = deo u - deo(u,)*
R, m 6 M l e t d ( p ) o r d(m) be t h e h i g h e s t
degree a p o e a r i n o i n
a homoqeneous d e c o m p o s i t i o n f o r p o r in, t h e n :
* m*
k
= T (pin)*
v)
p
vi)
I f x, y
c
, where
k = d(n)
+
d(p)
+
d(m) - d(pm)
k * M a r e such t h a t d ( x ) > d ( y ) , t h e n T (x+y) = x*
+
T1y*
where 1 = d ( x ) - d ( y ) and k = d ( x ) - d ( x + y ) . i s an R-submodule o f ?I t h e n N* s t a n d s f o r t h e graded submodule
If
o f M [ T I g e n e r a t e d by t h e n*, of
E.
Note t h a t each x
c (N*)
n
6 N.
We say t h a t N* i s t h e homogenized
i s o f t h e f o r m TV n* f o r some n 6
y.
~
To a graded R [ T I - submodule L o f
M
[ T I we a s s o c i a t e L,
-
The p r o p e r t i e s m e n t i o n e d above e n t a i l t h a t L,
-
=
tu,
u C h(L)i.
i s an R-submodule o f
y.
It. Some General Techniques
y
The correspondence
+
N* s a t i s f i e s t h e f o l l o w i n g :
2
N*.
121
l*. N = (N*),. 2*.
n
N* -
M = (N) .
L2
3*. I f
(1
t h e n L*
4*. I f N c M i n R-gr t h e n 5*.
=
N*[ T I .
If L i s a l e f t i d e a l o f R then (L
6*. I f x
c M
7*.
Ni)*
(,P 1
Cl
t h e n (N- : x)* = =
,n
1C
The correspondence
* N
y)*
=
(lJ* (!)*.
Wi)*.
l
L
+
L,
satisfies :
1,.
@*I* ' L
2,.
I f L c L ' t h e n L,
4*.
If
5,.
I f u F h ( M [ T I t h e n [ L . u1,
L
:
* x .
c L'*.
i s a graded l e f t i d e a l of R [ T I
Note t h a t i n c o m b i n a t i o n w t h (
=
,
t h e n (JL),
= J*L*.
L* : u*.
)* we o m i t t o w r i t e t h e
(-)
indicating
t h a t we a r e l o o k i n a a t an unqraded module. From t h e above p r o p e r t i e s it i s
e v i d e n t t h a t we may d e f i n e a f u n c t o r
g i v e n by E ( M ) = R [ T I / ( T - 1 )
60 M 2 R [TI
E :
M/(T-1))
R [TI
-
gr
+
R-mod,
M, f o r M F R [ T I - g r
W i t h t h e n o t a t i o n we have :
1 1 . 8 . 1 . Lemma. 1. E i s an e x a c t f u n c t o r . 2. I f _. N F R-mod ( o f f i n i t e type, of f i n i t e p r e s e n t a t i o n ) , t h e n t h e r e i s an M F R [ T I - g r ( o f f i n i t e t y p e , of f i n i t e p r e s e n t a t i o n ) such t h a t E ( M ) 2 N. 3. I f R has n e q a t i v e ccradino and if P i s a p r o j e c t i v e R-module, t h e r e e x i s t s a p r o j e c t i v e Q i n R c T 1 - y such t h a t E ( Q ) 2 P .
then
122
A. Some General Techniquesin the Theory of Graded Rings
Proof. 1. R i g h t e x a c t n e s s o f
E
i s o b v i o u s . Take F.1 c R [ T I - g r and c o n s i d e r
M ' c M i n R-gr. L e f t e x a c t n e s s o f E w i l l f o l l o w i f we can show t h a t M'
n
( T - l ) M = ( T - 1) M ' .
n (T-l)M then + ... t mo + . . . +
I f x F M'
m = m-t
+
m
S
M, where
...,
where deg m. = 0, i = -t,
_..s
0,
# 0, ms # 0. Now (T-1)m c M ' y i e l d s :
and we may t a k e m-t -m-t
x = (T-1)m F M ' w i t h m F
+
+ Tml-t
Tm-t - ml-t
S i n c e M ' i s graded m-t
c
.. .
Tml-t
-ms t Tm,
M ' f o l l o w s ; t h e n ml-t
PI'.
F
F M ' f o l l o w s , and so
on t i l l we f i n d t h a t m 6 M ' i . e . x F (T-1)M'. 2. We have N = R ( @ / L-., f o r some s e t V . The R [ T J
-
module
[T] i s
T I / L*. One e a s i l y v e r i f i e s t h a t M(T-l)M 2' N,
g r - f r e e . P u t M = R(')[
and a l s o M i s f i n i t e l y g e n e r a t e d ( p r e s e n t e d ) i f N i s f i n i t e l y generatc (presented). 3. I f P i s p r o j e c t i v e t h e n
P
Q
Q f o r some s e t V.
L e t R(')
graded i n t h e o b v i o u s way. An R - b a s i s t ev, v F V } f o r R(") y i e l d s an R [ T I
-
b a s i s f o r R(')
[ T I . W r i t e ev = xv
+
also
xv F
y,.
be
P, y,
(we s t i l l w r i t e e v f o r t h e i s o m o r p h i c copy i n P 8 Q ) . Chosing e v = (0, 0,
...,
a l l v c V , t h u s d ( x v t y,)
R(')
Q*
= (R('))*
[ T I = P* Q
11.8.2. 5
=
+
,
yv* and hence e,
[TI
Q*,t h e r e f o r e
( o r i n R [ T I - mod)
0) we see t h a t decl e v = 0 f o r
= d ( x v ) = 0 o r e l s e d ( x v + y,)
I n b o t h cases ( x v + yv)* = xv* Thus P* +
...
0, 1, 0
.
S i n c e P*
[TI /
Q*
n Q*
=
d(y ) = 0
.
= xt t
yv*
= 0 it
follows tha
i s projective i n R[T]-gr
and i t i s c l e a r t h a t E (R(')
[TI /
Q*)
R(')/(I
Theorem. L e t R be a qraded r i n q o f t y p e Z, t h e n g r . q l . dim R
gl.dim
R c 1 t g r . g l . dim
R.
II. Some General Techniques
1
.
Proof. The f i r s t inequality i s j u s t Corollary 1.2.7. then the followinq sequence i s exact in R [ T I mT- 1 0 + M[T] M[T] 5 M 0.
i s multiplication on the r i a h t by T-1, and s i s given by i
s (Z mi T ) i
that 5
2 m
=
12 E ( M ) .
gr.gl dim R [ T I
g l . dim R
Proof.
,
hence g l . dim R
5
1 +gr
5
ar.
. gl . dim
D.
R.
dim M 5 R [TI
If R i s stronqly graded of type Z , then : q l . dim R 0 c
5 1
+
g l . dim Ro.
By Theorem 1.3.4.
, or. 91.
dim R
=
gl dim Ro.
If R i s strongly qraded of type Zn then
11.8.4. Corollary. 5
If N 6 R-mod then l e t 11 < R [ T I -gr be such
i'
Exactness of E e n t a i l s t h a t p. dimR N
11.8.3. Lemma. 5
mod :
+
--f
where mT-l
-
If M F R [ T ] -or
gldim R
5
n + gldim Re.
Proof.
If n = 1, apply Lemma 11.8.3.. We proceed by introduction on n .
Consider H c Zn such t h a t H 2 Z"',
Zn/H 2 Z . If
canonical epimorphism, consider R
where ( R (Zn/H) Since Zn/H 2' Z we may apply 11.8.3. t o deduce t h a t gl .dim R
( Zn/H)
: Zn
+
.
:
c 1 + gl.dim R ( " ) .
The induction hypothesis implies t h a t : gl .dim R ( H ) 5 n Combining b o t h r e s u l t s y i e l d s g l . dim R c n 11.8.5.
Zn/H i s the
(Zn,H))e' TI
+ gl
-
1 + gl .dim Re.
.dim Re.
Lemma. Let R be a l e f t ~ o e t h e r i a npositively oraded rino and
assume gl.dim R
<
m
.
c R-mod be projective and of f i n i t e type,
Let
then there e x i s t projective Ro-modules of f i n i t e type, Q and Q ' say, such t h a t : Proof.
P
fD ( R 60 Q ) '2 RO
R rn
Q'.
RO
Let M F R [TI - gr be such t h a t E(P1) '2 P , a n d choose M of f i n i t e
type. Since g l . dim R
<
m
we obtain the followina exact sequence in R [ T I
124
A. Some General Techniques in the Theory of Graded Rings
(*) : 0 Qo,
Qn
--f
Q,, . ..,
...
+
Q,
+
+
Qo
PI
+
+
0
where
Qn a r e p r o j e c t i v e and o f f i n i t e t y p e i n R [ T I
-
The
gr.
graded v e r s i o n o f Nakayama's lemma may be used t o deduce t h a t each p r o j e c t i v e module o f f i n i t e t y p e i n R [ T I - q r i s o f t h e f o r m R [ T I Ro-module o f f i n i t e t y p e S. P u t t i n q , Qi
f o r some p r o j e c t i v e
R
=
RO
[ T I 60
Ro
i n (*) and a p p l y i n q E t o (*) y i e l d s i n e x a c t sequence i n R-mod,
E(*-) : 0
+
E(Qn) dn
+
where E ( Q . ) 2 Q . / ( T - 1 )
dn- 1
E(Qn-l)
+
Q . 1: R 60 Si. 1
...
d2+ E(Q1) d l
+
E(Qo)
S
60
d4
Si
0
p +
Projectivity o f P entails :
RO
P R
8
K e r do
R
60
Sn-l 2
Ker dn-l
@
R 60 Sn. So t h e s t a t e m e n t f o l l o w s .
RO
11.8.6.
RO
Theorem.
L e t R be a l e f t N o e t h e r i a n r i n q w i t h g l d i m R c
L e t P be a p r o j e c t i v e l e f t R [ T I ,
... , T n ]
m.
-module o f f i n i t e t y p e . There
e x i s t p r o j e c t i v e R-modules o f f i n i t e t y p e Q, Q ' such t h a t
Q [ T 1 ,..., T n l Proof.
8
P 2
Q' [TI
A p p l y Lemma 11.8.5.
,..., T n I .
t o t h e r i n q R [ T1,.
. . ,Tn 1 .
L e t us now p o i n t o u t some f u r t h e r a p p l i c a t i o n s o f t h e s e homoaenization techniques : Homogenization o f Presheaves and Sheaves. The r e s u l t s
i n c l u d e d i n t h i s s e c t i o n a l l o w t o r e l a t e c e r t a i n sheaves o f
( g r a d e d ) r i n a s o v e r Spec and P r o j .
I n t h i s book we do n o t want t o a0
i n t o t h e " g e o m e t r i c a l " aspects o f P r o j ; t h e i n t e r e s t e d r e a d e r may c o n s u l t [ 1 2 5 1 f o r some p r o j e c t i v e "non-commutative
g e n e r a l t h e o r y o f sheaves and presheaves i s [ 1231.
a l q e b r a i c aeometry".
The
w e l l documented i n [ 35
I,
125
II. Some General Techniques
L e t X be a f i x e d t o p o l o q i c a l space and l e t Open (X) be t h e c a t e a o r y o f open subsets o f X w i t h morphisms b e i n a g i v e n by t h e " c o n t a i n i n a " r e l a t i o n s L e t R b e a qraded r i n o o v e r X i . e .
a s h e a f o f graded r i n q s o v e r X such
U
q,
t h a t t h e s h e a f r e s t r i c t i o n morphisms
: R(U\
+
R(V), i s araded o f
degree z e r o f o r each p a i r V c U i n Open ( X ) . We i n t r o d u c e t h e f o l l o w i n g n o t a t i o n s : n ( R ) , r e s p e c t i v e l y o(R) i s t h e c a t e o o r y o f presheaves, r e s p e c t i v e l y sheaves, o f modules o v e r t h e s h e a f R ; gn (R), r e s p e c t i v e l y gu(R)
,will
be t h e c a t e g o r y o f presheaves, r e s p e c t i v e l y sheaves, o f
graded modules o v e r t h e s h e a f R, and i n t h i s case t h e morphisms i n t h e c a t e g o r y a r e p r e s h e a f , r e s p e c t i v e l y sheaf, morphisms w h i c h a r e l o c a l l y o f degree z e r o . The f u n c t o r " f o r g e t t i n g g r a d a t i o n s " i s denoted by :
- : gn(R)
+ n(R).
I f R i s a graded r i n q t h e n t h e r i n o o f
R
f i n e d as f o l l o w s :
T I (U)
t h e g r a d a t i o n on R ( U ) morphism
*u ( P )v
( o* ) ( Tm+n+P where xi
: R X-m+.
[TI
f o r e v e r y U F Open(X)
T I i s as i n s e c t i o n A . I I . 2 . ,
(U) T I
.
= R;U)
p o l y n o m i a l s , R [ T I say, i s de-
and t h e r e s t r i c t i o n
i s g i v e n by : m+n+p U Tp xn) = T P ~ ( x - ~ ) +. .+ . Tp o;txn)
R(V) [ T I
+
.+ Tn+p
xo
+...+
i s homogeneous i n R(U) o f degree i . I t i s n o t
that R [ T I
i s a sheaf, i . e .
, where
hard t o v e r i f y
R [ T I i s a araded r i n q . F o r any Module
M c g n ( R ) t h e Module o f p o l y n o m i a l s M [ T I may be d e f i n e d s i m i l a r l y and we have t h a t M [ T I
c
TI
( R [ T I ) . Now l e t N c n ( R ) be a subpresheaf of M.
F o r a l l U F Open(X) p u t N*(U) = N(U)* c M(U) [ T I where N(U)* i s t h e homogenization o f N(U), as d i s c u s s e d i n A . I I . 2 . then (N*);
* u . (N ) v i s
If V c
g i v e n by :
(Tm+n+P x -m +...+ Tn+p xo +...+ Tp xn) - Tm+n+p MV U (x-,,) +. .+ Tn+p MVU (x,) +. . .+ Tp A!:
where M;
U i n Open (X)
.
(x,)
i s t h e r e s t r i c t i o n morphism f o r M c o r r e s o o n d i n o t o V c U, and
A. Some General Techniques in the Theory of Graded Rings
126
xi
i s homogeneous o f deqree i i n N(U). I t i s c l e a r t h a t N* i s a oraded
subpres h e a f o f M [ T ] 11.8.7.
.
Proposition.
N* c g u
W i t h n o t a t i o n s as above,
i f N 6 o(R) t h e n
(R [ T I ) .
Proof. ___
1. L e t U
e
i c I } an open c o v e r i n o o f U. I f n C N(U)
Open ( X ) and { Ui,
i s such t h a t (N*)Uui(n*)
=
(n*) = (MUi(n)) u
0 f o r a l l i c I t h e n o = (M*)'
*
'i =
(N:i(n))*.
S i n c e (x*),
= x h o l d s f o r any x
c N(U), i t f o l l o w s t h a t
U N ( n ) = 0 f o r a l l i c I . S i n c e N i s a s e p a r a t e d w e s h e a f n = 0 and 'i n* =
o
f o l lows.
c
2. C o n s i d e r n i Each n ! = 1
*
V.
T
n.
1
Ui such t h a t (N )ui
h(N*(Ui))
c N(Ui).
f o r some ni
We o b t a i n :
=
Tj
(Nuj
ui n u .
(nj))*
J
Dehomogenization y i e l d s NU'i i
(nil
j
"j
= NUi
s h e a f i t i s p o s s i b l e t o f i n d an n F N(U) such
'i i F I . The degree of (Nui
(ni))*
j
'i
component o f h i g h e s t degree i n NUi that
'i deg (Nu,
,,.(ni))*
1 entails v =
V .
1
= deg (NUiU i
( n . ) and s i n c e N i s a J t h a t NU U (n) = ni f o r a l l
uj
i
e q u a l s t h e degree o f t h e homooeneous
,
J
(ni).
uj
The above e q u a l i t y e n t a i l s
(nj))*
and c o n s e q u e n t l y (*)
J =
V .
J
f o r a l l i, j c I .
S e t t i n g n ' = TV n* p r o v i d e s an e l e m e n t o f N*(U) under t h e c o r r e s p o n d i n g
*u (N )ui.
mappinq t o t h e o i v e n n;
=
127
II. Some General Techniques
I f the n;
6
N*(Ui)
are n o t necessarily homogeneous the proof may be
completed by checking the homogeneous decomposition of the n! ' s . 1
If L i s a qraded subpresheaf of M [ T I then L,(U)
U c Open y
a n d (L,)"
(X),
U
h ( L ( U ) ) such t h a t y,
6
f o r each U equal t o (Lv(y)), f o r some
( x ) with x c L,(U) =
=
L(U),
x , defines the dehomooenized presheaf L,.
I I .8.8. Proposi t i on.
With notations as above , i f X L c g a ( R [ T l ) then L,
c u(R).
Straiqhtforward. Compactness of X i s n o t needed t o prove t h a t
Proof.
L,
i s a compact topoloaical space and
i s a separated presheaf; i n checkin? the sheaf axiom one has t o
reduce the problem t o a f i n i t e coverinq however. Compactness cannot be dropped; f o r a routine example c f . [63] If i : o(R)
+
.
n ( R ) i s the canonical inclusion then there e x i s t s a
s h e a f i f i c a t i o n functor a_- : n ( R )
+
o ( R ) which i s the l e f t adjoint of i
This functor i s the r e f l e c t o r with respect to the Giraud subcateoory u(R)
of the
Grothendieck catepory n ( R ) ; act u ally in
[
1231 i t i s made
a,' be the
c l e a r how a_- may be viewed as a localization functor. Let s h e a f i f i c a t i o n functor n ( R [ T I ) 11.8.9. Lemma.
Proof.
Easy
+
o(R [ T I ) .
If M c g n ( R [ T I ) then -2 ' ( M ) c q o(R [ T I ) .
(a'
i s given by cer t ai n d i r e c t and inverse consecutive
limits). 11.8.10.
Theorem.
1. Let N c n ( R ) be a subpresheaf of an M
C
9 n(R)
then -a ' ( N * )
= (g(N))*.
2 . Let X be a compact topoloqical space and l e t N c o n(R [ T I be a praded
subpresheaf of M [ T I with M c n(R),
then ( a , ' ( N ) ) *
=
a,
(N,).
128
A. Some General Techniques in the Theory of Graded Rings
Proof. 1. Because o f t h e f o r e g o i n g , b o t h ( p ( N ) ) * and $ ' ( N * )
a r e i n g u(R [ T I )
and t h e r e f o r e i t w i l l be s u f f i c i e n t t o e s t a b l i s h isomorphisms between t h e s t a l k s , and t o t h i s end i t w i l l be s u f f i c i e n t t o e s t a b l i s h t h a t ( N * I X = (NX)*. t h e r e i s a neiqhbourhood o f x, U say,
Given a f h((N*),), y F h($(U))
r e p r e s e n t i n q a, say y = y-,
degree i. D e f i n e f : (N*)x
U
Mx(Y-,)
Tmtn+p
+ .. . +
+
+ ... +
Tmtn+P
y,
and an e l e m e n t Tp w i t h yi
c hf(U)
(hix)* by s e n d i n a a t o
U Mx(yn) Tp. One e a s i l y checks t h a t f does n o t
depend on t h e c h o i c e o f U and y . If
f(a)
= 0
U t h e n Mx(yi)
= 0 f o r a l l i ; hence
U neighbourhood W c U o f x such t h a t MW(yi)
=
-s
there
a
0 f o r a l l i . R e p l a c i n g U by
W i n t h e d e f i n i t i o n o f f, g i v e s a = 0 and t h u s f i s i n j e c t i v e .
I f on t h e o t h e r hand y F h(Nx)*) t h e n y i s o f t h e f o r m y = T " ( y ' ) * f o r some y l F Nx, say y ' = y ' -
+ . .. +
each i t h e r e i s a neiqhbourhood Ui
y',
where y,' i s o f deqree i. F o r
o f x and an xi
U-m P u t U = fli Ui and c o n s i d e r b ' = MU (y-J b = TP(b')*
F h(N*(U))
t...t
F h(N(Ui))
'Jn
MU
(y,)
f
X. F o r an a f (N,)x
then
y i e l d s a preimaqe f o r y .
2. Again i t w i l l s u f f i c e t o p r o v e t h e isomorphism ( N * ) x
x
r e p r e s e n t i n g yi.
t h e r e i s a neighbourhood
Z' (Nx)* f o r each
U o f x and a
y F N,(U)
r e p r e s n t i n g a. P i c k z F h ( N ( U ) ) such t h a t z* = y and d e f i n e f : (N*)x by f ( a
=
U (Nx(z)),
.
+
I t i s e a s i l y v e r i f i e d t h a t t h i s d e f i n i t i o n i s independent
o f t h e c h o i c e s made. If f(a
=
U 0 t h e n N x ( z ) = 0 and i t f o l l o w s t h a t z, hence y, r e p r e s e n t s
zero, i . e . a = 0 and f i s i n j e c t i v e . On
t h e o t h e r hand, i f y ' F (Nx)* t h e n y l = z ' * f o r some z ' E h(Nx). P i c k
a v c h(N(U)) r e p r e s e n t i n g z and p u t a = (N,)x f i s surjective.
(Nx)*
U
( v ~ ) ;t h e n f ( a ) = y, hence
129
II. Some General Techniques
The above theory can be modified so as t o o b t a i n a s i m i l a r theory in t h e general case o f Giraud s u b c a t e q o r i e s of R-mod and correspondinq s u b c a t e q o r i e s of R-gr. The obvious a p p l i c a t i o n o f the above techniques w i l l be the cases where X i s Spec R o r P r o j R [ T 1 ; a c t u a l l y i f R i s p o s i t i v e l y q r a ded then P r o j ( R [ T ] 2 P r o j ( R ) U Spec(R), where Spec(R) i s then homomorphic
t o t h e open s u b s e t of P r o j R d e s c r i b e d by- the i d e a l aenerated
by T .
130
A. Some General Techniques in the Theory of Graded Rings
11.9. Torsion Theories over Graded Rings.
Let us r e c a l l some d e f i n i t i o n s and basic elements of oeneral torsion theory. Consider a Grothendieck cateqory b (more oeneral one may consider abelian c a t e g o r i e s ) . A torsion theory
in b i s a couple ( T , F ) of nonvoid classes
of objects of b such t h a t the followinq properties h o l d .
n
F
, 0 denites the i n i t i a l object f o r b.
TT 1.
: T
_ TT_ 2 .
: T i s closed under homomorphic images in b.
__ TT 3.
: F i s closed under taking subobjects i n b.
__ TT 4.
:
{O}
=
For an M
f
b t h e r e i s a subobject t(M) o f M
such t h a t t ( M ) F T
, M / t ( M ) c F.
A torsion theory i s said t o be hereditary i f
T i s closed under takina
subobjects as well. We will only use hereditary torsion theories here. A kernel functor on b i s a l e f t exact subfunctor of the i d e n t i t y in b. A kernel functor
for M
K
i s s a i d t o be idempotent
i f and
only i f K(M/K(M))=O
c b.
Elementary r e s u l t s i n torsion theory y i e l d t h a t there i s a one-to-one correspondence between idempotent kernel torsion theories i n b
.
functors
K
on b and hereditary
These a l s o correspond b i j e c t i v e l y t o idempotent
Gabriel- f i l t e r s ( o r , -topoloqies) i n some chosen generator fi f o r b. Such a G a b r i e l - f i l t e r in G will then consist of the subobjects L of G such t h a t G/L
<
T i . e . K ( G / L ) = G/L f o r the correspondino kernel functor.
Indeed, the torsion c l a s s of a torsion theory associated kernel functor
K
i s given as the c l a s s o f M
while M c F i f and only i f K ( M )
=
c b
t o an idempotent
such t h a t K ( M )
=
M,
0. For more d e t a i l about torsion theory
c f . [ 321, [ 36 1 , [ 40 1 . Notation
.
I n the sequel , ( T , F ) will always be a hereditary torsion
II. Some General Techniques
theory in band
K
131
will be the correspondina idempotent kernel functor o n b
Evidently we wish t o apply these techniques t o b
=
R-gr in case R i s a
graded ring of type G . Since R-qr i s indeed a Grothendieck cateqory t h i s presents no fundamental problems. However R i s
n o t a oenerator f o r
R-gr so the theory developped f o r R-mod does not completely aeneralise
t o the graded case. Moreover we will encounter (solvable) problems in r e l a t i n g torsion theories in R-at- and torsion theories in R-mod. 11.9.1. Lemma.
Let R be a qraded ring of type G ,
and l e t ( T , F ) be a
torsion theory in R-gr. The following statements are equivalent :
1. If M c T then M(u)
c T f o r every
u C 6
2. I f N c F then N ( T )
c F f o r every
T
Proof.
c G.
Straightforward
11.9.2. Example.
Let R be a graded rina of type Z which i s positively
graded. Consider f o r T the c l a s s of M
c R-gr
such t h a t Mi = 0 i f
and take F t o be the c l a s s of M c R-ar such t h a t Mi = 0 i f i
3
i
<
0
0. I t i s
e a s i l y v e r i f i e d t h a t (T,F) i s a hereditary torsion theory which does not have the properties mentioned in Lemma 11.9.1.. On the other hand i t i s easy t o cjenerate torsion theories wich do s a t i s f y these properties by addinq t o T a l l objects obtained as u suspensions
objects with respect t o a l l
u
O f
the Original
c G.
A torsion theory s a t i s f y i n g the equivalent conditions of Lemma 11.9.1.
i s said t o be a r i g i d torsion theory. A kernel functor i s said t o be i f the correspondinq torsion theory i s . For any torsion (T,F) in
R-qr
theory
we have t h a t HomR-gr(M,N) = 0 f o r every M c T , N c F.
However i t should be noted t h a t H O M R ( M , N )
= 0 f o r a l l M 6 T, N 6 F i s
exactly equivalent t o ( T , F ) being a r i g i d torsion theory.
.
132
A . Some General Techniques in the Theory of Graded Rings
1 1 . 9 . 3 . Lemma.
A kernel functor
K(M(u))
= K(M)(u)
Proof.
If
on R-gr i s r i q i d i f a n d only i f
K
f o r a l l M c R-gr, u c G .
i s r i a i d then K(M)(u)c (M(u)) i s obvious. On the other
K
hand ( M / K ( M ) ) ( u )
=
EI(u)/K(M)(u)
i s x-torsion f r e e i . e . i n F. Therefore
K(M(o)/K(M)(cT)) = 0 . Conversely, i f M c T then K ( M ) =
M(u)
follows. Thus M ( u )
A non-empty s e t
graded f i l t e r __ F.l.
If L
=
F1 i . e . K ( M ( c J ) )
=
c T f o r a l l u C G.
of qraded l e f t i d e a l s of R , say 6 , i s said t o be a i f the followinq properties h o l d :
F 6
and L1 i s a qraded l e f t ideal of R such
t h a t L c L1 then
L1 c 6.
n L2 c
F.2.
If L1L2 c L then L1
__ F.3.
If L
F.4.
I f L1 c 6 and [ L : x ] c 1 f o r a l l x c h ( L 1 ) then L c 6 .
<
d: then ( L : x ]
1.
e L for all x c
L(R).
Since R i s not a qenerator f o r R-ar we cannot
expect t h a t graded f i l t e r s
i n R a n d ( h e r e d i t a r y ) torsion theories i n R-ar correspond b i j e c t i v e l y . However we do have the following : 11.9.4. Lemma.
sets ( ! ) -
There i s a b i j e c t i v e correspondence between the following
:
1. (Hereditary) r i g i d torsion theories i n R-gr. 2 . Rigid (idempotent) kernel functors on R-gr.
3. Graded f i l t e r s i n R . Proof.
A t r i v i a l modification o f the proof of the s i m i l a r statement
in the ungraded cases. The sequel
of t h i s section i s concerned with the characterization of
kernel functors on R-gr which can be induced by kernel functors on R-mod.
An idempotent kernel functor K_ on R-mod i s said
t o be
graded i f
the
I I . Some General Techniques
filter
133
t(5) o f 5 possesses a c o f i n a l s e t o f graded l e f t i d e a l s .
11.9.5.
Lemma.
L e t R a qraded r i n a o f tyDe G and l e t 5 be a araded
r(M) and M/k(M) a r e araded morphism M -M/K _ -( M -) may be c o n s i d e r e d
k e r n e l f u n c t o r on R-mod. I f M F R-qr t h e n l e f t R-modules and t h e c a n o n i c a l
+
as a graded morphism o f degree z e r o . F u r t h e r m o r e , functor
K
induces a kernel
on R-gr w i c h i s r i g i d and i d e m p o t e n t . The t o r s i o n c l a s s
T(K) consists o f
N
N C R-gr such t h a t Iff ( K )
5
t R-gr such t h a t
5(N)
=
r\l
whereas F ( K ) c o n s i s t s o f
~(1) = 0.
i s t h e graded f i l t e r o f
n
i n R t h e n we have : S ( K ) = L ( R ) 4
K
L (-K ) ,
where L ( R ) denotes t h e s e t o f graded l e f t i d e a l s o f R. 9 Proof.
I f we e s t a b l i s h t h a t K_(!)
t h a t -M/K_
(y)
i s graded and t h a t
i s graded t h e n i t w i l l +
!/K(!)
may be viewed as a araded
r(M) L = z
morphism o f degree z e r o . Since'K- i s qraded, x F f o r some L F Z ( K ) f o r each u T
c G
n
L (R). q
Write
c
x =
ocF
y i e l d s LuxT = 0 f o r a l l
6 G and hence xT
F
K(M)
for all
T
f o l l o w immediately
U,T
xu,
e n t a i l s t h a t Lx = 0 Lo. Then L
x = 0
OFG
F F. Thus L x
E C-, i . e .
?(M)
= 0 for all
is a graded
submodule o f M , w h i c h we w i l l t h e r e f o r e denote by ~ ( 1 1 ) . F u r t h e r m o r e , i f
T ( K ) and F ( K ) a r e as s t a t e d t h e n i t i s s t r a i g h t f o r w a r d t o v e r i f y T . T . l . t i l l T.T.4.
and so ( T ( K ) , F ( K ) ) d e f i n e s a t o r s i o n t h e o r y i n R-qr w h i c h
i s e a s i l y seen t o
be r i a i d . The k e r n e l f u n c t o r
K
corresponding t o t h i s
t o r s i o n t h e o r y i s a r i q i d k e r n e l f u n c t o r on R-or, h o l d s f o r e v e r y M c R-qr. The s t a t e m e n t
such t h a t
K
(M)
=
5(M)
S ( K ) = Lq(R) fl Z ( 5 ) f o l l o w s
e a s i l y from t h i s . 11.9.6.
Theorem.
correspondence,
L e t R be a araded r i n g o f t y p e
5
ft
K ,
between :
5
1" g r a d e d k e r n e l f u n c t o r s 2" r i g i d kernel f u n c t o r s
K
on R-mod.
on R-qr.
G.
There i s a b i j e c t i v e
134
A. Some General Techniques in the Theory of Graded Rings
Proof.
Consider a r i o i d kernel functor
graded f i l t e r in R . We define l l
= {
K
on R - g r a n d l e t I ( K ) be i t s
L a l e f t ideal of R , L
q
c l ( ~}, )
a n d i t i s straightforward t o check t h a t L 1 i s a f i l t e r . Let K~ be the
kernel functor on R-mod associated t o l l . By the d e f i n i t i o n of I 1 i t i s c l e a r t h a t l 1 i t i s c l e a r t h a t f 1 n L ( R ) = L ( K ) . We claim t h a t q
k1(M) =
K
(M).
Indeed, f o r a r i g i d
kernel functor, l i k e
equivalent t o a n n R x c I(%), thus K(M) c
K ,
~ ~ ( 3Conversely, ).
x c K(M) is if x c
~~(3)
c l1 i . e . (ann x ) c I ( K ) Writing, . x = x +.. .+xc f o r R g 9 n the homoqeneous decomposition of x , we have a xo, = 0 f o r a l l i = l , . . . , n , then
annR x
f o r every homoaeneous a c h ( a n n R x ) . Hence xo, c
1 K
( M ) and thus x c K(M)
1
Since L l obviously has a cofinal s e t of qraded l e f t i d e a l s , i t i s obvious from Lemma 11.9.5. t h a t we may write K~ = K , i t induces R-gr, and so the correspondence K
++
K
on
i s indeed b i j e c t i v e .
As in any Grothendieck cateqory, we define,
cogenerated i n R-gr by a n M E R-gr
K
the torsion theory
by puttinq T ( K ~ )equal t o the c l a s s
of N c R-gr such t h a t Hom (N, E g ( M ) ) = 0 , where E q ( M ) i s the i n j e c t i v e R-gr h u l l of M in R-gr. I f K # i s r i q i d then N C T ( K ~ e) n t a i l s H O M R ( N , E g ( M ) ) = O o r , Horn ( N , 8 E g ( M ) ( , ) ) R-gr u ~ G
=
0.
Hence the r i q i d kernel functors amonqst the
K~
a r e associated t o the
torsion theories cogenerated by a suspension-invariant M F R-qr. Let
t o the torsion theory on R-qr
K ~ ( u denote ) the kernel functor associated
cogenerated by M(o),
u
c G. Recall t h a t
K
=
inf { K ~ , i E I } i s by
d e f i n i t i o n the torsion theory obtained by takinq f o r Of the T ( K ~ ) i, F I . We will w r i t e
K
= A { K ~ ,i F
J(K)
.the i n t e r s e c t i o n
I } i n t h i s case. With
t h i s notation i t i s then c l e a r t h a t A { K ~ ( u )u, c G} i s a r i g i d kernel functor which corresponds t o the torsion theory cogenerated by E9( 8 M ( u ) ) u6G
135
II. Some General Techniques
We define
K
1z
as follows :
via the l a t t i c e of hereditary torsion theories i n R-or
K~
5
K~
i f and only i f T ( K ~c) T ( K ~ ) I. t follows
K~
r i g i d torsion theory K
5 A
u F G}.
{KM(u),
K~
-
K~~
M
C
R-ar, i f and only i f
{ K ~ ( u )5,
F G } will be
K;.
Let R be a qraded ring of tyoe G and consider M t R-qr.
11.9.3. Theorem.
Let
5
The kernel functor A
R-gr; i t will be denoted by
6
K
kernel functor associated (or cooenerated by) t o
called the r i g i d
M
satisfies
K
that a
fi F
be the kernel functor on R-mod coaenerated by
R-mod, then
1. Z ( K ~ =) { J , l e f t ideal of R such t h a t t h a t f o r a l l 0 # x c M and a c R , (3:a) 4 annR x . } 2. Z
r
( K ~ )=
(L:a) 3. d:(rcM)
{
L F L ( R ) such t h a t f o r a l l 0 # x
Q annRx}. n L (R)
=
9
z(K;).
.
4 4 . If X F R-gr then K L ( X ) =
~~(5).
Proof.
we have :
-
_
By d e f i n i t i o n o f
L : ( K ~ )= { L F L ( R ) ,
9
_ K;
t h(M) and a C h(R),
-
H o m R ( R / L , Eg(M)) = 0 }
Let L c l ( x i ) and suppose there e x i s t s x # 0 in h(M) toaether with
an a t h ( R ) such t h a t ( L : a ) c a n n R x . Since (L:a)
6 L(K$)
i t then
follows t h a t annRx c Z ( K ~ ) .The l a t t e r e n t a i l s the existence of a nonzero
f in HOMR(R/annRx, E g ( M ) ) and t h i s contradicts R/annRx c T(K;).
Conversely, i f H O M R ( R / L ,
Eq(M)) # 0 f o r some L t L,(R)
havinq the
property t h a t for a l l x # 0 i n h(M) and a l l a c h ( R ) , (L:a) then there e x i s t s a graded morphism f : R/L
E'(M)
F
h ( R ) such t h a t x
=
f(a)
annRx,
which i s not the
i d gr-essential in Eq(M) there e x i s t s a n
zero map. Since M represented by a
--f
4
a
c R/L,
# 0 i s in h ( l 1 ) . Clearly
(L:a) c annRx, a contradiction. This proves 2 and t h a t L ( K ~ )i s as s t a t e d i n 1. i s well known. Further, the inclusion d : ( ~ n~ L) a ( R ) c i s obvious.
~(KL)
136
A. Some General Techniques in the Theory of Graded Rings
For the converse inclusion consider L c Decompose x as xo
... +
+
1
c 6 , j = 1,..., m. I f x
T.
J
X1 all
6
L
. ..
h2 X1,
c
ui c G , i = 1, ...,n, and a = a
n # 0 then there
-~
Since X/ K;(M)
X1 aT
$,,...A2
x2 e
m
h ( R ) such X2X1 a
T2
m
c L
c L and j,... X2 x1 xn # 0. P u t
then X x # 0 while h a F L , hence (L:a)
(5); l e t
K~
~
+...+aZ
T1 i s a X1 c h ( R ) such t h a t
Q annRx i . e .
proves 3 . An immediate consequence of 3. i s t h a t
C d : ( ~ ~ )This .
r (X) K~
x # 0 in M and a c R .
# 0. Using a recurrence argument we end up with a n element
n
X,,, 6 h ( R ) such t h a t X
On
,
(K;),
1 and X1 x on # 0 . Similarly, there i s a
and X2X1xo
=
xo
d:
us prove the converse.
i s Kh-torsion f r e e there e x i s t s a
monomorphi sm
n
( II E 4 ( M ) ( o ) ) ( d i r e c t products in R-qr). UCG Since Eg(M)(u) c E(M) f o r a l l u c G , there i s thus a monomorphism of
X/K;(X)
+
icl
X / K ~ ( C L ) i n t o a product of copies of E ( M ) . Consequently X / K L ( X ) i s
torsion f r e e . Hence 11.9.10. Remark.
K,(:)
-
=
-
-.
K ~ ( X ) .
___
I n general f o r a r b i t r a r y PI c R-gr,
kernel functor. Let us now t r y t o bring
K~
K~
-
i s n o t a graded
t h i s a b s t r a c t theory down to e a rth by constructinq
some expl i c i t e examples of we1 1 behaved
t o r s i on theories in R-ar .
Consider a m u l t i pl i cat i v el y closed subset S of R . To S we associate a kernel functor L ( K ~ )=
{
L
~ t _on~
R-mod given by i t s f i l t e r
“3)where
:
l e f t ideal of R such t h a t (L:r) f‘ S # 0 f o r a l l r c R 1 .
11.9.11. Proposition.
Let R be a graded ring of type G ,
S a m ultipli-
c a t i v e l y closed subset of h ( R ) n o t containinq zero. Then K~ i s a araded kernel functor on R-mod, i . e . t o S there al s o corresponds a r i o i d
137
II. Some General Techniques
kernel f u n c t o r
on R-gr with qraded f i l t e r by :
K~
f(xS) = { L F Lg(R),
f o r a l l r c h ( R ) }.
(L:r) 0 S # @
f ( R ) t h i s y i e l d s t h a t s r c L f o r some s c S ,
hence s r
( L g : r ) n S # 0. I f r i s not homogeneous, w r i t e r = r Pick s n F ( L : ro )
n
s1
6 (L:sz...S
and
I f r i s in
Take L c f(5,) i . e . (L:r) n S # @ f o r a l l r F R .
Proof.
r
Ol
n S , s ~ c -( L ~: s n r o
) n- 1
F
u1
n S ,...,
Lo and
+ ... + r . %
) n S . I f we p u t s equal t o s l s 2 . . . ~ n then s r
e L9
s F S i . e . Lg F f ( ~ ~ )o , r i n o t h e r words F~ i s graded. The o t h e r
s t a t e m e n t s follows d i r e c t l y from Theorem 11.9.7. 11.9.12. Remark. I f S s a t i s f i e s _______
K ~ ( M )=
{
m c M, s m
=
t h e l e f t Ore c o n d i t i o n s then
0 f o r some s F S } f o r any 14 C R-qr.
Let us now i n t r o d u c e t h e gradea
Lambek t o r s i o n t h e o r y , i t i s t h e
t o r s i o n theory i n R-gr
coaenerated by R i n R-or, i . e . we s e t
T = { M F R-gr, HOMR(M,
Eg(R))
=
0
}
.
I t s graded f i l t e r i s qiven by
Q annRb f o r a l l a , b 6 h ( R ) , b # 0 3 . 9 This d e f i n i t i o n p a r a l l e l s t h e d e f i n i t i o n of Lambek's t o r s i o n theory
f = {
L F L ( R ) , (L:a)
i n R-mod which i s t h e t o r s i o n theory cogenerated by R i n R-mod. 5
11.9.13. Note.
Let R be a q r - f i e l d of type Z which i s
graded. The kernel f u n c t o r a s s o c i a t e d t o
not t r i v i a l l y
Lambek's t o r s i o n theory in
R-mod i s n o t a graded kernel f u n c t o r . Now l e t R be a qraded r i n q of type F where G i s M c R-gr d e f i n e t h e s i n g u l a r
an ordered qroup. For
submodule Z ( M ) of M t o be { m F M , annRx
i s an e s s e n t i a l l e f t i d e a l of R 1 . 11.9.14. Lemma.
Let R be a araded r i n g of type F, G beino an ordered
group . I f M c R-gr then Z(M) i s a araded R-submodule of M .
A. Some General Techniques in the Thcrory of Graded Rings
138
+...+
P i c k x # 0 i n Z(M) and decompose x as x
Proof.
assuming x
# 0 for i
ui
x
w i t h u <...a 1 n' "n P u t L = annRx, t h e n L-c annR xu . n
,... ,n.
= 1
S i n c e L i s an e s s e n t i a l l e f t i d e a l , L- i s an e s s e n t i a l l e f t i d e a l because o f Lemma 11.7.6. B u t t h e r e annRxu x
n
C
n
i s a n e s s e n t i a l l e f t i d e a l , hence
Z(M) f o l l o w s . The lemma t h e n f o l l o w s by r e c u r r e n c e on n.
I t may e a s i l y be checked t h a t Z d e f i n e s a r i o i d p r e r a d i c a l ( t e r m i n o l o o y
o f [ l o 2 1 ) on R-gr. The a s s o c i a t e d r a d i c a l ZG s a t i s f i e s ZG(M)/Z(M) = Z(M/Z(M)).
To a r a d i c a l t h e r e corresponds an i d e m p o t e n t k e r n e l
c f . [ 36 1 . The k e r n e l f u n c t o r
functor,
=
r a d i c a l ZG i s r i q i d and we r e f e r t o
K~
K&
on R-qr c o r r e s p o n d i n o t o t h e
as b e i n q G o l d i e ' s qraded k e r n e l
f u n c t o r . The f o l l o w i n g i s an i n t e r e s t i n g a p p l i c a t i o n o f t h i s c o n s t r u c t i o n . 11.9.15. ZA
Proposition.
( M [ X I ) = ZA(M) [ X I ,
ZA [ x , x - 1 ] Proof.
I E A-mod. L e t A be an a r b i t r a r y r i n q , 4
(M [ X , X - l I
then
and a l s o :
.
= ZA(M) [ X , X - ' I
L e t us s k e t c h t h e p r o o f o f t h e f i r s t s t a t e m e n t , t h e second
s t a t e m e n t f o l l o w s i n e x a c t l y t h e same way. Proof.
w i t h m # 0.
Consider m Xn 6 ZA(M) [ X I
f o l l o w s t h a t annRm i s an e s s e n t i a l
l e f t i d e a l o f R, e n t a i l i n q t h a t
(annRm) [ X l i s an e s s e n t i a l l e f t i d e a l o f A [ X I = (annAm) [ X i ]
,
t h u s m Xn 6 ZAlx1
i s c o n t a i n e d i n ZA
S i n c e m 6 ZA(M) i t
.
B u t ann
A [XI
xn =
( M [ X I ) and c o n s e q u e n t l y ZA(M) [ X I
(M [ X I ) . I n v i e w o f Lemma 11.9.14.
t h e converse
i n c l u s i o n w i l l f o l l o w i f we show t h a t each homogeneous element o f
zA r X I ( ~ [ x q ) t h e n annA
i s i n z ~ ( M )I X ~. I f m
XlmXn
immediately t h a t
xn
i s in
zA
r x l ( ~[XI),
= (annAm) [ X I and t h e r e f o r e i t f o l l o w s
(annAm) [ X ]
i s an e s s e n t i a l l e f t i d e a l o f A [ X I
and t h a t annAm i s an e s s e n t i a l l e f t i d e a l o f A i . e . m F ZA(M) and
139
II. Some General Techniques
m
xn
F z*(M) [
XI
.
Yet another example of a torsion theory in R-or i s obtained i f one l e t s ( T , F ) be the torsion theory in R-qr aenerated by the c l a s s of simple
objects in R-gr. I t i s an easy exercise a
t o verify t h a t t h i s defines
r i g i d torsion theory.
Recall t h a t a torsion theory i n R-mod i s
symmetric
i f i t s associated
f i l t e r i n R allows a cofinal system of ideals of R . Similarly we now say t h a t a r i g i d kernel functor
(or r i g i d torsion
K
i s symmetric ( i n R-gr) i t i t s graded
theory) on R-qr
f i l t e r has a cofinal s e t consisting
of graded ideals of R . 11.9.16. Lemma. functor
rinq of type 6 . Then a kernel
on R-gr i s symmetric i f and only i f K_ on R-mod i s symmetric.
i s symmetric. I f L F L ( K ) then L < f ( ~ ) hence , 9 contains an ideal I of R such t h a t I c f ( ~ ) B.u t then a l s o I < f ( K )
Proof.
L
K
Let R be a graded
Suppose t h a t
K
9 Suppose, conversely, t h a t
-
i s symmetric. If L F I ( K ) then L 6 f ( 5 )
K
and thus there e x i s t s an ideal J o f R such t h a t J c L and J
B u t then J
i s an ideal of
9 Since L contains J
g
6
f(y).
R and Ja F f ( ~ by) Theorem 11.9.7.
as well, the
lemma follows.
Let R be a l e f t Noetherian rinq an l e t 5 be a kernel functor on R-mod.
TO K we may associate
a symmetric kernel functor 5'
f i l t e r : L ( K-' ) = { L l e f t ideal of R such t h a t L and I
c
L
(K )
R-mod
is
I , I an ideal o f R
}.
Since R i s l e f t Noetherian the above does indeed and one may verify t h a t 5'
3
given by the
defi,le a f i l t e r in R
the l a r g e s t symmetric kernel functor on
which i s smaller than K.
11.9.17. Proposition.
Let R be a n l e f t Noetherian qraded ring of
type G.. Let 5 be a graded kernel functor on R-mod introducing the
140
A. Some General Techniques in the Theory of Graded Rings
r i g i d kernel f u n c t o r R-mod and t h e r i g i d
K
K O
on R-gr. T h e n induced
f u n c t o r on R-gr s m a l l e r than Proof.
KO
i s a qraded kernel f u n c t o r on
on R-gr i s t h e l a r o e s t symmetric kernel (what j u s t i f i e s t h e n o t a t i o n
K
KO).
Straightforward.
Concluding t h i s s e c t i o n we c h a r a c t e r i s e t o r s i o n
t h e o r i e s in R-gr i n
case R i s s t r o n g l y graded of type G . Let us i n t r o d u c e the f o l l o w i n q . I f R i s a qraded rinq of type I; then a kernel
11.9.18. D e f i n i t i o n .
f u n c t o r y on R -mod i s I;-stable, o r s t a b l e under conjugation, e y ( R o 60 M) = Ro QO y(M) f o r every M f Re-mod.
Re 11.9.19.
Re Proposition.
if
Let R be a s t r o n g l y qraded r i n g of type G , then :
1. There i s a b i j e c t i v e correspondence between (idempotent) kernel f u n c t o r s
on R-gr and (idempotent) kernel f u n c t o r s on Re-mod. 2 . In t h e b i j e c t i v e correspondence of 1. the r i q i d kernel f u n c t o r s on
R-gr correspond bi j e c t i v e l y t o the G-stable kernel f u n c t o r s on Re-mod. Proof.
To an idempotent kernel f u n c t o r
functor
K
e
: Re-mod
--f
R -mod given by e
K
By Theorem I . 3 . 4 . , one e a s i l y s e e s t h a t
K
on R-@rwe a s s o c i a t e the
e( X ) K~
=
(K(R
i s an
60
X))e,
f o r X f R-mod.
Re idempotent kernel f u n c t o r .
Using t h e same theorem one e a s i l y s e e s t h a t t o an idempotent kernel f u n c t o r K'
on R-mod one may a s s o c i a t e
f o r a l l M f R-gr.
K
on R-gr defined by ~ ( 1 1 )= R
60
Re
K'
(Me)
141
II. Some General Techniques
11.10. Graded Rings and Modules of Quotients.
cateaory may be
The general theory of l o cal i zat i o n i n a Grothendieck applied so as t o obtain objects of quotients
with respect to kernel
functors. I n the case o f R-pr the i n t er es t i n g point i s t h a t the objects of quotients with respect t o r i q i d kernel functors r e l a t e in a very d i r e c t sense t o c e r t ai n objects of quotients constructed in R-mod. The exposition of t h i s l i n k between l o cal i zat i o n i n R-mod and loc a liz a tion in R-gr i s the subject of t h i s section.
Some of the r e s u l t s are
r e a l l y i n t r i n s i c f o r the qraded s i t u a t i o n , the inte re ste d reader may
on t h i s point by comparing these r e s u l t s t o aeneral
convince himself
r e s u l t s concerning r e f l e c t o r s , c f . [ 1231, o r r e l a t i v e loc a liz a tion, c f . [ 1 2 2 ] , or the use of the Gabriel-Popescu theorem as in [ 1 2 7 1 . Throughout t h i s section R wi17 be a graded an idempotent kernel functor ( T , F ) and
K
will be
on R-gr with associated torsion theory
with associated graded f i l t e r L ( K ) in R . the object of quotients in R-gr with respect
I f M c R-gr, we define
to
ring of type G ,
( M ) = li+m H O M R ( L , M / ~ ( M ) ) . L C a: (K) As i n the unqraded s i t u a t i o n , one e a s i l y es t ab l ishe s t h a t Q: ( R ) i s in K
of M , t o be the graded R-module
Q:
a natural way a graded rinq o f type 6 whereas f o r every M Q:(M)
turns out t o be
a graded Q:(R)-module
i . e . Q:(M)
Let K_ be the graded kernel functor on R-mod inducing Theorem 1 1 . 9 . 6 . ) . l o c a l i z a t i o n of
4
K
R-w,
Q!(R)-qr.
on R-gr (see
Then f o r every M c R-gr we a l s o have 0, ( P I ) , the at i n R-mod. Let us polish up our memory and
r e c a l l t h a t K_ ( r es p . L c L ( K _ ) ( r e sp .
C
C
K )
i s said t o be of f i n i t e type
L ( K ) ) there
i f f o r every
e x i s t s an L ' c L ( 5 ) (re sp. L ( K ) ) such t h a t
L ' c L and L ' i s f i n i t e l y generated in R-mod ( r esp. in R - g r ) .
A. Some General Techniques in the Theory of Graded R jngs
142
11.10.2. P r o p o s i t i o n .
Let
on R-mod. I f M F R-qr then Proof.
By d e f i n i t i o n :
be a graded k e r n e l f u n c t o r
K
:
Q:(M)
Q:
(MI
of f i n i t e type
QK@)
-
HOMR (L, M / K ( M ) ) .
= LFE ( K )
But, Q:(M)
since K , hence
L EE ( K )
'
li+m
=
has f i n i t e type, we o b t a i n :
HOMR(L, M/ K ( M ) )
li+m
=
K ,
HornR (L,
M/ .(I7))
LFE(K) Where
e ( ~ ) i 's
t h e subset o f
f(K)
COnsistinq
t h e l e f t i d e a l s of
Of
f i n i t e type. Therefore, as unqraded modules : Q:(M)
Q,
-
(M)
11.10.3. C o r o l l a r y . L e t R be a qraded r i n g of tYDe G such l e f t gr-Noetherian and consider M
R-qr. Then @(M)
every graded k e r n e l f u n c t o r K On R-mod.
-
_2
a,(!)
that R i s holds f o r
II. Some General Techniques
graded o b j e c t s o f q u o t i e n t s . L e t K
be a araded k e r n e l f u n c t o r on R-mod
and c o n s i d e r M C R-ar. F o r each u F G NG
{ x E Q,
=
-
for all
(tj) , t h e r e i s L T
F J ( K ) such t h a t LT x c ( M /K(M))~~
E G }.
t h e a d d i t i v e groups o f
QK(F).The -
can f i n d a r e l a t i o n :
x
t h e n we a l s o f i n d Li
=
x
“2
t
sum 2 No i s d i r e c t . Indeed i f we UCG
... +
i
nn
Li,
... #
u
E G
... , n .
1
i=1
#
xG w i t h o1 # n
F L ( K ) such t h a t :
x D c ( M / K _ ( M ) ) ~ ~,, i = 1,
Take L =
put
(M) i s i n d e e d a qraded module). The No a r e subqroups o f
(Note t h a t M/K
(Li)T
143
t h e n on one hand L
c ( M / E ( M ) ) ~ ~ b u t on t h e o t h e r
1
However, ( M / K ( M ) ) ~ n~ ~( , . Z ~ ( M / ~ ( M ) =) ~0,~ , 1 =2 1 = 0 for all
T
F
S i n c e L c L ( K ) t h i s would i m p l y x
6 , hence L x
u1
E
=
(M/K(M))
0. = 0
F u r t h e r m o r e , i f a F R X and x F N o t h e n a x F NAG. So,
0
GFG
No i s a graded R-module; we w i l l denote i t by
11.10.4.
gQ,(M). -
L e t R be a qraded r i n q o f t y p e G and l e t
Proposition.
K -
be a
graded k e r n e l f u n c t o r on R-mod. Then q QK(R) i s a araded r i n a c o n t a i n i n q
-
R/K-( R ) as a graded s u b r i n o .
The qraded r i n g s t r u c t u r e o f o
QK
(R) i s t h e
-
u n i q u e r i n g s t u c t u r e c o m p a t i b l e w i t h i t s graded R-module s t r u c t u r e . For every
M
f R-gr,
g Q,
(E)i s
-
a graded q
P r o o f . I t i s f a i r l y o b v i o u s t h a t gQJR)
Qk
(R)-module.
-
i s a graded l e f t R-module con-
-
t a i n e d i n Q, (R) and c o n t a i n i n g R&(R) x, y
c
-
gQ,(R)
-
and c o n s i d e r xy
as a graded R-submodule. P i c k
c QK(R) ( t h i s i s a r i n g by a e n e r a l -
l o c a l i z a t i o n t h e o r y i n R-mod) w i t h o u t l o s s o f o e n e r a l i t y we may assume
144
A. Some General Techniques in the Theory of Graded Rings
x 6 ( g QK(R))o, y F ( g Q K ( R ) ) T f o r some
in
R/c(R) and
put L =
(5
: x ) n~
such t h a t a x
c 5 1.
where y i s a
r e p r e s e n t a t i v e f o r bx 6 R/K_(R).
We o b t a i n : L g xy c
L e t I, J 6 Z ( K ) be
Jp y c (R/c(R)JpT f o r a l l X,p F G . L e t
such t h a t Ix x c (R/h(R))x,, t h e imaqe o f J
G.
F
O,T
-
-
Then L
<
Indeed
f ( K ) .
(R/Y(R))~.,
,
I, i.e.
5
be
L = 1 a F I
i f b 6 I t h e n (L:b)
3
(J:y)
and t h u s xy c ( n Q K ( R ) ) m .
-
I t f o l l o w s t h a t qQK(R) i s a qraded r i n q and t h i s r i n n s t r u c t u r e i s
-
d e t e r m i n e d by t h e qraded R-module s t r u c t u r e .
Actually this rinq structure
i s i n d u c e d by t h e r i n g s t r u c t u r e o f Q K ( R ) and t h e l a t t e r i s u n i q u e l y -
d e t e r m i n e d by i t s R-module s t r u c t u r e ; c o n s e q u e n t l y t h e r i n o s t r u c t u r e on gGK(R) i s u n i q u e l y d e t e r m i n e d by t h e R-module s t r u c t u r e ( n o t e t h a t -
Q K ( g Q K ( R ) )= Q K ( R ) ) . F o r m a l l y s i m i l a r
- -
-
a r g u m e n t a t i o n y i e l d s t h a t 9QK(M)
-
i s a graded qQK(R)-module as d e s i r e d . -
11.10.5.
Theorem. L e t K- be a qraded k e r n e l f u n c t o r on R-mod and l e t
K
be
t h e r i g i d k e r n e l f u n c t o r i n d u c e d on R-ar. For every
R-qr we have t h a t Q:(M)
M c
2
gQK(F).
-
P r o o f . I t i s w e l l known t h a t : QK
(E)
-
=
Hom (L,EI/K-(M)) =
lim Le S ( K )
li+m
Horn ( L , W K (PI)).
LF 6 (K)
( t h e l a t t e r e q u a l i t y f o l l o w s s i n c e 5 i s qraded). The c o n s t r u c t i o n o f gQ,
(E)
-
i s such t h a t x F (9Q, (bJ))o i f
-
x may be r e p r e s e n t e d by a graded morphism o f degree
U,
and o n l y i f
fx : L
+
M/c(M)
f o r some L F X ( K ) i . e . f x F HOEIR ( L , M / K ( M ) ) ~ . 11.10.6. Moreover degree.
Corollary.
QZ
!Q
i s a c o v a r i a n t l e f t e x a c t e n d o f u n c t o r i n R-ar.
i s f u n c t o r i a l w i t h r e s p e c t t o oraded morphisms o f a r b i t r a r y
II. Some General Techniques
We o n l y have t o v e r i f y t h e second s t a t e m e n t . C o n s i d e r N, M C R-gr
Proof.
and a graded morphism o f degree o E G, f : N
I f x E (Q:(N)),,
i s R linear. that
145
I
CL
+
M. Then Q K ( f ) : QK(N) -
-
--f
QK(M) -
1 E G, t h e n t h e r e e x i s t s an I C f ( ~ )such
x c ( N / K ( N ) ! , ~ f o r a l l v f G. Thus f o r a l l
E G we nave
f(y
(wK(M))pxo.
From f ( I x ) = Q K ( f ) ( I x ) = I LL
-
I-L
Q ( f ) ( x ) we o b t a i n t h a t I v Q K ( f ) ( x ) E ( M / K ( M ) ) ~ , ~
iJ.5
-
f o r a l l p E G. Hence Q , ( f ) ( x )
and i t f o l l o w s t h a t Q K ( fi )s a c t u a l l y a graded
E Q(M : ,)
-
-
morphism o f degree u f r o m Q:(N)
t o Q:(M)
( u s i n g Theorem I I . 1 0 . 5 . ,
Proposition
11.10.2.). We now f o c u s on some p r a c t i c a l p r o p e r t i e s o f graded l o c a l i z a t i o n . 11.10.7.
M, N, S E R-gr
L e t R be a graded r i n g o f t y p e G. L e t
Proposition.
be such t h a t N c M and K ( H / N ) = !I/N f o r some r i g i d k e r n e l f u n c t o r Any graded morphism f o f degree
0
E G, f : N
way t o a graded morphism, h o f degree Let
Proof.
K
U,
h:M
+
--f
Q:(S),
K
on R-gr.
extends i n a unique
Q:(S).
be t h e graded k e r n e l f u n c t o r on R-mod, i n d u c i n g
K
on R-gr
Consider t h e f o l l o w i n g diagram :
S i n c e !-(M/N) M/N, f e x t e n d s t o a u n i q u e R - l i n e a r h : - = K(M/N) = If x
c
M,i
A
some
L c
C(K
c
G t h e n t h e assumption o n M/N e n t a i l s t h a t
) . Hence L
I-L
x
L h(x) = h(L x) = f ( L x) c i-1
thus [;(a
LL
w
t h e r e e x i s t s an i d e a l
c N
and t h e r e f o r e :
Q:(S)vao.
So f o r a c L
LLX
L'
i-1
E L ( K ) such t h a t :
h ( x ) ) c ( S / K ( S ) ) ~ , ~ , ~f o r a l l
e
c G.
L
+
QK(S). -
x c N for
we have a h ( x ) E
Q:(S)pxo,
A. Some General Techniques in the Theory of Graded Rings
146
Denoting L ' L by L " we o b t a i n for a l l
L" c
Z ( K ) such t h a t
c G; hence h ( x ) c Q;(S),,,
T
:L h ( x ) c ( S / K ( S ) ) ~ ~ ~
because o f Theorem 11.10.5.
Thus
h i s as s t a t e d i n t h e p r o p o s i t i o n .
A t this point recall that
M 0
<
QK(M),for -
R-mod, may be c h a r a c t e r i z e d by t h e f o l l o w i n g e x a c t sequence i n R-mod
+
where
M/E(M) E(M)
+
QK(M)
--
K(E(M)/M)
+
0
T l ( y
i s an i n j e c t i v e h u l l of
t h e canoncial epimorphism
M
f o r a kernel f u n c t o r K on R-mod
.
i n R-mod and v(M) : E(M)
--f
E(M)/M
In t h e graded s i t u a t i o n , i f we c o n s i d e r
t h e n we have t h e f o l l o w i n g exact diagram i n R-mod :
C R-gr,
where t h e bottom row may be d e r i v e d from t h e corresponding e x a c t row i n R-gr. With these n o t a t i o n s we have :
11.10.8.
P r o p o s i t i o n . L e t R be a graded r i n g o f type G. L e t
r i g i d kernel f u n c t o r on R-gr and take M E R-gr.
Then, Q:
K
be a
(M) E R-gr
i s c h a r a c t e r i z e d by t h e f o l l o w i n g exact sequence i n R-gr :
where Eg(M) i s an i n j e c t i v e h u l l o f M i n R-gr and ng(M) i s t h e canonical epimorphism E9(M) Proof.
+
Eg(M)/M.
Mere v e r i f i c a t i o n .
With n o t a t i o n s and assumptions as b e f o r e we have :
11.10.9. f : M
Proposition.
+
Q:(R)-linear
Q:(N)
Consider N C R-gr, M C Q:(R)-gr.
I f a map
i s R - l i n e a r and graded o f degree u then i t i s a l s o and
graded o f degree
u.
147
I I . Some General Techniques Proof.
Easy from Theorem 11.10.5.
and t h e c h a r a c t e r i z a t i o n o f gQK.
-
Recall t h e f o l l o w i n g d e f i n i t i o n . A kernel f u n c t o r on R-mod, 5 say, has p r o p e r t y T o r i s a T - f u n c t o r or t h e corresponding t o r s i o n t h e o r y i s s a i d t o be p e r f e c t , ifone o f t h e f o l l o w i n g e q u i v a l e n t c o n d i t i o n s i s fulfilled. i s exact and commutes w i t h d i r e c t sums.
P.T.l.
Q,
P.T.2.
For a l l
P.T.3.
For a l l L c L ( 5 )
-
fi
morphism R P.T.4.
The
6 R-mod,
+
= Q,(R)
a(,)!
-
-
, QK(R) = -
@?.
R
QK(R) jK ( L ) where jKi s t h e canonical
-
-
-
QK(R). -
q u o t i e n t category w i t h r e s p e c t t o K ( i n t h e sense o f [ 3 2
1)
c o i n c i d e s w i t h QK(R)-mod.
-
P.T.5.
Every QK(R)-module i s K - t o r s i o n f r e e as an R-module. -
For more i n f o r m a t i o n concernin'g p e r f e c t t o r s i o n t h e o r i e s c f . [ 36 c o n d i t i o n o f P.T.3.
1.
The
e n t a i l s t h a t any kernel f u n c t o r having p r o p e r t y T
has f i n i t e t y p e . L e t us now i n v e s t i g a t e t h e graded v e r s i o n of t h e above properties. 11.10.10.
Theorem. L e t R be a graded r i n g o f t y p e G. L e t
K
a r i g i d kernel
f u n c t o r on R-gr w i t h associated graded kernel f u n c t o r K on R-mod. The f o l l o w i n g statements are e q u i v a l e n t : jK (L) = Q:(R),
1. For a l l L E L ( K ) , Q:(R)
where j K :R
--f
Q:(R)
i s the
canonical graded l o c a l i z i n g morphism.
M c Q:(R)-mod, 2. For every 3. For every
c Q:(R)-gr,
4. The f u n c t o r s :Q Proof.
?
@
R
-
E Q:(R)-mod
f o r some L c L ( K ) hence Q;(R)(Lx) x = 0, by 1.
= 0
K ( M ) = 0.
and Q:(R)
1 = 2. Take
K(?)
a r e n a t u r a l l y e q u i v a l e n t i n R-gr. and suppose x E = 0 b u t then Q:(R)
k(M). Then
L x = 0
j K ( L ) x = 0 entails
A. Some General Techniques in the Theory of Graded Rings
148
2.
=
3.
5
3. Obvious from Theorem 11.9.6.
4 . Let M c R-gr.
The canonical R - l i n e a r
+
a,(!) -
factorizes as
follows :
Since Q:(M)
the above sequence r e s t r i c t s t o a
i s a Q:(R)-module,
sequence of graded morphisms :
> Q;(R)
M
where ag(m)
=
6O
R
M
-+
pg ( z ’ q i
160 m and
e n t a i l s t h a t : K(Q:(R)
Q,(M) i
M)
60
R
i s c l e a r because Ker a c
=
“h
mi)
=
C ’ 9.m.. The assumptiom 3. 1 1
0 hence Ker aq
5(M);t h u s Ker
oq =
2
K ( M ) . That K ( M )
K ( M ) . Then
morphism of degree e Q:(M)/Img9 that
4.
pg i s
Ker a‘
pq i s a mono-
morphism. Because of P r o p o s i t i o n 11.10.9. pg i s Q!(R)-linear,
hence
Note t h a t p g i s a graded
submodule o f Q:(M).
Im pg i s a graded Q:(R)-
3
c G ! Since Im pg c o n t a i n s M/K ( M ) i t follows t h a t
i s 1:-torsion and a graded Q:(R)-module.
T h u s by 3. we o b t a i n
s u r j e c t i v e , hence an isomorphism i n R-gr.
1. For every L c
we have
“ K )
Q:(L)
=
Q:(R)
because R / L i s
K - t o r s i o n . From 4. i t follows then t h a t :
Q:W
=
=
P~(Q:(R)
11.10.11. D e f i n i t i o n . property T i f Q:
P
R
L ) = Q ~ ( R )j K ( L ) .
A r i q i d kernel f u n c t o r
Let
a s s o c i a t e d kernel f u n c t o r
K K-
be a r i g i d kernel f u n c t o r on R-qr with on R-mod, then t h e following a s s e r t i o n s
are equivalent : K
on R-or i s s a i d t o have
i s e x a c t and commutes w i t h d i r e c t sums.
11.10.12. C o r o l l a r y .
1.
K
has p r o p e r t y T.
2 . 5 has p r o p e r t y T.
I f t h i s i s the case then Q:(M) -
-
=
QK(M)
14s
II. Some General Techniques
~
Proof. 1. = 2 . From the foregoinq theorem we r e t a i n t h a t , f o r a l l L c
&)(K),
L contains a graded l e f t ideal of f i n i t e type L ' which i s s t i l l i n L ( K )
we have : Q:(M)
By Proposition 11.10.2. I f now H Thus Q:(R)
c
.C(K_)
then H
9
j K ( H g ) = Q:(R)
_
c
=
_
a,-(!)
L ( K ) by d e f i n i t i o n of
f o r a l l M < R-gr. K -.
and from the above i t i s then obvious t h a t
Q K ( R ) j K ( H ) = Q K ( R ) follows i . e . K has property T . -
-
-
2. = 1. I f K- has property T then i n p a r t i c u l a r 5 has f i n i t e type and
then Lemma 11.10.1. y i e l d s t h a t Khas f i n i t e type on R-qr. Proposition 11.10.2. then implies t h a t Q:(R) ~t h i s t h a t K has property T.
=
Q K ( R ) and one e a s i l y
deduces from
Some p a r t i c u l a r r e s u l t s concerning l o c a l i z a t i o n of qraded rinqs of type
Z will be given i n Section C . I . l . , a f t e r we have deduced the araded version of Goldie's theorem(s).
150
A. Some General Techniques in the Theory of Graded Rings
11.11. I n j e c t i v e Modules, Prime Kernel Functors and L o c a l i z a t i o n a t Prime I d e a l s , In t h i s s e c t i o n we use the n o t a t i o n s of S e c t i o n 11.9. and i n p a r t i c u l a r Theorem 11.9.7. i s very u s e f u l . For any M F R-qr, E q ( M ) denotes the i n j e c t i v e h u l l of M i n R-or; i t i s c l e a r t h a t Eg(M(o)) = Eq(M)(,) all u
F
G and consequently : Ass M
=
for
Ass ( E g ( M ) ( o ) ) f o r o C G (terminolosy
of 11.7). The followin9 Lemma i s a s l i q h t s t r e n g h t e n i n g of a s t a t e m e n t of S e c t i o n I I . g . ( o n e of t h e remarks 1 1 . 9 . 7 . ) . Let R be a qraded r i n g of type G .
11.11.1. Lemma.
c R-gr i s s h i f t i n v a r i a n t i . e .
If M
r i g i d . Every r i g i d kernel f u n c t o r
0
6 G
on R-gr i s of the form
then K~
is
K~
f o r some
c R-gr.
s h i f t invariant M Proof.
K
M Z M(o) f o r a l l
The f i r s t s t a t e m e n t i s e a s i l y checked. Conversely, suppose
i s a r i g i d kernel f u n c t o r on R-gr. That
K
=
f o r some N
K~
c l e a r by general t o r s i o n t h e o r y . Now f o r any
u
F E we
F
K
R-gr i s
o b t a i n , f o r every
L F R-gr : = fl
KN(,)(L)
=
n
i Ker f , f {
F
Horn
R-qr
=
K
N ( a ) =K~
@N(o)
}
(Ker f ) (u), f - c Hom ( L ( o - l ) , E q ( N ) ) R-qr
= (KN(L(u-l)))
Hence
( L , Eg(N(o)))
=
}
KN(L).
f o r a l l o F 6 . This implies t h a t
K~
=
A i K N ( r r ) 30 6
=
and we may t a k e M = 0 N( 0). 0
U
Recall from 11.9. ( s e e Theorem 1 1 . 9 . 7 . ) t h a t t o M 6 R-or t h e r e corresponds a r i g i d kernel f u n c t o r such t h a t E ( K ~ 1)1 L ( R ) (1 I t follows t h a t t h a t K ~ ( M ) = 0.
K ;
I(;
defined by
= E(K$
K:
= A { K ~ ( o ) ,o F
6 1 , which i s
and a l s o , f o r X F R-qr : K ~ ( X ) = K,(?). -
i s the larqest r i q i d
kernel f u n c t o r on R-qr such
151
II. Some General Techniques
An M c R - g r i s s a i d t o be a araded s u p p o r t i n g module functor
K
f o r the kernel
on R-gr i f K ( M ) = 0 b u t ti(M/N) = M/N f o r e v e r y araded submodule
N o f M ( d e l e t i n g "graded" everywhere y i e l d s t h e d e f i n i t i o n o f a s u p p o r t i n q
I).
module f o r a k e r n e l f u n c t o r 5 on R-mod, u t i l i z e d i n [40 functor
K
on R-gr i s s a i d t o be qraded p r i m e
a graded s u p p o r t i n g module. K~
-
i s a graded p r i m e
A kernel
i f i t i s coqenerated by
I t i s c l e a r t h a t i f M F R-qr i s such t h a t
k e r n e l f u n c t o r on R-mod w i t h s u p p o r t i n q module M
t h e n z M i s a qraded p r i m e k e r n e l f u n c t o r on R-or w i t h s u p p o r t i n a module El. i . e . M F R-?r does n o t make
Warning : remember Remark 11.9.10. By d e f i n i t i o n o f c o g e n e r a t i o n i n
(see 11.9.)
R-clr
tiM
-
araded!
it i s clear that
tiM
depends on Eg(M), n o t on M. T h e r e f o r e l e t us i n v e s t i q a t e Eg(M) j u s t a be an i n j e c t i v e h u l l o f
l i t t l e more. L e t E(4)
M
i n R-mod. Suppose M C R-ar
i s such t h a t t h e s i n g u l a r r a d i c a l Z(M) equals z e r o . , LT x c MTo P u t E d = { x C E(M)
l e f t ideal L
for
T
c
G, f o r some nraded e s s e n t i a l
.
of R }
E' = 0 E l c i s i n R-or and i t c o n t a i n s M a s a qraded e s s e n t i a l
Obviously
0
R-submodule. 11.11.2.
Lemma.
M c
R-qr be such t h a t Z(M) = 0, t h e n E ' i s t h e l a r q e s t
R-submodule o f E(M) - w i t h t h e p r o p e r t y t h a t E l i s graded and c o n t a i n i n a
M as a graded submodule. Proof.
Suppose we have M c N i n R-gr such t h a t
x E No t h e n
( f j:
Since ( M : X ) ~ x c 11.11.3.
M c N
_. -.
c
E(M) - . If
x ) i s a graded l e f t i d e a l o f R w h i c h i s a l s o e s s e n t i a l .
PI
n
Proposition.
NTO = MTo,
x
c E' follows.
I f M F R-gr i s such t h a t Z(M) = 0 t h e n Eq(M)
E'
i n R-gr. Proof.
I n an a r b i t r a r y
Grothendieck category, i n j e c t i v e h u l l s o f o b j e c t s
may be c h a r a c t e r i z e d as b e i n g maximal e s s e n t i a l e x t e n s i o n s o f t h o s e o b j e c t s .
152
A. Some General Techniques in the Theory of Graded Rings
Eg(M) i s a g r - e s s e n t i a l o f M,then by Lemma I . Z . B . ,
However, i n R-gr,
&is
an e s s e n t i a l e x t e n s i o n o f M i n R-mod i . e . we may assume
c E(M).
The f o r e g o i n g lenuna then y i e l d s Eg(M) c E ' and t h e m a x i m a l i t y o f E a ( M ) w i t h r e s p e c t t o being p r - e s s e n t i a l then y i e l d s E ' = Eq(M). 11.11.4.
A kernel functor
Remark.
i n R-gr i s coqenerated by M i f i t
K
i s cogenerated by Eg(M), i f and o n l y i f K i s coaenerated by E(M). We say t h a t a r i g i d kernel f u n c t o r functor i f
K
= A {
f u n c t o r on R-gr, i . e . module f o r
K
~
u F G }
K,(u), K
=
K;
K
on R-gr
, where
i s a rigid-prime kernel i s a graded prime k e r n e l
K~
f o r some M F R-qr which i s a araded s u p p o r t i n q
.
denote by ( E ) ~
L e t K be an a r b i t r a r y k e r n e l f u n c t o r on R-mod, then we the r i g i d k e r n e l f u n c t o r on R - o r a i v e n
by t h e araded f i l t e r
X ( (-K ) ~ )=
5 K b u t e q u a l i t y o n l y holds i f K_ i s I ( K ) rl L (R). Note t h a t ( K ) 9 9 a graded k e r n e l f u n c t o r on R-mod. By Theorem 11.9.7. i t f o l l o w s t h a t f o r =
r ( K ~ = ) K ~~ For . X F R-gr we i n t r o d u c e the f o l l o w i n q n o t a t i o n s :
M E R-gr,
Since
need n o t be a qraded kernel f u n c t o r on R-mod t h e f o l l o w i n a
K~
-
m o d i f i c a t i o n o f t h e techniques o f P r o p o s i t i o n 11.10.4. By g y,
a,(?)
we now mean t h e
-
=
UF
I x
F Q,
-
(J),
Lz x c ( X h M ( X ) & 11.11.5.
qraded module 0
Theorem.
c-
Yo,
i s necessary
where
t h e r e i s an L 6 I(K;) such t h a t for all
T
F G
1.
L e t R be a graded r i n g o f type G. Then w i t h n o t a t i o n s
as b e f o r e : 1. g QM(R) i s a graded r i n g c o n t a i n i n q R / K ~ ( R )as a qraded s u b r i n g -
.
II. Some General Techniques
2 . For every X c R-or, g
Q,(Z) -
153
i s a gradedgQ, (R)-module. Moreover : -
f o r a l l X t R-gr :
Qi(x)9 QM(&). -
___
=
Proof. 1. As i n Proposition 11.10.4. 2 . Consider the following diagram i n R-mod : 0
If x c
=X -Qi(x)
x/rM(x)
+
Qi(X)o then,
U Q M (- Z )
__
because coker
ax
i s Kp,-torsion, we have -
x F q Q,(X)".
O n the other hand, the construction of 9QM(X) i s such t h a t coker p x i s Kh-torsion i s an e s s e n t i a l i n R-qr. Proposition 11.10.7. and and the f a c t t h a t Q;(X) extension of X / K , ~ ( X ) i n R-mod, e n t a i l t h a t gQ,(X) c hence LT x
S Q (X) -M -
c ( X / K ~ ( X ) ) f~o~r every T
=
c
G, i . e .
Qi(X),
Qi(x).
__
Now consider a graded prime ideal P of a l e f t Noetherian graded rina R of type G. P u t M = R / P and write C.I.3.
5
i t will be shown t h a t E g ( R / P )
I n the Z-graded
= K
~
/
-
~K ; ,
=
K:/~.
I n Section
i s a P-cotertiary waded R-module.
case we will be able t o deduce a more complete r e s u l t ,
see Section C . I . l . ,
here i n the oeneral G-graded case we have :
11.11.6. Proposition.
Let R be a l e f t Noetherian ring graded of type G and l e t P F Spec ( R ) , then K~ and
1.
q
K ;
may be described by the followino f i l t e r s :
= { J l e f t ideal of = { J l e f t ideal
of R / P 1
R , ( J : a ) n G(P) # 0 f o r a l l a F R
}
of R , J maps t o an e s s e n t i a l l e f t ideal
154
2.
A. Some General Techniques in the Theory of Graded Rings
~(KF)
=
I J graded l e f t i d e a l o f
R such t h a t J maps t o an e s s e n t i a l
graded l e f t i d e a l o f R/P } Proof.
By Theorem 11.9.7.
and t h e f a c t t h a t
(I<
=
-
i t follows
K;
t h a t 2 . i s an immediate consequence o f 1.
1. If ( J : a ) n G(P) # 0 i s t h e imaqe of some x
for all a
c
c
x # 0 i n R/P,
R, c o n s i d e r
R- P. P i c k s F G(P) r i
Q
and s x # 0, t h e r e f o r e (J:a)
(J:a).
where
c
Hence sa
x J
annR X.
( l ) , J 6 l(.xp) follows.
By 11.9.7.
Conversely, suppose t h a t P u t L = ( J : a ) . Hence L
4
c
J
J ( K ~ ) , and a F R.
annRx f o r a l l
x
L i n R/P i s an e s s e n t i a l l e f t i d e a l o f R/P. then from (L:s)
annR s
t h a t X 1 annRS, hence 1 s
c
R/P,
Then (J:a)
x#
L and X
s#
(L~).
d:
0. The imaqe
Indeed, i f
?, c
R/P,
i t follows that there exists a 1
<
c
c
s#
c (L:s)
of 0 such
0. S i n c e R/P i s a l e f t N o e t h e r i a n
p r i m e r i n g , an e s s e n t i a l l e f t i d e a l has t o c o n t a i n a r e g u l a r element,
?
say, i . e .
5
F
L,
But a representative t f o r
o f G(P)) i n G(P) and t h u s L 11.11.7.
f
i s then (by d e f i n i t i o n
n G(P) # 8 .
Remark.
1. The f o r e g o i n g p r o p o s i t i o n s t a t e s t h a t
.xP i s
the kernel functor
a s s o c i a t e d t o t h e m u l t i p l i c a t i v e s e t G(P) ( i n t h e sense o
Section I.6.,
b u t ungraded). The e x a c t graded analoae can o n l y be i n c l u d e d a f t e r we d e r i v e d a qraded v e r s i o n o f G o l d i e ' s theorems. T h e r e f o r e we w i l l i n c l u d e t h i s r e s u l t i n Section C . I . l . E x e r c i s e . L e t R and S be two graded r i n a s and f : R
4
S an homomorphism
o f graded r i n a s ( f ( 1 ) = 1). Then f i s an epimorphism o f r i n g s i f and o n l y i f i n t h e c a t e g o r y o f graded r i n g s .
f i s an epimorphism
155
CHAPTER B: COMMUTATIVE GRADED RINGS I: Some Commutative Algebra Revisited
I.1. Graded F i e 1 ds . Throughout p a r t B o f t h i s book a l l r i n a s a r e commutative a s s o c i a t i v e r i n g s w i t h u n i t unless
o t h e r w i s e s t a t e d . Moreover
a l l graded r i n g s w i l l be
graded o f t y p e 2 . Some more g e n e r a l r e s u l t s , i n p a r t i c u l a r f o r qraded r i n g s o f t y p e G where C: i s t o r s i o n f r e e a b e l i a n , w i l l be h i n t e d a t i n t h e e x e r c i s e s . The r e s u l t s o f S e c t i o n A . I . 4 . o f type
Z
entail
where k
a r e o f t h e form k o r k [ T , T - l ]
t h a t araded f i e l d s i s a field, T a
v a r i a b l e o f degree t
c
L e t R be a q r - f i e l d ,
l e t K be i t s f i e l d o f f r a c t i o n s and l e t Q be an
Z. We w i l l w r i t e q r - f i e l d f o r oraded f i e l d .
a l g e b r a i c c l o s u r e o f K. An u 6 Q i s _ g r - a l a e b r a i c o v e r R i f t h e m i n i m a l o f a o v e r K has t h e form,
monic p o l y n o m i a l w i t h ai
c
h(R)
and deg(ai)
- dea(ai+l)
*
: Xn
+
an-l
xn-l
c F Z f o r a l l i = 0,
=
+...+
a
0'
..., n - 1 .
T h i s c o n d i t i o n s t a t e s e x a c t l y t h a t t h i s m i n i m a l p o l y n o m i a l , f ( X ) say, i s a homoqeneous e l e m e n t o f R [ X I when t h i s r i n g i s c o n s i d e r e d w i t h t h e q r a d a t i o n g i v e n by R [ X I n =
z
i+jc = n
R~ XJ f o r a l l n F
Z.
A graded r i n g S c o n t a i n i n g R as a graded s u b r i n g i s s a i d t o be algebraic over R 1.1.1.
Lemma.
ifevery element o f S i s g r - a l a e b r a i c over R.
Let a
c
S be q r - a l a e b r a i c o v e r R. I f g(X) i s a monic
p o l y n o m i a l i n R [ X I w i t h minin;al degree g(X) i s a m i n i m a l p o l y n o m i a l f o r Proof.
z-
I f g ( X ) = hl(X)h2(X)
a
i n X such t h a t q ( a ) = 0, t h e n
over K.
i n R [ X I then t h e leadino c o e f f i c i e n t s o f
hl and hg a r e u n i t s i n R and t h u s i n h ( R ) . S i n c e R i s a o r - f i e l d we may assume t h a t hl(X)
and h2(X) a r e b o t h monic p o l y n o m i a l s . The m i n i m a l i t y
assumption on degXg(X) e n t a i l s t h a t e i t h e r hl o r h2 e q u a l s 1
156
6. Commutative Graded Rings
Hence g ( X ) i s i r r e d u c i b l e in R [ X I because R
Ro [ T , T - l ]
,
However R i s a f a c t o r i a l rinq
with Ro a f i e l d ,
therefore g(X) will a l s o remain
i r r e d i c u b l e in K [ X I . T h u s g ( X ) i s a minimal monic polynomial f o r a over K. 1.1.2. Remark:
An u c Q i s gr-alqebraic over R i f i t i s inteqral over R
and the monic polynomial of
minimal deqree s a t i s f i e d by a has the form
*.
This j u s t i f i e s the d e f i n i t i o n of gr-algebraic. Indeed, a weaker d e f i n i t i o n a s s e r t i n g only the existence of a polynomial of the form ( * ) , s a t i s f i e d by a
, will not lead t o the desired r e s u l t s . If a c Q i s qr-alaebraic over R then
1.1.3. Corollary.
the subrinq
R [ a ] of D i s a q r - f i e l d . __ Proof.
Let f(X) be the minimal monic polynomial over R
f(X) i s of the form (*) by assumption.
s a t i s f i e d by a
,
By the above lemma R [ a ] 2 R [ X ] / ( f ( X )
as r i n g s . I f R [ X I i s graded by qivinq X dearee c, where c i s
the constant
in Z appearing in (*), then ( f ( X ) ) i s a qraded ideal so we obtain a gradation on R [ a ] such t h a t deg a = c and R i s a graded subrino of R [a]. Let us proceed t o show t h a t ( f ( X ) ) i s a qr-maximal ideal of R [ X I
=
If
then i t follows from Lemma 1.1.1.
h(X) i s any homogeneous element of R [ X I that either K[X] ( h ( X ) , f(X))
.
K [ X ] or K [ X ] ( h ( X ) , f ( X ) )
=
K[XI(f(X)).
I n the f i r s t case ( h ( X ) , f ( X ) ) = R [ X I follows ; in the second case r h ( X ) c ( f ( X ) ) f o r some r
c R . Hence r i h ( X ) c ( f ( X ) ) f o r each homoqeneous
component ri of r . Since R i s a in R, hence h ( X )
R [XI
c ( f ( X ) ) follows. So ( f ( X ) ) i s a or-maximal ideal of
a n d thus R [ a ] 2 R [XI / ( f ( X ) ) i s a g r - f i e l d .
1.1.4. Lemma. Let a
p
g r - f i e l d , each nonzero r i i s i n v e r t i b l e
F 51
be gr-algebraic over R and p u t dea a = ca. Let
t S be gr-algebraic over R of deqree c
F'
Then p i s qr-alqebraic over
R [ a ] and of degree cp with respect t o i t s minimal polynomial over R[aI.
157
I. Some Commutative Algebra Revisited
Proof. Let h be a monic polynomial of minimal deoree in X
p over R
Since R
[a].
[a]
is a gr-field,
s a t i s f i e d by
Corollary I . 1 . 3 . , R [ a ][ X I
cf.
i s a f a c t o r i a l ring and thus h i s i r r e d u c i b l e and i t i s a l s o the minimal monic polynomial f o r p over K(a), where K(a) i s the f i e l d of f r a c t i o n s of R
.
[a]
If g F R [ X I i s the minimal monic Dolynomial f o r p over R
t h e n g i s homogeneous i n R
[X]
by deg X = c
gradation determined
P‘
Embed R [ X ] i n t o
obvious way. I n R [ a ][ X I we have t h a t q I,, =
considered with the
when t h i s rina i s
=
h 5
+
p.
R [ a ][ X I i n the
B u t ~ ( p )= 0 e n t a i l s
0 by the minimality condition on degXh. The qradation on R
being determinied by the oradation of R from g = h 5
t h a t h and
[a]
and by deq X
a r e homoqeneous i n R
form (*) over R [a] with the same constant c gr-algebraic over R [ a I of degree c :
P
[a] 6
=
c
P’
[a]
[ X I
i t follows
[ X I i . e . h has the
Z. By d e f i n i t i o n p i s
P’
The subset of Q consisting of the qr-aloebraic elements
1 . 1 . 5 . Corollary.
over R forms a qraded r i n q , a c t u a l l y i t i s a q r - f i e l d . If a s a t i s f i e s of polynomial of the form (*) with constant c F Z
Proof.
t h e n we write deg(a) = c; this i s j u s t i f i e d by the above lemma. P u t E(R),
Take a, that R
R
[a,
c 52,
= { a
p
[a,
p].
a
gr-algebraic over R and deg(a) = c 1
.
E ( R ) and consider R [a, p ] i n 52. From Lemma 1.1.4.
F
p
i t follows
i s a q r - f i e l d and thus a + P i s homoqenous of degree c i n
]
cn r i
If
polynomial f o r
i =O
a+P
(a+p)j =
0 i s the evaluation of the minimal monic
over R then we may s e l e c t terms of hiohest degree
r . . (a+P)’ = 0 , where i n the gradation of R [ a . P J and obtain : 2 j+ic=N l Y J
r . . i s the homogeneous component of deqree j of r i . Up t o multiplication 1 .J
by a unit of R we obtain a monic polynomial
*
which i s s a t i s f i e d by
Hence
a+p F
E(R),.
z r X1 of the form j+ic=N i , j
a+P.
I n much the
same way i t i s Possible t o show t h a t ,
B. Commutative Graded Rings
158
graded r i n g and a g r - f i e l d i s c l e a r . 1.1.6.
Remarks.
1. E(R)
i s c a l l e d t h e q r - a l g e b r a i c c l o s u r e o f R i n 8.
2. S i n c e graded modules o v e r g r - f i e l d s a r e q r - f r e e i t i s p o s s i b l e t o use t h e r a n k (we w i l l n o t w r i t e g r - r a n k
s i n c e t h e r e i s no a m b i q u i t y p o s s i b l e ) .
F o r example : i f t h e degree o f t h e m i n i m a l p o l y n o m i a l ( o f f o r m * ) f ( x ) f o r a gr-algebraic
a F 51,
o v e r R,
i s equal t o n, t h e n n =
[ R [a] : R ]
.
We l e a v e t o t h e r e a d e r t h e v e r i f i c a t i o n o f some v e r y e l e m e n t a r y p r o p e r t i e s o f gr-algebraic extensions, ( t h e p r o o f o f these p r o p e r t i e s i s j u s t l i k e t h e ungraded c a s e ) , w h i c h we w i l l Now c o n s i d e r a f i e l d e x t e n s i o n L o f = { a
c L,
G
use f r e e l y i n t h e s e q u e l .
K
w i t h [ L : K ] = n. P u t EL(R)c =
,
i s g r - a l q e b r a i c o v e r R, deg (a) = c }
and EL(R) =
0
EL(R)c.
CFZ
Them EL(R) i s a g r - f i e l d and we r e f e r t o i t as t h e q r - a l q e b r a i c c l o s u r e o f R i n L. I t i s c l e a r t h a t [EL(R) : R ]
5
LL:K]
=
n, b u t e q u a l i t y need
n o t h o l d g e n e r a l l y . A f i n i t e f i e l d e x t e n s i o n L o f K, n = [ L : K l realised
i f n = [EL(R):R]
.
Z
I f we can f i n d a t ' 6
,
i s graded
such t h a t R = Ro[T,T
-1
1,
w i t h deg T = t, may be chanaed t o R ' , w h i c h i s t h e r i n a R w i t h a new g r a d a t i o n g i v e n by deg T = t ' , such t h a t [ E L ( R ' ) : R ' ] that L i s
= n t h e n we say
graded r e a l i s a b l e . We w i l l r e t u r n t o t h e s e concepts i n S e c t i o n
1.3. F i r s t l e t us now p r e s e n t t h e graded v e r s i o n s o f two well-known theorems i n commutative a l g e b r a .
I . 1.7. Theorem.
(Graded v e r s i o n o f Z a r i s k i ' s theorem).
L e t R be a g r - f i e l d and l e t S be a g r - f i e l d e x t e n s i o n o f R q e n e r a t e d as an R - r i n g by a f i n i t e number
o f homogeneous elements, xl,...,xn
Then S i s a g r - a l g e b r a i c e x t e n s i o n of R and S i s i n R-gr.
say.
f i n i t e l y qenerated
159
1. Some Commutative Algebra Revisited
Proof. Cm
i=o
I f n = 1 then R [ x l ]
i
ri x1 w i t h : d e q ( r i )
+
can o n l y be a q r - f i e l d = - deq(xl).
i dea (xl)
-1 . i s a polynomial 1 The monic p o l y n o m i a l if x
( zm r . xi) = 1 has t h e f o r m * and i t i s e a s i l y seen 1 i=o 1 t o be t h e m i n i m a l p o l y n o m i a l o f x1 o v e r K. Thus x1 i s q r - a l q e b r a i c o v e r
'
f ( x ) deduced f r o m x
R, and by Remark I . 1 . 6 . ,
2 we have
[R
[x,]
: R]
=
deqx f ( x ) . We proceed
by i n d u c t i o n on n, assuming t h e s t a t e m e n t s t o be t r u e f o r a r - f i e l d e x t e n s i o n s q e n e r a t e d by l e s s t h a t n elements. As i n t h e unqraded case one f i r s t p r o v e s t h a t R [ Xl,...,~n] t h e n , knowing t h a t R [xl,
..., x n ]
i s an i n t e q r a l e x t e n s i o n o f R and i s a q r - f i e l d i t i s easy t o e s t a b l i s h
t h a t i t i s a qr-algebraic extension o f
R.
The above theorem may be used i n d e r i v i n q t h e f o l l o w i n q qraded v e r s i o n o f Hilbert's Nulstellensatz.
1.1.8. Theorem.
If R i s a gr-field,
polynomial r i n g i n n variables
c o n s i d e r S = R [X1,
. . . ,X n ] , t h e Y
qraded by p u t t i n a Sn =
2 it" t. . . t v
1
E v e r y p r o p e r graded i d e a l
I
o f S has a z e r o (xl,
..., x),
=m n
RiXl
where t h e xi
"1 .. ,. . Xn"
lie
i n a s u i t a b l e g r - a l g e b r a i c e x t e n s i o n o f R. Proof.
As i n t h e ungraded case, u t i l i z i n g Theorem 1.1.7..
Note t h a t
t h i s i s n o t t h e p r o j e c t i v e - n u l s t e l l e n s a t z used commonly i n p r o j e c t i v e a l g e b r a i c geometry. B a s i c a l l y t h i s i s due t o t h e f a c t t h a t t h e v a r i a b l e T appears i n R t o g e t h e r w i t h i t s i n v e r s e so t h a t P r o j R [ X I may be c o n s i d e r e d as an open subspace o f P r o j R [ T , X ] 0 t o t h e h y p e r p l a n e , T = 0.
i . e . t h e complement
160
6. Commutatiw Graded RinGA
1 . 2 . Integral Extensions of Graded Rings.
Consider a graded ring R which i s a subrina of a commutative domain S ( n o t necessarily graded). 1.2.1.
Definition.
R [sl
5
c
i+jc=n n,(X)
=
An s c S is gr-algebraic over R of deqree c
if
R [ X I / Ker u s , where R [ X I i s graded by putting R [ X I n =
RiXJ,
a n d where
n
S
: R
[XI
+
S
i s the rinq morphism defined by
s , n , ( r ) = r , r E R . i . e . Ker n s i s generated by homogeneous
polynomials. We say t h a t s i s qr-integral over R of deqree c , i f s i s gr-algebraic of degree c and Ker nS contains a monic Dolynomial. I t should be c l e a r t h a t t h i s d ef i n i t i o n i s d i f f e r e n t from : s s a t i s f i e s a homogeneous monic polynomial in R [ X I ! Indeed, even in the case where R i s a g r - f i e l d the " r i g h t " d ef i n i t i o n should imply t h a t the minimal
monic polynomial of a gr-inteqral element 1.2.2.
Lemma.
homogeneous
S'
Zm-' ai Xi = f ( X ) , a i c R , be a monic polynomial in i =O s i s of degree c, give X degree s , and selectinq homogeneous
Let Xm
Ker n s . I f
.
I f s i s a gr-integral over R then there i s a
monic polynomial in Ker n Proof.
homoqeneous
t
components of degree mc i n f(X) y i e l d s a homogeneous monic polynomial i n Ker nS. 1.2.3.
Lemma.
Let s E S be such t h at R [ s ] (constructed within S
is
a graded ring such t h a t s i s homogeneous of deqree c. If t E S i s graded inteqral over R of deqree d , then t i s graded integral over R [ s l of degree d. Proof.
By our hypothesis n t : R [ X I
+
S has qraded kernel Ker n t (note
t h a t R [ X I i s now graded such t h a t X has deqree d ! ) a n d i t contains a
161
1. Some CommutativeAlgebra Revisited
We
monic homogeneous p o l y n o m i a l . for
: R [ X I
nt
+
have t o e s t a b l i s h s i m i l a r p r o p e r t i e s
S, g i v e n by nt (X) = X f o r X F R [ s ] , n t ( X ) = t . L e t
h(X) be t h e monic p o l y n o m i a l i n Ker n t . Then a l s o h(X) t Ker nt; we now show t h a t t h e homooeneous components o f any f ( X ) 6 Ker n t
proceed t o
w i l l a g a i n be i n Ker n t , t h i s b e i n a t h e case, h ( X ) F Ker
iit
w i l l finish
the proof. F i r s t l e t us p o i n t o u t t h a t i f R ' i s any domain, f ( X ) , g ( X ) t R [ X I such t h a t deqX f ( X ) 5 degX q ( X ) , t h e n t h e r e e x i s t s = f(x) q(x)+ r(X) with q(x) F R " X 1
P a r t 1.
a
c R ' such t h a t aa(X)
=
and degX r ( X ) < deg X f ( X ) .
Take g ( X ) F Ker n t such t h a t g ( X ) has m i n i m a l dearee i n X as T a k i n g t h e homoqeneous component o f h i q h e s t deqree appearinq, w i l l
such.
by p u t t i n g an i n d e x ( - ) " w h i l e t a k i n g components of l o w e s t
be denoted
degree w i l l be
(3,.
denoted b y
g(X)H F ( R [ s l [XI),.
Case 1.
I f a l s o g(X),
t h e n g ( X ) i s homoqenous and t h e n t h e r e i s
F (R [ s ] [
n o t h i n g t o w o v e . So l e t us assume t h e s t a t e m e n t t o be t r u e i f g(X),
c
f o r 0 3 1 > k . N e x t c o n s i d e r t h e case where q(X),, t (R[ s ] [ 1 R [ s ] be such t h a t a h(X) = q(X) q(X) + r ( X ) . Then a h f t ) = 0 and
(R [ s ] [ X ] ) Let a
c
t h e n deqXr(X) < degXg(X) e n t a i l s r ( X ) = 0
g(t) = 0 y i e l d r ( t ) = 0 but
and ah(X) = q ( X ) q ( X ) . S i n c e h ( X ) i s homoqeneous t h i s y i e l d s ab,h(X) = t h u s a g a i n q r 4 ( t ) = 0 o r q,(t)
= gM(X) q,(X)
Then degX q,(X) Therefore a',qM(X)
2 deqX OM( X)
t h e r e is a n
a'
F R [ s ] such t h a t a'qbz(X)
degX qh(X) 2 degX q,(X),
<
= 0.
.
= gM(X) q I M ( X ) . I f gM(X)
and s i n c e now degX q',(X)
= 0. Suppose t h a t q,,(t)
t
=
q(X)ql(X),
hence
R [ s l t h e n we may r e p e a t t h e p r o c e d u r e
degX qM(X) we w i l l r e a c h t h e s i t u a t i o n where
t h u s q M ( t ) = 0. On t h e o t h e r hand, if qM(X) F R [ s ]
t h e n aMh(X) = qM(X) qM(X) y i e l d s ( p u t t i n g gM(X) = p ) :
162
(*)
B. Commutative Graded Rings
e
ah(X) = g ( x ) ( P qH(x) +. . .+ aM h ( X ) )
I f h ( X ) has degree r i n X t h e n cannot c a n c e l o u t i n p qH
p
aM
t...+
Xr appears i n
aMh(X) and t h i s t e r m
aMh(X) (where aMh(X) = pqM(X)), because
i s homogeneous and terms d i f f e r e n t f r o m
p qF1 (X)
g r a d a t i o n degree d i f f e r e n t f r o m t h e deqree o f
aM
y i e l d terms i n X o f
Xr
! T h e r e f o r e , comparing
degrees i n (*) l e a r n s t h a t g(X) c R [ s ] , c o n t r a d i c t i o n .
So f a r we have
p r o v e d t h a t gM( t ) = 0. B u t t h e n (g-gM) ( t ) = 0 and o(X)-gb,(X)
c (R [ s I [ X
with 0 > 1 > k. The i n d u c t i o n h y p o t h e s i s t h e n y i e l d s t h a t t h e homooeneous p a r t s o f g(X)-gM(X) i . e . t h e homogeneous components o f q(X) a r e i n Ker n t . Case 2 : g ( X ) H
c (R [ s ]
w i t h i 2 0.
[
I f i = 0 t h e n we a r e i n Case 1, so we proceed by i n d u c t i o n on i. R e p l a c i n g
M by H i n t h e p r o o f f o r Case 1 y i e l d s - a c o m p l e t e l y s i m i l a r p r o o f . P a r t 2 : Now c o n s i d e r p ( X ) F K e r nt o f a r b i t r a r y deqree i n X. L e t p,(X) be t h e sum o f t h e homogeneous components o f p(X) w h i c h a r e i n Ker nt. p(X)-p,(X)
=
0 t h e n we a r e done, o t h e r w i s e q ( X ) = p(X)-pl(X)
If
i s i n Ker n t
and no components i n q ( X ) i s i n Ker n t . L e t q ( X ) be as i n P a r t 1. t h e n degXq(X)
?
degXg(K), hence we may s e l e c t y 6 R [ s
1
such t h a t
y q(X) =
g(X)qr ( X ) + r ( X ) w i t h d e g X r ( X ) < deqXg(X). Then q ( t ) = 0 = r ( t ) i m p l i e s t h a t r ( X ) = 0 hence yMqM(X) = gM(X) ql(X).
The c h o i c e o f q(X) and t h e
f o r e g o i n g p a r t o f t h i s p r o o f y i e l d t h a t g M ( t ) = 0 hence yM(t) = 0 hence yM q M ( t ) =
1.2.4.
0 o r qp,(t) = 0 b u t t h i s c o n t r a d i c t s qM(X) ,iKer r t .
rn
C o r o l l a r y . L e t R be a qraded r i n g w h i c h i s a s u b r i n q o f t h e domain S.
P u t I(R), I(R) = 0
= { x c S,
x graded i n t e p r a l o f degree c o v e r R } and w r i t e
I(R)c.
Then I ( R ) i s a graded r i n o , w h i c h i s c a l l e d t h e
CCZ
graded i n t e g r a l closure
of
R i n S . I f S i s qraded, t h e n an s F Sn which
i s g r - i n t e g r a l o v e r R i s n e c e s s a r i l y o f degree n. T h i s shows t h a t t h e g r a d a t i o n o f I ( R ) i s i n d u c e d by t h e a r a d a t i o n o f S i n t h i s case.
163
1. Some Commutative Algebra Revisited
1.3. Graded V a l u a t i o n Rings i n Graded F i e l d s . F i r s t l e t us r e c a l l some f a c t s a b o u t p r i m e s i n r i n g s . Primes i n r i n g s and a l g e b r a s o v e r f i e l d s have been i n t r o d u c e d i n [ l l l ] , [ 1 0 7 1 , [ 1081 and t h i s t h e o r y y i e l d s a u n i f i e d approach
t o several
t e c h n i q u e s used i n t r y i n a t o e x t e n d v a l u a t i o n t h e o r y t o f i r s t commutative r i n g s and l a t e r a l s o non-commutative r i n a s . i s a prime o f R
L e t R be any r i n g w i t h u n i t . A c o u p l e (P,R') a s u b r i n g o f R, P i s a p r i m e i d e a l t h e n x c P o r y F P.
o f R ' and if x R ' y c
P
i f R' i s w i t h x, y F
R
I n t h e commutative case t h i s c o i n c i d e s w i t h t h e
p r i m e s s t u d i e d by F. Van Oystaeyen i n [110] ,[ 1111. The i d e a l P i s c a l l e d t h e k e r n e l o f t h e p r i m e . The s e t o f k e r n e l s o f p r i m e s o f a r i n g R i s denoted by P r i m ( R ) . To a f i n i t e s u b s e t F o f R = { P F Prim(R),
on P r i m ( R ) .
P
n F
=
6 }
:The
a s s o c i a t e D(F) =
s e t s D(F) f o r m a b a s i s f o r a t o p o l o q y
Prim thus defines a contravariant f u n c t o r from t h e category
o f commutative r i n g s t o t h e c a t e g o r y
o f t o p o l o g i c a l spaces, and t h i s f u n c t o r
i s a n a t u r a l t r a n s f o r m a t i o n o f t h e Spec f u n c t o r . The k e r n e l o f a p r i m e does n o t d e t e r m i n e t h e p r i m e c o m p l e t e l y ; however i f o n l y t h e k e r n e l o f a p r i m e i s s p e c i f i e d , say P, c o n s i d e r i n g t h e p r i m e (P,Rp) A prime P o f R i s said t o
t h e n i t w i l l be u n d e r s t o o d t h a t we a r e where R p i s t h e i d e a l i z e r o f t h e s e t P i n R .
be s e m i r e s t r i c t e d
e x i s t nonzero elements X and p i n R
P
i f f o r a l l x c R-R P t h e r e
suh t h a t X x li. F Rp. These p r i m e s
seem t o be t h e r i g h t q e n e r a l i z a t i o n o f v a l u a t i o n
R i s a f i e l d t h e p r i m e s o f R do c o r r e s p o n d F o r a d e t a i l e d account o f t h e theory t h e s i s [ 1071.
rinqs ; actually i f
t o v a l u a t i o n r i n a s . o f R.
o f p r i m e s we r e f e r t o J . Van G e e l ' s
F o r a commutative r i n a R, t h e f o l l o w i n o lemma w i l l be
rather useful : 1.3.1.
Lemma.
P 1. I f P i s t h e k e r n e l o f a p r i m e o f R t h e n R i s i n t e g r a l l y c l o s e d i n R.
B. Commutative Graded Rings
164
2. The i n t e g r a l c l o s u r e o f a s u b r i n g S o f R i n R i s o f a l l R f o r a l l p r i m e s (P,R) P r o o f . c f . [ 107
of R
such t h a t S
the intersection
c R.
I.
Now we r e t u r n t o t h e case where R i s a g r - f i e l d . A
araded s u b r i n q V o f
i f f o r e v e r y homogeneous x c R
R i s s a i d t o be a a r - v a l u a t i o n r i n g R e i t h e r x o r x - l i s i n V. 1.3.2.
Lemma.
If V i s a qr-valuation r i n g o f the q r - f i e l d R
then t h e
graded i d e a l s o f V a r e l i n e a r l y o r d e r e d by i n c l u s i o n . Proof. 1.3.
Easy. As i n t h e ungraded case.
..., an c
I f al,
Lemma.
,
h(R)-V t h e n t h e r e i s an i such t h a t aT1 a j
c V
f o r a l l j = 1, ..., n .
Proof.
Look a t t h e i d e a l s A . q e n e r a t e d by a - l i n V. Because o f Lemma J j 1.3.2. we may p i c k an i such t h a t Ai i s m i n i m a l amonast t h e A . , j = 1,..., n . J - 1 a . c V because o t h e r w i s e a -1 -1 -1 . = ( a . ai) ai y i e l d s A . = Ai, hence Then ai J J J J -1 -1 -1 r a. = a. f o r some r c V and a a a i n ai a . F V f o l l o w s . J J From Lemma 1.3.2.
i t i s evident t h a t a gr-valuation r i n o i s a or-local
r i n g ; l e t MV denote t h e u n i q u e gr-maximal i d e a l o f V . I n Lemma 1.3.3. i t i s t h e n c l e a r t h a t a i l a . c MV i f and o n l y i f Ai J
1.3.4.
2
A.. J
If V i s a gr-valuation r i n q o f the o r - f i e l d R then
Proposition.
(MV, V ) i s a semi r e s t r i c t e d p r i m e o f R . Proof.
Pick a = a
Suppose ai
,..., ai
1
-1
1 t
+
... + 6 R-V.
such t h a t x = a.1 . a = aT1a 1 . 1 J J x
c
an c R-V,
where dep a . = mi.
By Lemma 1.3.3.
+...+ 1 + ... a:n;.
1
we may s e l e c t an ij. 1 c j 5 t, i s i n V.
Suppose t h a t
MV then, t a k i n g p a r t s o f degree z e r o , we o b t a i n 1 c M
T h e r e f o r e a:;
V
a contradiction.
a 6 V-MV and t h u s (Mv,V) i s s e m i r e s t r i c t e d ( t h a t i t i s J indeed a prime o f R i s obvious !).
165
I. Some CommutativeAlgebra Revisited
P r o p o s i t i o n . If (P, Rp) i s a prime o f t h e q r - f i e l d R then P . P . Ro) i s a q r - v a l u a t i o n r i n g ( P g , R ) i s a prime o f R too. Moreover (P 9 (" o f R w i t h gr-maximal i d e a l P g' P P Proof. Take x t h(R). Since x x - l F R i t f o l l o w s t h a t e i t h e r x c R o r P P Consequently R p i s a q r - v a l u a t i o n x - l F R and t h e r e f o r e e i t h e r x - l F R q' 9 1.3.5.
.I
r i n g o f R. Now P
q
o b v i o u s l y contains t h e i d e a l qenerated by t h e y C h(R)
. Indeed i f t h i s I t i s n o t n e c e s s a r i l y so t h a t P f! Rp = 9 pq were t h e case then (P,Rp) would dominate (Po,R ) i n t h e sense o f [ 107 I . 1.3.6.
Remark.
However a s e m i r e s t r i c t e d prime i s maximal w i t h r e s p e c t t o t h e domination relation i.e. P = P Due t o Lemma 1.3.1.
9
and Rp = Rp 9
would f o l l o w .
i t i s n a t u r a l t o t r y t o a p p l y primes t o t h e study
o f i n t e g r a l i t y and i n t e g r a l extensions. Theorem. L e t S be a graded domain w i t h graded f i e l d R. Consider
1.3.7.
a g r - f i e l d extension T Then
S
over R and l e t
5
i s a graded s u b r i n g o f T and thus
be t h e i n t e g r a l c l o s u r e o f S i n T.
S
equals t h e q r - i n t e g r a l c l o s u r e
o f S i n T. Proof. By Lemma 1.3.1.,
2.,
5
primes o f T c o n t a i n i n q S, say f o l l o w s . Hence 1.3.8.
5
Corollary.
i s the intersection o f =
n
t h e domains o f
R ' . Since S i s oraded
n R'
= fl
Rb
i s a graded r i n g . The use o f primes y i e l d s an easy p r o o f o f t h e f a c t
t h a t t h e i n t e g r a l c l o s u r e o f a graded domain i n i t s f i e l d o f f r a c t i o n s i s again a graded r i n g (a well-known f a c t c f . [ 129 1 ) .
i n commutative algebra
,
166
6. Commutative Graded Rings
Extending f u r t h e r on the r e s u l t s of Sections 1.1. and 1 . 2 . : 1.3.9. Theorem. be a f i n i t e f i e l d
Let R be a g r - f i e l d with f i e l d of f r a c t i o n s K . extension of K with [ L : K ]
=
Let L
n.
Then the followin9 statements are equivalent : 1. L i s graded r e a l i s a b l e over R . 2 . If R
=
k [ T , T - l ] , where k i s f i e l d , then the inteoral closure
R
of
R i n L i s a graded ring containinq R as a qraded subrina Drovided the gradation o f R i s changed by givinq T a s u i t a b l e degree. Proof.
By notation : K = k ( T ) , where k
=
Ro i s a f i e l d
1 = 2 . The gradation of R may be chanoed by puttincl deq T = t ' such
t h a t [ E L ( R ) : R l = n, (note t h a t over a g r - f i e l d E L ( R ) the f i e l d of
=
I L ( R ) ) Consequently
f r a c t i o n s of E L ( R ) i s L . Theorem 1 . 3 . 7 . e n t a i l s t h a t the
integral closure
in L i s graded , b u t since E L ( R ) i s a o r - f i e l d
( c f . remarks following 1 . 1 . 6 . ) i t has t o be i n t e q r a l l y closed in i t s f i e l d of f r a c t i o n s . Thus 2
=
=
E L ( R ) i s a graded r i n g .
1. I n the s u i t a b l e gradation on R we obtain t h a t
i s a qraded rino
containing R as a graded subrinq. Any x c h ( R ) s a t i s f i e s a homogeneous f(X) c R [ X I
(where R [ X I i s qraded by puttino dea X = deq x . ) which
may be assumed t o be monic since R i s a q r - f i e l d . such a polynomial of minimal polynomial f o r x
I t i s then c l e a r t h a t
degree in X will also be the minimal
over K i . e . x c E L ( R ) a n d x has degree c as a n element
of E L ( R ) . 1.3.10. Corollary. Theorem 1.3.9. extends 1.3.8. in t h a t i t allows graded r e a l i s a b l e extension f i e l d s L of K. where F i s a f i n i t e extension of the f i e l d k closure of R in L i s a graded r i n g ,
I n particular, i f L = F(T) =
Ro then the integral
t h i s p a r t i c u l a r case i s the
generalization of Theorem 11 mentioned in
1291.
167
I. Some Commutative Algebra Revisited
1.3.11.
Examples.
L e t R be k [ T , T - ' ]
1. P u t L = K ( a ) w i t h a2
=
w i t h deq T = t; K = k ( T ) .
T. C l e a r l y L i s qraded r e a l i z a b l e i f T i s g i v e n
even degree. However i f t i s odd t h e n EL(R) = R; 2. P u t L = K ( a ) w i t h a2 = T t 1; t h e n L i s n o t oraded r e a l i z a b l e ;
3 3. p u t L = K ( a ) w i t h c2 t Ta = T ; t h e n L i s n o t graded
realizable.
Moreover E L ( R ' ) = R ' f o r e v e r y a d m i s s i b l e change of a r a d a t i o n on t h e u n d e r l y i n g r i n g R. T h i s i s o f
course due t o t h e f a c t t h a t t h e p o l y n o m i a l
X2 + TX
-
1.3.12.
Remark. The g r - v a l u a t i o n r i n o s o f a f i x e d
3 T c a n n o t be homopeneous i n R [ X] , whatever deg
X may be.
o r - f i e l d R f o r m a dense
s u b s e t o f Prim(R), w i t h t h e i n d u c e d Z a r i s k i t o p o l o o y . T h i s dense s u b s e t may be
a s s o c i a t e d t o R.
c o n s i d e r e d t o be t h e compact Riemarn s u r f a c e
Note t h a t a copy o f t h e Riemann s u r f a c e o f t h e f i e l d Ro, yo say, may may be f o u n d i n Oo [ T , T - ' ]
,
by c o n s i d e r i n g t h e g r - v a l u a t i o n r i n g s
where 0, i s a v a l u a t i o n r i n o o f Ro.
L e t us f o c u s o n t h e r e l a t i o n between g r - v a l u a t i o n s and v a l u a t i o n s . T h i s r e l a t i o n becomes e v i d e n t t h r o u g h t h e s t u d y of c e r t a i n v a l u a t i o n f u n c t i o n s . 1.3.13.
Theorem. L e t V a g r - v a l u a t i o n r i n q o f t h e a r - f i e l d R .
corresponds a v a l u a t i o n f u n c t i o n v:R* function
v e x t e n d s t o a v a l u a t i o n v: K*
K o f R. I f x 6 R then
Proof.
+
On h(R*),
as f o l l o w s . F o r a, such t h a t a = xlb
x
c
V
r
To V t h e r e
f o r some o r d e r e d group T. T h i s
+
r
o f the f i e l d o f fractions
i f and o n l y i f v ( x )
?
0.
where R* = R - { O } , we d e f i n e an e q u i v a l e n c e r e l a t i o n b F h(R*)
write a
and b = x2a. L e t
r
V
b
i f t h e r e e x i s t xl,
x2 6 V
be t h e t o t a l l y o r d e r e d group
obtained from the equivalence classes w i t h respect t o how we d e f i n e a v a l u a t i o n f u n c t i o n v : h(R*)
--t
r.
y. I t
i s obvious
L e t us check t h a t r
does d e f i n e a " v a l u a t i o n f u n c t i o n " as d e s i r e d o n t h e whole o f R*.
168
8. Commutative Graded Rings
I f a c R*,
say a = al +...+an J
=
dl
<
... <
dn. P u t v ( a ) =
j = 1,..., n }. L e t us v e r i f y t h a t f o r any a, b 6 R* we
= min. { v(a.),
J have v ( a b )
w i t h deg ai = di,
v(a)
+
v(b) ; the other properties o f a valuation function
a r e even more e a s i l y v e r i f i e d .
... +
w i t h deq b . = ej, j = 1,..., m. J L e t ai ,...,ai , r e s p . b ,..., b j be t h e homogeneous components o f a 1 r jl S r e s p . b, where t h e v a l u a t i o n reaches i t s l o w e s t v a l u e and assume t h a t
Write b
il <
=
...
= v(ai
bl +
b,
< irand j, <
...
< js. We have t o check whether
v(ab) =
) t v(b. ). J1
1
Clearly
, (ab)i
= b. 1' Jl ail J1
w i t h k # il, 1 # j,,
+
2 k + l = i +j ak bl' 1 1
b. then 1 J1 e i t h e r v ( a k ) = v(ai ) and v(bl) = v ( b . ) , o r v ( a k ) = v ( b . ) and 1 J1 J1 v ( b ) = v(ai ) , by t h e c h o i c e o f il and j,. The l a t t e r i m p l i e s 1 1 If akbl
has t h e same v a l u a t i o n as ai
) hence v ( a k ) = v(ai ) and v ( b . ) 1 1 J1 t h e second p o s s i b i l i t y reduces t o t h e f i r s t . v(ak)
5
v(ai
) entails k whereas v(bl) il 1 t h i s c o n t r a d i c t s k + 1 = il + j,. Consequently
Now v ( a k ) = v(ai
t e r m i n t h e d e c o m p o s i t i o n of ( a b ) l+jl
=
v(a. ); '1
therefore
= v ( b . ) e n t a i l s 1 > j,;
J1
ai b . is t h e unique 1 J1
w i t h minimal v a l u a t i o n
) + v(b. ) . 1 J1 T h a t t h e above v a l u a t i o n v a l u e i s m i n i m a l amonqst t h e v a l u a t i o n s o f t h e Hence : v ( ( a b ) i
tj ) = v(ai 1 1
bj,)
= v(ai
homogeneous components o f a b i s o b v i o u s , so by d e f i n i t i o n s v on R* i t follows t h a t v(ab) = v ( a i ) t v(b. ) = v(a)+v(b). 1 J1
V = I x 6
R*,
V(X)
2 0 } U {O}.
From t h i s i t i s a l s o c l e a r t h a t
L e t M be t h e qr-maximal i d e a l o f V. I t
i s easy t o o b t a i n f r o m v an e x p l i c i t e v a l u a t i o n us i n c l u d e a d i r e c t p r o o f o f t h e f a c t t h a t Q,(V)
: K*
--+
r;
however l e t
i s a valuation r i n g o f K
169
1. Some Commutative Algebra Revisited
(QM denotes t h e l o c a l i z a t i o n f u n c t o r w i t h r e s p e c t t o V-M). P i c k z F -1 f o r c e r t a i n nl, ...,nr, dl,...,dS say z = (xl t . . . t x ) (dl +...+d S ) By Lemma 1.3.3.
. . .tEnr)(Edl+
t h e r e i s a 5. F h(R) such t h a t z = (Enl+
K, c
h(V).
. . . + E do)
-1
E d . c h(V) - { O } w h i l e some Eni o r E d . i s equal t o 1. J J Thus e i t h e r t h e n o m i n a t o r o r t h e denominator o f z, i n t h i s r e p r e s e n t a t i o n , and such t h a t Eni,
l i e s o u t s i d e M and t h i s means t h a t e i t h e r z o r z - l i s i n QM(V). T h a t t h e
'? may
valuation associated t o t h i s valuation r i n g i s e x a c t l y
be e a s i l y
checked. 1.3.14.
Proposition. Let
R
be a g r - f i e l d and l e t V be a
gr-valuation
r i n g o f R w i t h qr-maximal i d e a l M. Then b1 does n o t c o n t a i n Vc0
unless
e i t h e r V i s t r i v i a l l y qraded and t h e n so i s R, o r V = k [ T I o r V = k [ T - l ]
).
(where R = k [ T,T-'] P r o o f . Take y a F
c V-l
with 1 >
Vo such t h a t v ( a )
=
6 and
n # 0 t h e n ~ ( a yn) - ~ = 0 i.e.
o f V b u t t h i s c o n t r a d i c t s V-ln M
n k
suppose v ( y ) = m >
= 0 hence k c V and V,
a
t r i v i a l l y graded. Suppose Vo = k
Thus M
.
3
V
40
-m
I f t h e r e i s an x # 0 i n Vtl
u n i t of V b u t s i n c e y i s n o t a u n i t , by assumption, = 0
# 0, y
Vtl
c
hence T
V-l
c
with 1 > 0 i.e. V i s positively V i .e. k [ T ] c V .
f o r some
t
x is a
= 0 follows
i t f o l l o w s t h a t Vt = 0 f o r a l l t
If V i s n o t n e q a t i v e l y qraded t h e n t h e f o r e q o i n q i m p l i e s all y
= 0 implies
e n t a i l s Vo = k o r V, and hence R , i s
t > 0 t h e n we o b t a i n : y t x F V o = k. Hence, s i n c e yt x
t f o r a l l t > 0. Now f r o m V1 c Vtl
yn i s a u n i t
n Vo
c M u n l e s s n 5 0. B u t M Xk.
I f t h e r e i s an
0.
that y = 0 for
qraded. I n t h i s case T - l f V
S i n c e k [ T ] i s a q r - v a l u a t i o n r i n o and a
p r i n c i p a l i d e a l r i n g one e a s i l y checks t h a t V = k [ T I . S i m i l a r l y i f V i s n e g a t i v e l y graded t h e t V = k [ T - ' ] 1.3.15.
R
follows.
C o r o l l a r y . A n o n - t r i v i a l l y graded v a l u a t i o n r i n g V o f a g r - f i e l d
possesses elements o f n e g a t i v e v a l u a t i o n and o f non-zero
degree.
0.
B. Commutative Graded Rings
170
The construction in Theorem 1.3.13.
shows t h a t a praded valuation has
value g r o u p Z i f and only i f the associated valuation of the f i e l d of f r a c t i o n s has value g r o u p Z . I f t h i s i s the case, we may say t h a t the gr-valuation i s a d i s c r e t e gr-Val uation. 1.3.16. Proposition. Let V be a gr-valuation ring of the p r - f i e l d R and l e t v be the associated valuation function. Then v i s d i s c r e t e i f and only i f V i s Noetherian. Proof.
I f V i s Noetherian then so i s QM(V) and then QM(V) i s a d i s c r e t e
valuation ring of K , hence v i s d i s c r e t e .
2
For the converse i t will s u f f i c e t o check t h a t i f I ideals o f V then Q,(V)I Indeed, i f QM(V)I
=
2
J
c
11 are oraded
QM(V)J.
QM(V)J then any x c h ( J ) may be multiplied by some
c c V-M such t h a t cx c I. Write c = c l t
...t c n
with deg ti
=
t i $ t l <...< t and c i
M. From cx C I ,
cix c I follows ; b u t ci i s a u n i t o f V hence x c I . Because of Proposition 1.3.4. a qr-valuation rina V i s i n t e q r a l l y closed in R, b u t a l s o in K . Therefore a d i s c r e t e gr-valuation rinp will be a Noetherian i n t e g r a l l y closed domain i . e . also a Krull domain. The Kdim V=l, V-or hence V has c l a s s i c a l Krull dimension a t most two ( c f . A.II.5.). ideal P of height one i s e i t h e r M, or such t h a t P
0
=
0 . So i f
A prime
P # M has
h t ( P ) = 1 then Q (V) i s obtained as a l o c a l i z a t i o n of R and a t the P i s a d i s c r e t e valuation ring prime ideal RP of R and therefore Q,(V) ( b u t i t s valuation i s not the
extension of a gr-valuation of R!).
Conclusion : a d i s c r e t e gr-valuation rinq i s a Noetherian Krull domain ( a n d a reqular gr-local r i n q ) of dimension a t most 2. Results about the class g r o u p and other arithmetical f e a t u r e s a r e included in the next section.
171
II: Arithmetically Graded Rings
11.1. Graded Krull domains. Class groups. Throughout t h i s section R i s commutative domain with f i e l d of f r a c t i o n s K . We say t h a t R i s a Krull domain i f there i s a family
of d i s c r e t e rank
one valuation rings of K , { V i , i F I } say, such t h a t R such t h a t f o r
=
n V i and
ic1
any nonzero r c R we have t h a t r i s a u n i t i n V i f o r a l l
b u t a f i n i t e number of i c I . Let us r e c a l l some of the well-known properties of Krull domains : 11.1.1. Proposition. Let R be a Krull domain in K. closed. I f k i s a subfield o f K then
Then R i s i n t e g r a l l y
R n k i s a Krull domain in k . If L
i s a f i n i t e extension of K then the integral closure of R i n L i s a Krull domain i n L . The polynomial ring R [ X i ,
i F I]
,
a s well as the ring of
formal power s e r i e s R [ [ X I ] , i s a Krull domain. 11.1.2. Proposition.
If R =
n V i and J i s a subset of I then n V
i tI
jcJ
j
i s a Krull domain.If S i s a multiplicatively closed subset of R then
QS(R)
=
n
jtJ
V . (where 3 = { j F I , S c U ( V j )
}) i s a Krull domain.
11.1.3. Proposition. The domain R i s a Krull domain i f and only i f the following
conditions a r e s a t i s f i e d :
1. For every prime ideal P o f height one i n R, the localized rinq Q p ( R ) i s a d i s c r e t e rank one valuation ring. 2 . We have R =
n Q p ( R ) , the i n t e r s e c t i o n ranginq over a l l prime ideals of
height one of R. 3 . Any nonzero r c R i s contained in a t most a f i n i t e number of prime
idea’ls of height one.
B. Commutative Graded Rings
172
F o r more d e t a i l s on K r u l l domains we r e f e r and Samuel [ 129 1 , Nagata [ 77
1,
t o K r u l l [ 60
Bourbaki [ 11
1,
Zariski
I and Fossum [ 29 1 .
I t i s n o t e w o r t h y t h a t t h e r e do e x i s t K r u l l domains p o s s e s s i n a h e i a h t
one p r i m e i d e a l s w h i c h a r e n o t o f f i n i t e t y p e , c f . E a k i n and H e i n z e r
26 1 . On t h e o t h e r hand, i f R
[
i s a N o e t h e r i a n normal
domain t h e n
R i s a N o e t h e r i a n K r u l l domain. 11.1.4.
Theorem ( K r u l l
dimension
-
A k i z u k i ) . I f R i s a N o e t h e r i a n domain o f K r u l l
1 then every intermediate
r i n g between R and i t s i n t e g r a l
closure i n K i s a l s o Noetherian.
11.1.5.
Remark. If R has K r u l l dimension two t h e n t h e i n t e g r a l c l o s u r e
o f R i n K i s Noetherian b u t there
may e x i s t non-Noetherian i n t e r m e d i a t e
r i n g s . We w i l l r e t u r n t o t h i s s i t u a t i p n i n t h e graded case.
11.1.6.
Theorem (Mori-Nagata).
L e t S be t h e i n t e g r a l c l o s u r e o f t h e
N o e t h e r i a n domain R t h e n :
1. S i s a K r u l l domain 2. Only a f i n i t e number o f p r i m e i d e a l s o f S l y o v e r
a given prime
i d e a l o f R.
3. I f P c Spec S , p = P
nR
t h e n Q(S/P)
i s a finite
dimensional f i e l d
e x t e n s i o n o f Q( R/p). Proof. [
F o r t h e p r o o f of t h i s v e r y i n t e r e s t i n g theorem we r e f e r t o Fossum
291.
I t i s a known consequence o f a theorem o f I . Beck t h a t a f l a t e x t e n s i o n o f t h e K r u l l domain
n Vi i n K
iC I
i s a subintersection
An R submodule o f K, I say, i s c a l l e d a
fl
j cJ
fractional ideal
Vj,
J c I. of R if
t h e r e e x i s t s r # o i n R such t h a t r I cR. I f I i s a f r a c t i o n a l i d e a l o f R, p u t I* =
**
[ R : I ] = { h F K, h I c R } and w r i t e I
f o r (I*)*.
173
II. Arithmetically Graded Rings f*
If I = I
then I i s s a i d t o be a d i v i s o r i a l i d e a l o f R. A f r a c t i o n a l
ideal I o f R
such t h a t t h e r e e x i s t s a f r a c t i o n a l i d e a l J o f R such
t h a t I J = R i s s a i d t o be i n v e r t i b l e . Any i n v e r t i b l e
i d e a l of R i s
divisorial. I f R i s a K r u l l domain then t h e s e t D(R) o f d i v i s o r i a l i d e a l s may be
i n t r o d u c i n g t h e product 1.J = ( I J )
made i n t o a group by
** . The
subgroup
o f p r i n c i p a l ( d i v i s o r i a l ) i d e a l s i s denoted by P(R) and t h e q u o t i e n t group Cl(R) = D(R)/P(R) i s c a l l e d t h e _ d i v i s o r c l a s s group o f R. I t i s e a s i l y seen t h a t t h e i n v e r t i b l e f r a c t i o n a l i d e a l s o f R from a group Inv(R) c o n t a i n i n g P(R), and Inv(R) / P(R) i s isomorphic t o t h e P i c a r d qroup, P i c ( R ) , o f R which i s d e f i n e d t o t h e qroup o f o f p r o j e c t i v e R-modules o f rank 1. The group D(R) i s isomorphic fo
isomorphism classes
w i t h r e s p e c t t o t h e sensor product.
the f r e e a b e l i a n group generated by
the prime i d e a l s o f h e i q h t 1. Also, P(R) i s isomorphic t o U(K)/U(R) v i a t h e mapping d i v : U(K) o f prime i d e a l s o f
+
P(R) g i v e n by d i v ( X ) = d i v ( x R ) = t h e d i v i s o r
h e i g h t one c o n t a i n i n a xR.
For a K r u l l domain R t h e p r o p e r t y Cl(R) a f a c t o r i a l r i n g i.e.
=
0 is
e q u i v a l e n t t o R beinq
i f every element i n R i s , up t o m u l t i p l i c a t i o n
by a u n i t , u n i q u e l y a f i n i t e p r o d u c t o f i r r e d u c i b l e elements. L e t us mention t h e f o l l o w i n g r e s u l t o f Storch [ 103 I 11.1.7.
Proposition.
.
For a K r u l l domain R, t h e f o l l o w i n g statements
are e q u i v a l e n t :
1. Cl(R) i s a t o r s i o n group. 2. Every
subintersection o f R i s a r i n g o f quotients.
3. I f f, g
ideal
c
R then t h e r e i s an n
c N such t h a t R f n n Rg" i s a p r i n c i p a l
.
4. I f P i s a prime i d e a l o f h e i q h t 1, then Pn i s o r i n c i p a l f o r some n
c N.
6. Commutative Graded Rings
174
The f o l l o w i n g r e s u l t g i v e s t h e r e l a t i o n between d i v i s o r c l a s s groups o f subintersections o f R 11.1.8.
Theorem ( N a g a t a ) . L e t R be a K r u l l domain and l e t S be a s u b i n t e r -
s e c t i o n o f R, say R =
n Vi,
S =
ic I
fl V . jcJ
w i t h J c X'(R), the s e t o f
p r i m e i d e a l s o f h e i q h t one o f R, t h e n : 1.
TT
: Cl(R)
+
Cl(S) i s a surjection.
2. Ker n i s g e n e r a t e d by t h e c l a s s e s o f t h e p r i m e i d e a l s o f h e i o h t one not i n J.
I I . l . 9. Coro 1 1a r y . 1. I f S i s a m u l t i p l i c a t i v e l y c l o s e d s e t i n R t h e n C l ( R )
-f
C1(QS(R)) i s
s u r j e c t i v e and t h e k e r n e l i s g e n e r a t e d by t h e c l a s s e s o f p r i m e i d e a l s w h i c h meet S. 2. I f S i s g e n e r a t e d by p r i m e elements t h e n C l ( R )
+
C1(QS(R)) i s b i j e c t i v e ;
i n p a r t i c u l a r i f QS(R) i s f a c t o r i a l t h e n R i s f a c t o r i a l . L e t us now t u r n t o t h e case o f qraded domains i . e . f r o m hereon we assume that R
is a
graded commutative domain w i t h graded f i e l d o f f r a c t i o n s
Kg and f i e l d of f r a c t i o n s K. We w r i t e X = Spec(R), Xa = Spec ( R ) t h e s e t 9 1 o f graded p r i m e i d e a l s o f R, X1 = { P c X, h t (P) 5 1 1 , X1 = { P C X ,P F X 1. 9 9
A g r - f r a c t i o n a l i d e a l o f R i s a a r a d e d R-submodule o f Kg, I say,such d I c R f o r some nonzero d
c
that
R; o b v i o u s l y one may assume t h a t d c h ( R ) - { O } .
The elements o f D(R) w h i c h a r e qraded f o r m t h e suboroup o f w a d e d d i v i s o r s
D ( R ) . The subgroup o f graded p r i n c i p a l d i v i s o r s i s denoted by Po(R) . 9 t h e graded d i v i s o r c l a s s group o f R i s d e f i n e d t o be C1 (R) = D (R)/P,(R) (I 4 The g r a d e d P i c a r d g r o u p o f R, P i c ( R ) , is d e f i n e d t o be t h e a r o u p of
9 isomorphism ( n o t n e c e s s a r i l y degree z e r o ! ) c l a s s e s o f w a d e d p r o j e c t i v e
R-modules o f r a n k 1. S i m i l a r l y , I n v (R) denotes t h e qroup o f i n v e r t i b l e 9 graded f r a c t i o n a l i d e a l s of R. I t i s easy t o v e r i f y : P i c ( R ) = I n v (R)/P (R), 9 9 9
II. Arithmetically Graded Rings
175
and consequently we may consider P i c ( R ) and Pico(R) a s suboroups od C l ( R ) and C1 ( R ) r e s p e c t i v e l y . 9 A graded domain R is s a i d t o be a gr-Krull domain
11.1.10 D e f i n i t i o n .
i f t h e r e e x i s t s a family of d i s c r e t e g r - v a l u a t i o n r i n o s V i , i c I , such that 1. R
: =
n v.
'
ic1
2 . For a l l x
c h(K9) -
11.1.11. Lemma.
{O},
i c 1 , x n o t a u n i t of Vi } i s a f i n i t e s e t .
{
Let I c Kg be a nonzero graded f r a c t i o n a l i d e a l of R ,
then I* c Kg and I* i s graded. h* ( K 4 ) .
T = [ R:I]
Furthermore I*
=
n R x - l , where
XfT
I n p a r t i c u l a r , i f I i s i n v e r t i b l e then I-'= [ R : I ]
i s graded. Proof. Since I i s nonzero, I 8-R # 0 follows. Pick s F h*(InR). I f t
then t s c R hence t = r s-'
F
6
h(I*)
Kq f o r some r i n R . I t f o l l o w s t h a t I*cK".
Now, f o r a r b i t r a r y t c I* we have t I c R . So, i f we write t = t l +. . . + t n w i t h ti of degree d i beinq t h e
find t i x
c R for i
= 1,
..., n
we o b t a i n t i I c R o r ti F I* remaininq s t a t e m e n t s of t h e 11.1.12. P r o p o s i t i o n .
homcgeneous components of t i n K g , we Since the l a t t e r holds then f o r a l l x F h ( 1 ) This e s t a b l i s h e s t h a t I*is graded. The
emma a r e now e a s i l y checked.
The qraded domain R i s a gr-Krull domain i f and
only i f i t i s a Krull domain i n K . Proof. I t i s c l e a r t h a t i f R i s a Krull domain then i t w i l l a l s o be a gr-Krull domain ; l e t us now e s t a b l i s h t h e converse. Since Kg i s of t h e form k[X,X-
1
1 i t i s a Krull domain i n K
=
k ( X ) . Let i u j j j C J be t h e
family o f v a l u a t i o n s of K d e s c r i b i n g K g i n K . Let
v i be the (qraded)
v a l u a t i o n of Kg correspondina t o vi a s i n Theorem 1 . 3 . 1 3 . . Extend v i
B. Cornrnutatiw Graded Rings
176
t o a valuation
7. on 1
K by puttina
Ti ( a b - l )
vi (a)-vi ( b ) .
=
Consider the family of valuation rings of K , { W ] } the valuation rinos associated t o {wj}jFJ
U
l c I u j consistinq of
fii}iFI.Obviously
a r e d i s c r e t e rank one valuation rings of K .
the W k
and
n w1 = ( n w i ) n ~g = n (wi n ~ 9 =) n vi = R . lcIUJ i FI iF I i FI I f x = ab-l i s a r b i t r a r y b u t nonzero in K then the s e t s { i 6 I , v i ( x ) # 0} and { j F J , w . ( x ) # 0 J
of W1 3 i s f i n i t e . Before
}
a r e f i n i t e , therefore { 1 F I U J , x i s n o t a u n i t
o
turninq t o the study of special graded Krull domains l i k e g r -
Dedekind r i n g s , we include some general Droperties of araded Krull domains. I f R i s a gr-Krull domain then Cl(R) 2 C1 ( R ) and g Pic(R) 2' Picq(R). 11.1.13. Lemma.
Proof.
We know t h a t K4 = k [ X,X-']
(we may assume
that R i s non-trivially
graded because otherwise there i s nothinq t o prove) where k i s a f i e l d . By Nagata's theorem C l ( R ) i s qenerated by the prime ideals of heiqht one containing an homogeneous
nonzero element of R . I f P
i s such a prime
i . e . P i s a qraded orime i d e a l . Consequently 9 Cl(R) 5 C1 ( R ) . The statement about Pico(R) i s an easy exercise, l e f t 9 t o the reader. ideal then P
9
# 0 hence P
=
P
Some consequences of t h i s lemma have been included
in the exercises.
Let us focus on positively qraded Krull domains f o r a moment. 11.1.14. Proposition. Let R be a positively qraded f a c t o r i a l Noetherian domain. Then the followinq properties hold : 1. Ro i s f a c t o r i a l 2 . For each n F N, Rn i s a reflexive Ro-module.
177
II. Arithmetically Graded Rings
Proof. -
1. S e l e c t a , b F Ro such t h a t ab # 0. Then, f o r some c E R , Ra 11 Rb = Rc.
I t follows immediately from t h i s t h a t c must be homogeneous of degree z e r o , hence Roc = Ro a fl Rob. I f t h e i n t e r s e c t i o n o f p r i n c i p a l i d e a l s i s p r i n c i p a l then i t follows t h a t Ro i s f a c t o r i a l . 2 . Let p be a prime i d e a l of R o , of height one. Then :
1 Taking t h e i n t e r s e c t i o n over a l l p F X ( R o ) y i e l d s :
Since Rn i s obviously a f i n i t e l y qenerated and t o r s i o n f r e e Ro-module,
R,
=
n
pcX1(Ro)
QR
o -I)
(R,)
y i e l d s t h a t R n i s a r e f l e x i v e Ro-module.
11.1.15. Example ( c f . [ 29 I ) . Let r , s , t
< N be p a i r w i s e r e l a t i v e l y
prime. Let R be a Krull domain. Then R [ x , y , z ] w i t h xr + ys +zt
=
0
then R [ x , y , z l i s
i s a Krull domain and i f R i s a r e q u l a r l o c a l r i n q
f a c t o r i a l . Actually i f p r o j e c t i v e R-modules a r e f r e e then we o b t a i n Cl(R) 2 ' C l ( R [ x , y , z ] ) . I t i s c l e a r t h a t R [ x , y , z ] may be qraded by p u t t i n g
R [x,y,zIn
=
c Ri
x"'
yv2
Zv3
where n
=
i +
v1
st
+
v2
rt
+
v3
rs.
( I f R i s t r i v i a l l y graded then Ri = 0 i f i # 0 and Ro = R ) . 11.1.16.
Example. Let C be a Noetherian Krull domain and l e t 11 be a
f i n i t e l y generated C-module. Construct a p o s i t i v e l y qraded r i n g SC(M) a s follows : ( s ~ ( M =) c) ,~ ( s ~ ( M =) M ) ~, ..., power of M ,
s,(M)),=
nth-symmetric
... .
A r e s u l t going back t o P. Samuel , [ 98 1 , s t a t e s t h a t i f each (Sc(M))n
i s a r e f l e x i v e C-module then SC(M) i s a Noetherian Krull domain and C 1 ( C ) 2 C1(SC(M)). Moreover i f M i s an i n v e r t i b l e C-module then one may
B. Commutative Graded Rings
178
define t h e Z-graded
S I C ( M )
nth symmetric power of M-',
as before b u t w i t h (SlC(M))-, equal t o the for n
?
0. Checking along the line s of [ 29
Theorem 10.11. and Proposition 10.7.c; y i el d s t h a t SIC(M) is a graded Krull domain too. Note however t h a t the proof given i n 1 29 I of the fact that C l ( C )
+
C1(SC(M)) i s i n j e c t i v e f a i l s f o r S I C ( M ) . The study of
the kernel of the map C 1 ( Ro)
+
C1 ( R ) f o r
returned t o i n the following s ect i o n .
the Z-graded ring R will be
I
II. Arithmetically Graded Rings
179
11.2. Generalized Rees Rings and Gr-Dedekind Rinqs
This section extends, in the commutative case, some problems and r e s u l t s hinted a t in Section A.II.4. Recall t h a t
a graded commutative domain i s called a qr-principal ideal
ring i f every graded ideal i s p r i n c i p a l . A qraded domain i s said t o be
a gr-Dedekind ring i f every qraded ideal i s a projective module. 1 1 . 2 . 1 . Theorem. Let R be a graded domain. THe following statements are equi Val e n t . 1. R i s a gr-Dedekind r i n g .
2. R i s a K r u l l domain a n d nonzero araded Drime idea s are maximal qraded ideals. 3 . R i s Noetherian and i n t e g r a l l y closed i n i t s f i e l d of f r a c t i o n s K , and
nonzero graded prime i d e a l s a r e or-maximal.
4. Every graded ideal of R i s i n v e r t i b l e . 5. Every graded ideal of R i s ( i n a unique way) a product o f graded prime ideals. 6 . R i s Noetherian a n d g r - i n t e q r a l l y closed ( i n the sense of I I . 1 . 2 . ) , and every nonzero graded prime ideal of R i s gr- maximal.
7 . The graded f r a c t i o n a l ideals of R form a m u l t i p l i c a t i v e oroup. 8. R i s Noetherian and each l o c a l i z a t i o n REl of R a t a pr-maximal ideal PI o f R i s a principal ideal ring.
9 . R i s Noetherian a n d each graded localization R g of R a t a graded prime P ideal P o f R i s a qr-principal ideal rinq. 10. R i s Noetherian and each qraded l o c a l i z a t i o n
R i of R a t a gr-maximal
ideal M of R i s a gr-principal ideal ring.
11. All graded R-modules which are qraded d i v i s i b l e are g r - i n j e c t i v e . Recall t h a t N c R-gr i s graded d i v i s i b l e i f a x = b with a c h ( R ) , b c h ( N ) has
a solution in N .
8. Commutative Graded Rings
180
Most implications may be established i n a way s i m i l a r t o the
Proof.
proofs of the corresponding statements in the ungraded case, taking care
t o i n s e r t the standard t r i c k s concerning gradation when necessary. I n p a r t i c u l a r 6 = 5 i s a f u r t h e r refinement of 3 = 5 which i s the s t r a i q h t forward "graded version" of the analoqous unqraded statement. 6
3 stems
from the f a c t t h a t the integral closure of a oraded domain in i t s f i e l d of f r a c t i o n s i s graded, see Theorem 1 . 3 . 7 . , Corollary 1.3.8. Let us aive
a more d e t a i l e d proof only f o r the implication 7 = 6 . If I i s a graded ideal of R t h e n by 7 , I i s i n v e r t i b l e and hence of f i n i t e type. Thus R i s gr-Noetherian, hence Noetherian. Now suppose x c h ( K q ) s a t i s f i e s xm + am-l xm- 1 + . . . + a o = 0 with deo a i - deq a i + l = deg x , i = 0,
...,m - 1 ,
of R generated by 1, x ,
ai c R for i
=
Then xm i s in the fractional qraded ideal J
a n d ai c R . 0, ...,m - 1 .
...,
xm-'.
I t i s c l e a r t h e t J 2 C J follows from
Therefore J L = J follows from R c J . However
J i s i n v e r t i b l e by assumption, thus J
=
R and i t has been proven t h a t R
i s g r - i n t e g r a l l y closed in Kg (and then a l s o in K). Consider P # 0 in Spec ( R ) , and take a c h ( R ) - P . Write H = P+Ra, H ' = Help. Obviously P c H ' . 9 On the other hand i f y t H ' then a y c HH-lP = P y i e l d s y F P . This e n t a i l s t h a t H ' = P and HP = P whence H gr-maximal i d e a l . The equivalence 1
-
=
R by 7 . Consequently P i s a
3 y i e l d s t h a t the notions of gr-local ar-Dedekind ring
a n d d i s c r e t e gr-valuation ring a r e equivalent ( i n view of the r e s u l t s
of 1.3.)
.
A n easy elaboration of t h i s f a c t y i e l d s : 11.2.2.
Proposition.
Let V be a qr-valuation rina of the a r - f i e l d R ,
then the following statements are equivalent : 1. V i s a d i s c r e t e g r - v a l u a t i o n r i n a . 2 . V i s a gr-Dedekind ring.
181
II. Arithmetically Graded Rings
3. V i s a gr-principal ideal domain. 4 . The gr-maximal ideal M of V i s qenerated by a simple homoaeneous
element of V . 5 . V i s a Krull domain with Cl(V)
=
0.
6 . V i s factorial
7. V i s a gr-Krull domain with Clg(V)
=
0
8. V s a t i s f i e s the ascending chain condition f o r principal i d e a l s .
9 . V i s Noetherian. 10. V i s a Krull domain and h t ( M )
11.2.3. Lemna.
=
1.
I f R i s a qr-Dedekind rina then Ro i s a Dedekind rinq
Proof. Let I ' be an ideal of R o ,
generated by e l ,
Re +...+Re i s a projective R-module, RI' i s a 1 n gr-free R-module L ( s e e A . I . ) ,
...,e n
say.
Since
d i r e c t summand of a
generated by elements v l ,
..., v,,
of degree
zero. I t i s c l e a r t h a t Lo i s a f r e e Ro-module and t h a t ( R I ' ) o = I ' i s a d i r e c t sununand of Lo. Hence I ' i s a projective Ro-module and therefore
Ro i s a Dedekind domain. 11.2.4. Lemma.If R i s a gr-Dedekind ring then there i s a n e E N such
that R = Proof.
8
kcZ Easy.
Rek with Re # 0.
11.2.4. Lemma. I f R i s a gr-Dedekind rinq t h e n every f r a c t i o n a l waded ideal of R can be generated by two homogeneous elements, one chosen a r b i t r a r i l y in the i d e a l . Proof.
Let I be a araded f r a c t i o n a l ideal of R . An aroument s i m i l a r t o
the ungraded case y i e l d s t h a t I may be generated by two elements a and b,
a E h(1) a n d b not necessary homoqeneous.
Write b = b l +...+ bn, d e q ( b i ) = t i ,
B. Commutative Graded Rings
182
tl < . . . < t nSince . bi c I we have : (*) b .
x.a + y . b + . . . + y . b . + . . . y . b 1 1 1 1 1 i n f o r each i = 1, ... ,n f o r c e r t a i n x i , y i c h ( R ) . I f bi = x i a then we may 1
=
r e p l a c e b by b - bi and r e p e a t t h e argument. If b # bi # xia then comparino degrees i n (*) y i e l d s e i t h e r y i and x.a 1
+
bl+ ,..+fjit...t b n
=
=
0 which c o n t r a d i c t s b i # x i a , o r y i = 1
0. In t h e l a t t e r case b-bi F Ra hence we
may r e p l a c e b by b i . F i n a l l y we o b t a i n t h a t I i s generated by a and some b i Now we determine t h e s t r u c t u r e of gr-Dedekind r i n o s , f i r s t
i n the s t r o n g l y
graded case i . e . when RR1 = R . 11.2.5. Lemma.
Let R be a or-Dedekind r i n g which is s t r o n g l y qraded
and l e t P be a graded prime i d e a l of R . Then Po
Proof.
P n e n t a i l s n = 1.
c
Since R i s a qr-Dedekind r i n g P # Pn i f n
>
1. Therefore t h e r e
4
PR-t R P t
c
P n . B u t then R-t P t c Po c Pn y i e l d s t Pn, hence R u t c P . However ( R - l ) c R-t then i m p l i e s R-l c P
but R R - l
=
R c o n t r a d i c t s P # R . Consequently i f n
e x i s t s a t c Z such t h a t P t
>
1, then Po
4
Pn.
1 1 . 2 . 6 . P r o p o s i t i o n . Let R be a s t r o n q l y qraded qr-Dedekind rino, then :
1. For each graded prime i d e a l P of R , P = R P o . 2. For every graded i d e a l of R i s generated by i t s p a r t o f deqree z e r o . 3 . I f Ro i s a p r i n c i p a l i d e a l r i n q then R i s a o r - p r i n c i p a l i d e a l r i n o
and R 2 Ro [ X , X - ' ] Proof.
where X i s an i n d e t e r m i n a t e .
1. and 2 . hold f o r any s t r o n g l y w a d e d rinq, c f . 1 . 1 . 3 .
3.
Immediate from 2. and the r e s u l t s of S e c t i o n A . I . 3 . 11.2.7. Theorem.
Let R be a s t r o n g l y qraded ar-Dedekind r i n g then there
e x i s t s a f r a c t i o n a l i d e a l I of Ro such t h a t R = H 0 ( I ) ( f o r t h e d e f i n i t i o n of t h e g e n e r a l i z e d Rees rinq, s e e A . I I . 4 . ) There i s a canonical group epimorphism
~i
: C1(Ro)
+
C l ( R ) , t h e kernel of
TI
i s e x a c t l y t h e suboroup
of C1(Ro) generated by t h e c l a s s of I . Poreover , TI i s an isomorphism
11. Arithmetically Graded Rings
183
i f and o n l y i f I i s a p r i n c i p a l i d e a l and i n t h i s case R
Also, every
fl0
.
Ro [ X , X - l ]
( I ) f o r some f r a c t i o n a l i d e a l I o f Ro i s a qr-Dedekind r i n q .
P r o o f . By A . I I . 4 . ,
Proposition A.II.4.2.,
the rinas x0(I) are a l l
N o e t h e r i a n r i n g s , so t o show t h a t N 0 ( I ) i s qr-Dedekind i t w i l l s u f f i c e to
establish t h a t i t i s i n t e a r a l l y closed i n i t s f i e l d o f fractions, w h i l s t
graded p r i m e i d e a l s a r e qr-maximal. Let
5
be t h e i n t e g r a l c l o s u r e o f S = f l 0 ( I ) i n i t s f i e l d o f f r a c t i o n s
c (5 -
Suppose y
S)n f o r some n C Z .
Then S-n y c
5
b u t S-n y
because SS-, = S f o r a l l n F Z. T h e r e f o r e t h e r e i s a z F
z s a t i s f i e s T~ Tn t (an-l)o
t
Tn-’
Dedekind r i n q , z
Tn- 1
an-l
+...+(a
c So
)
+...t
0 0
=
a.
w i t h ai
0 w i t h (ai)o
of
S,
- S)o. I f
F S then i t also s a t i s f i e s F
S o . S i n c e So = Ro i s a
follows, a contradiction.
us now show t h a t f o r any p r i m e i d e a l
(5
4
Q(S)
Therefore
5
= S. L e t
Po o f Ro, SPo i s a ar-maximal i d e a l
s.
Indeed S/SPo i s a s t r o n g l y graded r i n q o f t y p e Z w i t h (S/SPo) = Ro/Po 0
b e i n g a f i e l d . Thus S/SPo i s a q r - f i e l d and hence SPo i s ar-maximal i n S .
So f a r we have e s t a b l i s h e d t h a t e v e r y q e n e r a l i z e d Rees r i n q N 0 ( I ) i s a gr-Dedekind r i n g . C o n v e r s e l y i f R i s a qr-Dedekind r i n p w h i c h i s s t r o n a l y graded t h e n i t i s c l e a r t h a t , i n o r d e r t o i n v e r t a l l elements o f h(R) i t suffices
-1 f xdl
Kg
t o i n v e r t t h e elements o f Ro-tO}. Take x - ~F R-l-{O}.
y i e l d s t h a t K’I = KO [ x - ~ ,
XI:] where
Then
KO i s t h e f i e l d o f f r a c t i o n s
of Ro. L e t J be t h e l a r q e s t f r a c t i o n a l i d e a l o f Ro such t h a t J x - ~c R and I t h e l a r a e s t
J X - =~ R-l,
( f r a c t i o n a l ) i d e a l o f Ro such t h a t I
-1 I x - ~= Rl.
From R-l
XI: c R.
Then :
R1 = Ro i t f o l l o w s now t h a t IJ = J I = Ro
i . e . J = I-l.S i m i l a r l y , i f I2 i s t h e i d e a l o f Ro n i v e n by (R2 : xl)2
2
t h e n 121-1 c I f o l l o w s f r o m R-l R2 c R1, hence I2 I . On t h e o t h e r 2 2 2 hand, I x1 = (Ixl) c R2 y i e l d s I c 12. R e p e a t i n o t h i s arqument,
184
B. Commutative Graded Rings
we o b t a i n R 2 N0(I). Finally
,let
TI
: C1(Ro)
+
C l ( R ) be d e f i n e d as t h e c l a s s map o b t a i n e d
f r o m mapping an i d e a l o f Ro t o i t s e x t e n s i o n i n R. T h a t
If H i s
TI
i s surjective
Certainly the class o f I ,
f o l l o w s f r o m P r o p o s i t i o n 11.2.6.. contained i n Ker
TI
s i n c e No(I).X-'
=
I
say, i s
No(1).I.
an i d e a l o f Ro such t h a t RH i s p r i n c i p a l t h e n w r i t e RH = Rh f o r
some h c h(RH), o f degree t say. T a k i n q p a r t s o f degree n y i e l d s (RH) = n n
= HI X
= Rn-th'
where h ' = hX
Consequently : H I n = I"'h
t
(we i d e n t i f i e d R and No(I) h e r e ) .
and I? i s i n t h e subrlroup o f C1(Ro) g e n e r a t e d
by t h e c l a s s o f I . F i n a l l y i f I = R i i s p r i n c i p a l t h e n R 0
and C l ( R ) '2' C I R o ) .
Conversely i f
Ro [ i X , ( i X ) - ' ]
i s a l s o i n j e c t i v e t h e n R I = RX-'
TI
entails
that I i s principal. 11.2.8.
Proposition.
If R =
0
( I ) 2 N0(J) t h e n I and J
rewesent the
same e l e m e n t o f C1(Ro). Conversely, i f I and J a r e i n t h e same c l a s s t h e n
__ P r o o f . W i t h o u t l o s s o f g e n e r a l i t y we may assume t h a t I and J a r e Ro- i n t e g r a l i d e a l s . From R I = RX-' #,(J)
=
2
n
we o b t a i n
I
=
RIX
-1
=
JYX-'
because R1 = JY where
JnYn. T h e r e f o r e IJ-1 = ROY".
Conversely, i f I = J z f o r some z 6 Ro t h e n :
N0(I)
=
11.2.9.
2 ( J z ) ~Xn = 1 Jn ( z X ) ~2 2 Jn Y n = #,(J). nFZ ncZ ncZ Remark. A qr-Dedekind r i n q R w h i c h i s p o s i t i v e l y oraded i s t h e f o r m
k [ X I where Ro = k i s a f i e l d and X a v a r i a b l e .
Proof.
R+ i s a nonzero qraded p r i m e i d e a l o f R hence i t i s qr-maximal.
Therefore R
0
= R/R+
i s a g r - f i e l d hence a f i e l d .
graded p r i m e i d e a l o f R hence R i s a q r - o r i n c i p a l
So R,
i s t h e unique
i d e a l r i n q . I f R,
=
t h e n R = Ro [ a ] and i t i s c l e a r t h a t a i s t r a n s c e n d e n t a l o v e r Ro s i n c e d e g a f 0.
Ra
185
It. Arithmetically Graded Rings
Rme w i t h t h e g r a d a t i o n ( R ( e ) ) n = Re,., 8 m U I t t u r n s o u t t h a t i n t h e abscence o f t h e s t r o n g l y graded p r o p e r t y ,
Recall t h a t
i.e.
R ( e ) , e F Z, i s t h e
i f RR1 # R, t h e n f o r some e F
11.2.10.
is nice.
Lemma. I f R i s a qr-Dedekind r i n o t h e n R(e) i s a qr-Dedekind
r i n g f o r any nonzero e F Proof.
N t h e s t r u c t u r e o f Riel
N.
By C o r o l l a r y A . I I . 3 . 1 3 . ,
R(e) i s N o e t h e r i a n . We may c o n s i d e r
,(e)
as a s u b r i n g o f R, n e q l e c t i n g t h e a r a d a t i o n , and t h e graded f i e l d o f f r a c t i o n s Q q ( R ( e ) ) o f R ( e ) may be c o n s i d e r e d as a s u b r i n g o f Qq(R). T h e r e f o r e an x
Qg(R(e))
w h i c h i s i n t e q r a l o v e r R(e) i s i n t e o r a l o v e r R;
t h u s i n R. O b v i o u s l y Q q ( R ( e ) ) = s k ( ' ) =
Q g ( R ) ( e ) and R
n
+
0
Hence Q g ( R ( e ) ) =
T h i s y i e l d s t h a t x F R ( e ) . Moreover
Qg(R)(e) =
t h e correspondence P
where S = R - { O } .
correspondes between
P(e) defines a b i j e c t i v e
Spec (R) and Spec ( R ( e ) ) , t h e i n v e r s e correspondence b e i n q g i v e n by 9 9 Q r a d ( R ( 1 Qme)). T h e r e f o r e , t h e graded p r i m e i d e a l s o f R ( e ) a r e a r m
maximal. The lemma now f o l l o w s f r o m Theorem II.2.1,3. 11.2.11.
Lemma. I f R i s a qr-Dedekind r i n q and P i s
a graded p r i m e i d e a l
o f R t h e n t h e graded r i n g o f f r a c t i o n s a t P i s o b t a i n e d by l o c a l i z i n a a t the m u l t i p l i c a t i v e l y
c l o s e d s e t Ro-Po and t h e l o c a l i z e d r i n o i s a d i s c r e t e
gr-valuation ring. Proof.
By Theorem 11.2.1. i t i s c l e a r t h a t Q:(R)
r i n g . Now R/P i s a graded f i e l d . I f
i' C (R/P)-, 9 and j' i n
i
F (R/P),
i s a discrete gr-valuation
i s nonzero t h e n t h e r e i s an
w h i c h i s a l s o nonzero and i f y and y ' a r e r e p r e s e n t a t i v e s f o r R then i t s u f f i c e s
t o i n v e r t yy'
c
Ro i n o r d e r t o i n v e r t y .
T h i s p r o v e s t h e s t a t e m e n t i n case R/P i s n o n - t r i v i a l l y graded. I f R/P i s t r i v i a l l y qraded t h e n P c o n t a i n s a l l Rn w i t h n # 0.
B. Commutative Graded Rings
186
Then the gr-maximal ideal M of Q:(R) (note t h a t i n this. case R-P
=
Q;
impossible unless
a l s o contains a l l ( Q ; ( R ) ) n
Ro-Po!).
( R ) i s positively
with n # 0
By Proposition 1.3.14. t h i s i s o r negatively qraded. B u t then R
i s positively
o r neaatively graded hence of the form k [ X I because o f
Remark 11.2.9.
.
I n t h i s case P = ( X ) and the statement of the lemma i s
then t r i v i a l . Let R be a gr-Dedekind r i n g , then there e x i s t s a n e F N
11.2.12. Theorem.
and a f r a c t i o n a l ideal I of Ro such t h a t Proof. If RR
I V-
..Pnn.
RR1 = P y l .
of R i t follows
r, 0
<
r
<
R then e
=
Ho(I).
=
1 and Theorem 11.2.7. holds
.
I f RR1 # R write
Since R / P i s a g r f i e l d f o r every qraded prime ideal P # 0
t h a t each such P contains
8 Rme+r f o r some e F N, some mcZ e . The argument used i n the proof of Lemma 11.2.11. s t a t e s
t h a t P cannot contain a l l R t with t # 0, hence e # 0.
I n other words (ei ) such t h a t Pi does
f o r each Pi containing R1 we find an e i 3 0 in N (ei1 Let e be the smallest common multiple of the n o t contain ( R )l. ei i = l . . . n . On the other h a n d , i f P R1 then P Rm f o r any
+
rn
>
0 and P ( e ) $ ( R ( e ) ) l .
a gr-Dedekind rinq
+
By Lemma 11.2.10 i t follows t h a t
a n d since R ( e )
(R(e)),=
R(e)
follows from the f a c t
t h a t ( R ( e ) ) , i s not contained in any graded prime follows t h a t Theorem 11.2.7.
is
ideal of R ( e )
it
applies t o R ( e ) ; whence the statement of
the theorem follows. Before turning t o the study of extensions
of gr-Dedekind rings l e t us
introduce the graded R-length of a n M F R-gr. Let R be any graded domain and l e t
E F R-gr be f i n i t e l y qenerated. Denote by t E
of E consistino o f the torsion elements. Then M
=
c R-gr the submodule
E/tE is a finitely
generated t o r s i o n - f r e e graded module. Let t v l , . . . , v n }
be a maximal s e t
of homogeneous elements of M among a given s e t of homogeneous generators
fl. Arithmeticaffy Graded Rings {y l,...,ym}
such t h a t
{V
l,...,~n}
187
i s l i n e a r l y independent. L = Rvl
@...aRvn
i s g r - f r e e maximal submodule o f M. L e t ' s see t h a t t h e dimension o f L, i s u n i q u e l y determined. F o r each y . t tyl,
,..., bnj c
aj,blj
J
}, t h e r e e x i s t elements
...,y
h(R) n o t a l l 0, such t h a t a.y. J J
+
b l j vl+
...+ b nJ.v n
=
0.
E v i d e n t l y , a . # 0, w h i c h p r o v e s t h a t a . y . c L. L e t a = al...am. We have a J J J monomorphism o f aM i n t o L. L e t u l , ..., us be a n o t h e r maximal s e t o f l i n e a r l y i n d e p e n d e n t homogeneous elements o f FI and suppose s > r . Then t h e elements
..., auS
aul,
,...,s
j = 1 5
2 j=1
C.U. = J J
vi's
a r e n o t l i n e a r l y i n d e n e n d e n t and t h e r e e x i s t elements c . c h ( R ) , J S S n o t a l l such t h a t
2
j=l
c. a u J
j
= a
c c . u . = 0. T h i s y i e l d s j=l J J
0, c o n t r a d i c t i o n . S i m i l a r l y , by i n t e r c h a n g i n g t h e r o l e s o f t h e
and t h e u . ' s , one p r o v e s t h a t t h e case r > s i s i m p o s s i b l e t o o . So r = s . J
11.2.13.
D e f i n i t i o n . The d i m e n s i o n o f t h e g r - f r e e module L , j u s t d e f i n e d ,
i s c a l l e d t h e graded R - l e n g t h o f t h e f i n i t e l y g e n e r a t e d module E. I f E i s n o t f i n i t e l y generated, d e f i n e i t s graded R - l e n q t h as t h e supremum o f t h e graded r a n k s o f t h e elements o f t h e f a m i l y o f f i n i t e l y g e n e r a t e d submodules o f E. 11.2.14.
Lemma. L e t R be a graded n o e t h e r i a n r i n g w i t h o u t z e r o d i v i s o r s
i n w h i c h e v e r y nonzero graded p r i m e i d e a l i s qraded maximal and M a f i n i t e l y g e n e r a t e d graded R - t o r s i o n module. Then t h e graded R - l e n g t h o f M i s f i n i t e .
Proof. of M
As M i s a qraded t o r s i o n module, each a s s o c i a t e d graded p r i m e i d e a l
i s nonzero, hence maximal ( f o r i n f o r m a t i o n s a b o u t t h e qraded a s s o c i a t e d
p r i m e i d e a l s of a graded module, c f . s e c t i o n A . I I . 7 . )
.
The lemma f o l l o w s
t h e n f r o m t h e f a c t t h a t , s i n c e t h e c a t e a o r y o f graded R-modules i s a G r o t h e n d i e c k c a t e g o r y , t h e Jordan-HBlder theorem h o l d s i n i t .
B. Commutative Graded Rings
188
11.3.15.
Lemma. L e t R be a qraded r i n g , T a qraded R-module,
(Ti)
an
ascending, f i l t e r i n g f a m i l y o f submodules o f T, such t h a t t h e i r u n i o n i s T. Then t h e graded l e n g t h of T on R i s equal t o t h e supremum o f t h e qraded l e n g t h s o f t h e Ti on R . 11.2.16.
Lemma. L e t R be a qraded N o e t h e r i a n r i n o w i t h o u t z e r o d i v i s o r s
such t h a t e v e r y nonzero qraded p r i m e i d e a l of R i s w a d e d maximal,
M a
t o r s i o n - f r e e graded R-module of f i n i t e qraded r a n k n, and a a nonzero homogeneous element o f R . Then R/aR i s a qraded R-module o f f i n i t e l e n a t h and we have : l e n q t h (M/aM) i n . l e n g t h (R/aR). 11.2.17.
Theorem. ( " g r a d e d " K r u l l - A k i z u k i ) L e t R be a graded n o e t h e r i a n
ring without
z e r o d i v i s o r s such t h a t e v e r y nonzero qraded p r i m e i d e a l i s
graded maximal, Kg i t s graded f i e l d o f f r a c t i o n s Lo a qraded e x t e n s i o n o f Kg o f f i n i t e deqree and S
a qraded s u b r i n q o f Lq c o n t a i n i n q R. Then
S i s N o e t h e r i a n , and e v e r y nonzero graded n r i m e i d e a l o f S i s gr-maximal. The p r o o f s o f t h e p r e c e d i n g a s s e r t i o n s a r e r a t h e r s i m i l a r t o t h e ungraded e q u i v a l e n t s ( c f . [ 13 1 ) . C a u t i o n ! qraded f r e e modules a r e qraded and f r e e b u t t h e converse does n o t h o l d : as a consequence t h e graded r a n k can be d i f f e r e n t from t h e rank ! From t h e above theorem we o b t a i n t h e f o l l o w i n 9 two c o r o l l a r i e s .
11.2.18. C o r o l l a r y . T h e graded i n t e g r a l c l o s u r e o f a gr-Dedekind r i n g i n a graded e x t e n s i o n o f f i n i t e degree o f i t s graded f i e l d o f f r a c t i o n s i s a gr-Dedekind r i n g .
11.2.19.
C o r o l l a r y . L e t R be a gr-Dedekind r i n q and S a m u l t i p l i c a t i v e
c l o s e d s u b s e t o f h*(R). a gr-Dedekind r i n g .
Then t h e graded l o c a l i z a t i o n o f R a t S, Q z ( R ) , i s
II. Arithmetically Graded Rings
189
Remark. I f R i s a q e n e r a l i z e d Rees r i n g , say R = # o ( I ) , t h e n t h e
11.2.20.
R , i n an e x t e n s i o n of t h e f i e l d o f f r a c t i o n s i s a
i n t e g r a l c l o s u r e o f R,
g e n e r a l i z e d Rees r i n g t o o . T h i s f o l l o w s f r o m t h e f a c t t h a t , s i n c e R-l = Ro,l
c
(6)l(R)-1andhence
(R)l(R)-l
I f we denote by K = KO [ X,X-'] f r a c t i o n s o f R and
=
(R)o.
and L = Lo [ Y , Y - l ]
respectively
R1 =
t h e graded f i e l d s o f
!!
and i f we p u t
= (R)o(J).
then t h e class : Lo [ X , X - \
o f 61 i n C(a) i s equal t o t h e c l a s s o f Je, where e = dim(Lo [Y,Y-'] t h i s i s v e r i f i e d i n a s t r a i g h t f o r w a r d way. 11.2.21.
The Graded Norm.
L e t R be a gr-Dedekind r i n g i n Q ( t ) where t i s a v a r i a b l e o v e r t h e r a t i o n a l numbers Q. L e t S be t h e graded i n t e g r a l c l o s u r e o f
R
i n a f i n i t e extension
f i e l d L o f Q ( t ) . We may suppose t h a t L i s t h e f i e l d o f f r a c t i o n s o f S,
L
o t h e r w i s e we r e s t r i c t
( n o t e t h e c o n n e c t i o n t o qraded r e a l i z a b l e f i e l d s ) .
L e t P be a graded p r i m e i d e a l o f S and p u t D = R
n
P. We o b t a i n t h e f o l l o w i n g
inc7usion o f graded f i e l d s : R/p = ( R / P ) ~[ X , X - I ] hence deg X
deg Y.
?
+
S/P = (S/P)o [ y , Y - ' l
,
(We assume t h a t t h e graded f i e l d s have been w r i t t e n
down such t h a t deg X 2 0, deq Y
>
O(i.e.
up t o i n t e r c h a n g i n q X and X-',
e
Y
such t h a t X = XY P e q u a l s t h e r a n k [ (S/P)o[Y,Y-l] :(S/P)o[ X,X- 11 1 .
and Y - l i f n e c e s s a r y ) . So we f i n d a n a t u r a l number e with X
c
(S/P)o. Clearly t h i s
The norm o f P
0
e
P i n S o i s t h e number o f elements i n t h e r e s i d u e f i e l d
so/po = (S/P)o w h i c h i s a f i n i t e e x t e n s i o n o f t h e p r i m e f i e l d Ro/Po. i s c l e a r t h a t a d d i t i o n o f X,X-'
It
made S/P i n t o an i n f i n i t e r i n q , however
i t makes sense t o d e f i n e t h e graded norm o f P as f o l l o w s N(P) = No(Po)ep,N(P)n= = N(P)n and i f I = P1"1 ...Pr'r
t h e n we p u t N ( I ) = nr N(Pi)
'.
V.
i=l 11.2.22.
The graded
f u n c t i o n o f t h e (Fraded) i n t e q r a l c l o s u r e o f a. graded
Dedekind r i n g i n Q ( t ) . Definition.
t: ( s ) 9
=
z I
N(I)-',
( s a complex v a r i a b l e ) where t h e
sunwnation
190
B. Commutative Graded Rings
i s o v e r graded i n t e g r a l i d e a l s o f S. The d e f i n i t i o n o f t h e norm i m p l i e s t h a t
c0(s) i s
r: 9 ( s )
2.
Jg(S)
i s dominated by ~ ~ ( s where ) ,
t h e z e t a f u n c t i o n o f t h e p a r t o f deqree 0.
11.2.23. Proposition. 1.
r: 9 ( s )
F o r Re(s) > 1 we have :
converqes a b s o l u t e l y and a l m o s t u n i f o r m l y =
n
PcSpec ( S ) 9
(l-N(P)
-1 -1
1
Proof. 1.,2.
S l i g h t m o d i f i c a t i o n o f t h e p r o o f i n t h e un9raded case, c f . f o r
example [ 99 1 . Note t h a t t h e p r o d u c t a p p e a r i n g i n 3 i s f i n i t e . 3. Obvious f r o m 2. L e t us end t h i s s e c t on w i t h some r e s u l t s on t h e c l a s s group.
11.2.24. Proposition
I f R i s a gr-Dedekind r i n g t h e n t h e r e e x i s t s e F
N
such t h a t t h e f o l l o w i n g diagram o f group homomorphisms i s commutative.
c1 (R) Where
7
i s t h e c l a s s o f I i n C1(Ro) and I i s t h e f r a c t i o n a l i d e a l o f Ro
describing the structure o f R(e). Proof.
Take e and I as i n Theorem 11.2.12.
I d e n t i f y R(e) w i t h a s u b r i n a
o f R by f o r g e t t i n g t h e g r a d a t i o n . E x t e n s i o n o f i d e a l s f r o m Ro t o R ( e ) , "from"
t o R and f r o m Ro t o R d e f i n e s t h e arrows i n t h e diagram. One
e a s i l y checks c o m m u t a t i v i t y o f t h e diagram.
191
II. Arithmetically Graded Rings
I f RR1 # R t h e n we r e f e r t o t h e i d e a l RIR-l
o f Ro as b e i n g t h e
o f t h e qr-Dedekind r i n q R and i t w i l l be denoted by S ( R ) .
discriminator
1 1 . 2 . 1 5 . Lemma.
L e t R be a gr-Dedekind r i n g and
K
be an
k e r n e l f u n c t o r on t h e c a t e g o r y o f graded R-modules. Then,
dempotent f S denotes
Q j ( R ) , S i s a gr-Dedekind r i n g w i t h
{P,
Cl(S) = Cl(R)/ c
P 6 L(K)} >
where L ( K ) i s t h e f i l t e r o f
K ,
and P i s graded p r i m e .
S i n c e R i s a gr-Dedekind r i n g ,
Proof.
l o c a l i z a t i o n Q:
has p r o p e r t y T(A 11.9) and t h e
K
i s p e r f e c t . F o r e v e r y i d e a l L o f 5 , we t h e n have L = S.(LnR)
and t h e r e f o r e we o b t a i n an epirnorphism, Y = Cl(R)
+
Cl(5).
I f J i s an i d e a l o f R such t h a t S J = Sa f o r some homogeneous a c S, t h e n t h e r e e x i s t s an i d e a l L 6 L ( K ) such t h a t La c J . The g r a d e d Dedekind p r o p e r t y y i e l d s La = J.H where H i s such t h a t SH = S i . e . H F
L(ic).
H may
"1 " be w r i t t e n as H = P ... Pnn and each Pi i = 1,..., n belongs t o L ( K ) . So J = H-'
La and
-
J =H-'i.
prime belonqing t o
L(h)}>,
Now
thus
i 5
and
R
belong t o c {
belongs t o
P, P
i s a oraded
i t t o o , hence Ker y e q u a l s
t h i s subgroup. 11.2.26.
L e t R be a gr-Dedekind r i n n and S a m u l t i p l i c a t i v e
Corollary.
c l o s e d s u b s e t o f h*(R). and P
n
S #
Then : C l ( Q ? ( R ) ) = C l ( R ) / < I P ,
P praded p r i m e
@I>.
11.2.27.
C o r o l l a r y . L e t R be a gr-Dedekind r i n g and I a praded i d e a l o f R.
Then :
Cl(Q:(R))
= Cl(R)/<
P, ,..., Ps
>
where P1 ,..., Ps a r e t h e graded
p r i m e i d e a l s c o n t a i n i n g I. P r o o f . I n t h i s case L ( K ~ c) o n t a i n s o n l y a f i n i t e number o f graded rsrime i d e a l s c o n t a i n i n g I.
192
B. Commutative Graded Rings
11.2.28. C o r o l l a r y . W e have a commutative )d Cl(R)
where P1,.
.. ,Ps
>-
diagram.
C1
=
C l ( ? ) / < p l,...,ps>
a r e the graded prime i d e a l s c o n t a i n i n ? R,;
t h e prime i d e a l s of Ro containinq F ( R ) i . e . p i = Pi
n
p l , . . . ,ps a r e
Ro, and
6(R)
is
( R ) definin? the generalized t h e f r a c t i o n a l i d e a l of Qa 6(R) 0 Rees r i n g s t r u c t u r e of Q:R (R). 1
t h e c l a s s of
Proof.
Apply c o r o l l a r y 11.2.27. with I
=
RR1 and note t h a t Qg
aR)
a s graded r i n g s because of Lemma II.2:ll.
(R)
Q
RR1
(R)
11.2.29. P r o p o s i t i o n . Let R be a qr-Dedekind r i n g such t h a t there i s an N
C
N such t h a t t h e g e n e r a l i z e d Rees r i n g
C1(R(N))
+
R ( N ) has the property t h a t
Cl(R) i s i n j e c t i v e . Then
Cl(Qg ( R ) ) RR1
C1(Ro)/
7
>,
where I i s the i d e a l of Ro determining t h e s t r u c t u r e of t h e a e n e r a l i z e d Rees-ring R ( N ) and p i , i
=
1, ..., S , a r e the p r i m i t i v e f a c t o r s of the
discriminator in Ro. Corollary I I . 2 . 2 8 . , we w r i t e I , ( R ) = Q : ( R ) ( I ' )
-.In
f o r some i d e a l
Commutativity of t h e diagram y i e l d s t h a t I ' i s i n the kernel
I ' of Ro.
of t h e composition C1(Ro)
--f
C1(R(N))
v1 "SqY i f we w r i t e I ' = p1 ... ps ...qt
-f
Cl(R).
, then
(where e . , j = 1, ..., s a r e t h e r a m i f i c a t i o n J
r e s p e c t i v e l y ) and S I '
=
(SQ,)
C l ( Q & R ) ( R ) ) . Hence,
+
RI'
=
elVl P1 ...Psesv;
i n d i c e s of p l ,
. . . (SQt)Wt where
Q, . . . Q9 t
..., P,
S denotes Q:(,)
(R).
I/. Arithmetically Graded Rings w
Consequently, (SQl)wl...(SQt)
i s a p r o d u c t o f t h e P1,
p r i n c i p a l R - i d e a l s . S i n c e C1(R(N))
--f
193
...,P s up t o
Cl(R) i s i n j e c t i v e i t follows
0
t h a t the class whence
: I' 6
o f q y '...q t t i s <
pl,
...,p,,
- -
i n t h e group o e n e r a t e d by
I >.
pl,
...,p,.
-
-
I;
Therefore :
11.3. References, Comments, E x e r c i s e s .
11.3.1.
References.
- H. Bass, A l g e b r a i c K - t h e o r y , New York, Benjamin 1968. - I . Beck, I n j e c t i v e Modules o v e r a K r u l l Domain, J . o f A l g e b r a 17,1971, 116- 131.
- N. Bourbaki, AlgPbre Commutative I , 11, P a r i s , Hermann 1961. - L . Claborn, Note g e n e r a l i z i n q a r e s u l t o f Samuel's , P a c i f i c J . Math. 15, 1965, 805-808.
- L. Claborn, R . Fossum, G e n e r a l i z a t i o n o f t h e n o t i o n o f c l a s s qroup, Ill. J . Math. 12, 1968, 228-253. - L. Claborn, R. Fossum, C l a s s groups o f n - N o e t h e r i a n Rinos, J . o f A l g e b r a 10, 1968, 263-285.
-
P. E a k i n , W . H e i n z e r , Some open q u e s t i o n s on m i n i m a l p r i m e s o f K r u l l domains, Canad. J . Math. 20, 1968, 1261-1264.
- P. Eakin, W . H e i n z e r , Non f i n i t e n e s s i n f i n i t e d i m e n s i o n a l K r u l l domains J . o f A l g e b r a 14, 1970, 333-340.
- R. G i l m e r , M u l t i p l i c a t i v e
I d e a l Theory
- A. G r o t h e n d i e c k , L o c a l Cohomoloqy, L.N.M. Y 1, B e r l i n , S p r i n q e r V e r l a q 1967.
- W. K r u l l , I d e a l t h e o r y , E r g e b n i s s e d e r Math. 46, B e r l i n , S p r i n p e r V e r l a g 1968.
B. Commutative Graded Rings
194
- W . K r u l l B e i t r a g e z u r A r i t h m e t i k k o m m u t a t i v e r I n t e a r i t a t s b e r e i c h e , Math. Z . 41, 1936, 545-577. - Y . M o r i , On t h e I n t e q r a l C l o s u r e o f an i n t e r o a l domain, B u l l . Kyoto U n i v . Ser. B.,
7, 1955, 19-30.
- C. N x s t i s e s c u , F. Van Oystaeyen, Graded and F i l t e r e d Rinos and modules, L.N.M.
758, B e r l i n , S p r i n a e r V e r l a q 1980.
and Modules
- M. Nagata, A g e n e r a l t h e o r y o f a l g e b r a i c qeometry o v e r Dedekind domains I ( a l s o 11, 1 1 1 ) , Arner. J . Math. 78, 1956, 78-116. - M. Nagata, L o c a l Rings, I n t e r s c i e n c e T r a c t s i n Pure and A p p l i e d Math. 13, New York, 1962.
- P. Samuel, Anneaux graducs f a c t o r i e l s e t modules r e f l e x i f s , Math. France, 92, 1964, 237-249.
- P . Samuel L e c t u r e s on U.F.D., Bombay 1964.
-
B u l l . SOC
T a t a I n s t . f o r Fundamental Research no 30
J . P . Van Deuren, F. Van Oystaeyen, A r i t h m e t i c a l l y Graded Rings, I, R i n g Theory 1980, L.N.M.
825, B e r l i n 1981, S p r i n g e r V e r l a a .
- F. Van Oystaeyen, G e n e r a l i z e d Rees Rings and A r i t h m e t i c a l l y Graded Rinqs J . o f Algebra, t o appear i n 1982. - 0. Z a r i s k i , P . Samuel, Commutative A l q e b r a 1960.
,
I, 11, Van N o s t r a n d 1958,
195
11. Arithmetically Graded Rings
11.3.2.
Exercises.
a field.
1. L e t R be p o s i t i v e l y qraded N o e t h e r i a n and suppose t h a t Ro i s L e t M c R-gr be QR-R,
f i n i t e l y g e n e r a t e d . Then M i s f r e e i f and o n l y i f
(M) i s p r o j e c t i v e .
2. L e t R be a p o s i t i v e l y qraded N o e t h e r i a n r i n g and l e t M F R-gr be f i n i t e l y generated. Then M i s f r e e i f and o n l y i f Ro R -module and Torl(Ro,M)
=
0
60
R
M i s a free
0.
3. L e t R be a p o s i t i v e l y graded K r u l l domain such t h a t Ro i s a f i e l d k . Consider a f i e l d e x t e n s i o n k ' o f k . I f R ' = R 60 k ' i s a K r u l l domain then R ' i s a
k
f a i t h f u l l y f l a t R-module and C l ( R )
--f
Cl(R') i s injective.
4. ( I . Beck) L e t A be t h e i n t e ' a r a l c l o s u r e o f t h e N o e t h e r i a n i n t e g r a l domain B and l e t { a l,...,ar}
be a s u b s e t o f A; t h e n B : ( B : C B a.) c j J Deduce f r o m t h i s f a c t t h a t a d i v i s o r i a l ( p r i m e ) i d e a l
A:(A:zAa.). J o f A intersects B i n a d i v i s o r i a l (prime)ideal.
5. Consider K r u l l domains A and B , A c B . We say t h a t P 0 E
(Pas
d ' e c l a t e m e n t = no b l o w i n q up) i s s a t i s f i e d i f P F (Spec B ) l ht(P
n
A) c 1 . C o n d i t i o n
principal
ideals o f A
P.D.E
map
i s e q u i v a l e n t t o demandinq t h a t
t o p r i n c i n a l i d e a l s o f B under t h e map
( d i v ) i i n d u c e d by t h e i n c l u s i o n i : A
+
B.
Show t h a t PDE h o l d s i n any o f t h e f o l l o w i n a cases : a. B i s a f l a t A-module b. B i s an i n t e g r a l e x t e n s i o n o f A C.
B i s a subintersection o f A
Deduce t h e graded a n a l o
entails
o f t h e above s t a t e m e n t s .
6. Commutative Graded Rings
196
6. Examples. ( C f . [
29 ] and a l s o P. Samuel's T a t a l e c t u r e s . L e t k be a
f i e l d o f c h a r a c t e r i s t i c d i f f e r e n t from
z.
( a ) I f F i s a non-degenerate q u a d r a t i c f o r m i n k [X1,X2,X31/(F) C l ( k [X1,X2,X3] i s factorial. =
/(F)
i s c y c l i c o f order 2
o r e l s e k [Xl,X2,X31 /(F)
I f k i s a l g e b r a i c a l l y closed then
C l ( k [ X1,X2,X3]
/(F))=
2/22.
( b ) I f F i s a non-degenerate q u a d r a t i c f o r m i n k [X1,X2,X3,X4] t h e n C l ( k [ X1,X2,X3,X4]
/(F))
i s e i t h e r i n f i n i t e c y c l i c o r zero. I t i s
i n f i n i t e c y c l e f o r example i n t h e case where k i s a l q e b r a i c a l l y closed. ( c ) If F i s a non-degenerate q u a d r a t i c f o r m i n k [X,,...,X,] n 2 5, t h e n k [XI
,..., Xnl /(F)
with
i s a factorial ring.
2 2 2 2 /(Xl + X2 + X3 + 2X4) i s a f a c t o r i a l graded 2 2 2 2 . r i n g . However t h e f o r m X1 + X2+2X3 + 2X4 y i e l d s a graded r i n a 2 2 2 2 . Q [ X,,. . . ,X4 ] /X, + X2 + 2X3 + 2X4 w i t h c l a s s group Z. E x e r c i s e . Q [X1,X2,X3,X41
7. L e t D be a commutative graded domain. Check whether t h e f o l l o w i n g conditions are equivalent. a. D i s a gr-Dedekind r i n g b. A l l graded i d e a l s o f D a r e d i v i s o r i a l . 8. L e t C be a K r u l l domain and l e t I be a
d i v i s o r i a l i d e a l o f C. C o n s i d e r
the r i n g S Cl(C)
+
( I ) d e f i n e d i n 11.1.16. Check whether t h e k e r n e l o f C C l ( S l C ( I ) ) i s g e n e r a t e d by t h e c l a s s o f I.
9. L e t C be a K r u l l domain and l e t M be a
d i v i s o r i a l C-module. I s t h e
i n j e c t i v e h u l l of M i n C-mod a a a i n d i v i s o r i a l . What a b o u t maximal r i n g s o f q u o t i e n t s ? What a b o u t l o c a l i z a t i o n s a t t o r s i o n t h e o r i e s . Call a torsion
theory d i v i s o r i a l i f the associated f i l t e r o f i d e a l s
197
II. Arithmetically Graded Rings
has a f i l t e r basis c o n s i s t i n a o f d i v i s o r i a l i d e a l s . Is t h e l o c a l i z a t i o n o f a d i v i s o r i a l module a t a d i v i s o r i a l t o r s i o n theory d i v i s o r i a l over the l o c a l i z e d r i n g ?
R be a commutative qraded semiprime r i n q such t h a t
10. L e t
t h e o n l y idempotent elements o f of
0 and 1 a r e
R. Show t h a t t h e i n v e r t i b l e elements
R a r e homogeneous.
11.3.3.
Comnents
1. I f G i s a t o r s i o n f r e e a b e l i a n group s a t i s f y i n g t h e ascending c h a i n c o n d i t i o n on c y c l i c subgroups then t h e theory o f G-graded K r u l l domains p a r a l l e l s t h e Z-graded t h e o r y . The zealous reader may d e r i v e most o f the results i n
S e c t i o n 11.3.
f o r t h e G-graded case, u s i n g t h e t o o l s
o f chapter A.
2. The g e n e r a l i z e d Rees r i n g s i n t r o d u c e d h e r e may be f u r t h e r g e n e r a l i z e d t o t h e non-commutative case, e.g.
the P.I.
case, y i e l d i n g an i n t e r -
e s t i n g new c l a s s o f r i n g s . This c l a s s o f r i n g s c o n s i s t s o f subrings o f c e r t a i n crossed products R [ G,Q,c] group morphism, c : G x G a "nice" P.I.
L.
: G
+
Aut(R) i s a
U(Z(R)) a 2-cocycle and when R i s u s u a l l y
r i n g l i k e a tame order, an HNP r i n g , a maximal o r d e r
o v e r a K r u l l domain, by
-t
where
...
For a l l t h i s we r e f e r t o a r e c e n t paper
Le Bruyn and F. Van Oystaeyen, [ 64 1 .
198
Ill: Local Conditions for Noetherian Graded Rings
I n t h i s c h a p t e r , a l l r i n a s c o n s i d e r e d w i l l be commutative and N o e t h e r i a n u n l e s s e x p l i c i t e l y mentioned o t h e r w i s e . A l l graded r i n o s c o n s i d e r e d w i l l be o f t y p e Z . 111.1.
I n j e c t i v e dimension o f qraded r i n g s called a
L e t us i n t r o d u c e t h e f o l l o w i n g terminology : a r i n o R i s ~
local ring
i f i t has e x a c t ? y one maximal araded i d e a l ( t h e s e i d e a l s
a r e s a i d t o be gr-maximal) i n t h e s e t o f p r o p e r araded If R =
0
ic Z
2-
Ri
i d e a l s o f R.
i s g r - l o c a l w i t h M t h e maximal graded i d e a l t h e n Ro i s
a l o c a l r i n g and M
n
Ro i s t h e maximal - i d e a l o f Ro.
L e t P be a p r i m e i d e a l o f R and l e t S be t h e m u l t i p l i c a t i v e l y c l o s e d s e t S = h(R)-P. I f M 6 R-gr we denote t h e clraded module o f t h e f r a c t i o n s Qz(M) by Q i - p ( M ) . I f
Q
=
(P),
then : Qi-p(M) =
Qi-p(R) i s a r - l o c a l w i t h t h e maximal qraded
QaR-Q ( C ) .
Clearly
ideal (P)gQi-p(P). L e t
kg(R) denote t h e graded d i v i s i o n r i n g Q ~ - p ( R ) / ( P ) g Q ~ - p ( R ) .
111.1.1. Lemma.
L e t P be a p r i m e i d e a l o f R, t h e n :
1. F o r any M c R-gr, Q o n l y i f Q:(M)
=
R-P
(M) = 0 i f and o n l y i f Q R - ( p ) (M) - = 0 i f and 9
0.
2. P u t supp M = { P c Spec R, QR-p(M) = 0
1,
t h e n P c supp M i f
c supp M. 9 3. I f P i s a graded p r i m e i d e a l t h e n : Eg (ka(P)) Qi-p(R)
and o n l y
i f (P)
4. L e t P be a graded p r i m e i d e a l and l e t Ai
=
{x
Ei(R/P)
c Eq(R/P), Pix
then t h e f o l l o w i n g p r o p e r t i e s h o l d : a. Ai
i s a graded I?-submodule o f Eg(R/P) and E q ( R / P )
=
U Ai.
il.1
=
0
}
199
111. Local Conditions for Noetherian Graded Rings
b. Ai+l/Ai
i s a graded kg(P)-rnodule
c. A1
kg(P).
5. Any i n j e c t i v e o b j e c t i n R-qr i s a d i r e c t sum o f some E g ( R / P ) ( n ) , f o r some graded p r i m e i d e a l s P o f R and some n F 2 . 6. Denote by gr-inj.dimRI.c t h e i n j e c t i v e dimension o f M i n t h e c a t e g o r y R-gr. I f S
c
g r . i n j . dim
h(R) i s any m u l t i p l i c a t i v e l y c l o s e d s e t M 6 R-gr, t h e n S - l M C. q r . i n j . DimRM
S-'R
P r o o f . l . , 2., 3. a r e easy consequences o f t h e f o r e p o i n g chapters;4., 5.,
6.,
f o l l o w as i n t h e ungraded case.
L e t M F R-gr. and c o n s i d e r m i n i m a l i n j e c t i v e r e s o l u t i o n s i n R - q r and R-mod r e s p . :
0
+
F!
+
Eog
+
Elq
+
._.
0
+
Y
+
Eo
+
El
+
.i.
F o r a graded p r i m e i d e a l P o f R, l e t L{(P,M), number o f c o p i e s o f E i ( R / P ) i n E,: 111.1.2.
Propositions.
r e s p un(P,M),
be t h e
r e s p . o f ER(R/P) i n En.
L e t M F R-gr, P
be a graded p r i m e i d e a l o f R. functor
F o r n o t a t i o n a l convenience we w r i t e Q g f o r t h e l o c a l i z a t i o n on R-gr a s s o c i a t e d t o h(R)-P, t h e n :
i s a f r e e graded k'(P)-module
1. The group Qg Exti(R/P,M)
2. clq(P,M)
4. I f S
c
= wi(P,M)
h(R) i s any m u l t i p l i c a t i v e l y c l o s e d s e t such t h a t P
t h e n $(P,M) Proof.
o f r a n k LL:(P,M).
n S
= pn(S-lP,S- 1M) f o r a l l n 2 0.
We have :
Exti(R/P,M)
=
QgExti(R/P,M)
EXTi(R/P,M) =
=
HOMi(R/P,Qy) = HomR(R/P, QS),
QgHomR(R/P,Q;)
=
Horn
Q4(R) and t h e l a t t e r i s a f r e e kg(P)-module.
( k a ( P ) , Q:)
hence
=
d
8. Commutative Graded Rings
200
g
I f P l i s a graded p r i m e i d e a l o f R such t h a t P1 # P t h e n
Q tbm(R/P,El(P1))=O,
hence QgHomR(R/P,Qq) i s a f r e e kg[P)-module o f r a n k equal t o pY(P.M). S i n c e Exti(l?/P,M) i s a f r e e k ( P ) m o d u l e o f r a n k ki(P,M)
w h i l s t QExt(R/P,M)
a t P ( Q t h e l o c a l i z a t i o n i n R-mod
i s t h e l o c a l i z a t i o n o f QgExt(R/P,M)
a s s o c i a t e d t o P ) . 2 ) f o l l o w s i m m e d i a t e l y and 3) f o l l o w s f r o m 2 ) . F o r t h e s t a t e m e n t 4 ) we a p p l y t h e s t a t e m e n t 1). 111.1.3.
L e t M c R-gr,
Proposition.
P a graded p r i m e i d e a l o f R
a C h(R) a nonzero d i v i s o r on M and R, t h e n f o r a l l n pn(P/aR, 9
M/aM) =
(P;M)
E:
L e t O+M+
Proof.
d A
d-1
E:
be a m i n i m a l i n j e c t i v e r e s o l u t i o n o f
, E24 CI
0 :
-
M on R-gr and N
I f K 6 R-gr i s a r b i t r a r y , t h e n HornR (R/aR, K )
3
=
and
d
0
'
(E0')
(K:a) = { x
f
1
ax =O}
E2g)
+...,
K
From t h e commutative diaqram :
O+
M ? E o a A
N
iqa
1.a
1.a
M---t
Eoq-)N
---to where Q ~ ( X =) ax
O+
we have (N:a)
=
0
3
M/aM so Hom(R/aR,N)
h
C/aM.
From t h e e x a c t sequence
0-J
R
ip
a,R +R/aR
we deduce t h a t
-0
E x t i ( R / a R , M) = 0 f o r a l l i > 1.
T h i s i m p l i e s t h a t t h e sequence : (1):
0-
HomR(R/aR,N)+
i s e x a c t . Because HornR (R/aR,
HornR (R/aR, Elq)+HomR Eig)
h
(Eiq:a)
(R/aR,
i s an i n j e c t i v e module
o v e r t h e r i n g R/aR we have t h a t (1) i s a m i n i m a l i n j e c t i v e r e s o l u t i o n o f t h e R/aR-module M/aM = HornR (R/aR,N).
201
111. Local Conditions for Noetherian Graded Rings
Now, i f P i s a graded p r i m e i d e a l such t h a t a v e r i f i e d t h a t HornR (R/aR,
ki
(P/aR, M
111.1.4.
M) =
P, t h e n i t
i s easily
envelope o f t h e
i s the injective
E(R/P))
R/aR-module R/P = (R/aR)/(P/aR).
c
T h e r e f o r e , f r o m ( 1 ) we o b t a i n t h a t
(P,M). L e t P be a p r i m e i d e a l o f R such t h a t P #
Proposition.
t h e n f o r a l l n t N a n d M t R-gr we o b t a i n : P ~ ( ( P ) ~ , M )= P r o o f . By P r o p o s i t i o n 111.1.2.4. w h i l e untl(P,M)
= pn+l
Qg(R) and Qg(M) r e s p . , gr-maximal i d e a l
we have : u,,((P)~,M)
(Q4(P),Qg(M)).
(P),
(P,M).
= (Qq((P)g),Qg(M))
So we may r e p l a c e R and M by
we may suppose t h a t R i s g r - l o c a l and
i.e.
of R . I n t h a t case P i s a maximal i d e a l o f R. S i n c e
i s a qraded d i v i s i o n r i n q , t h e r e e x i s t s a t R such t h a t P = ( P ) +Ra. 9 9 From t h e e x a c t sequence :
R/(P)
( * ) O-Rl(P)g
a
4
R/(P)q-R/P
where ma denotes m u l t i p l i c a t i o n by a, we o b t a i n :
+ Extn(R/(P)g,M) 4 -+ Extn+l(R/(P)g,M)--t Note t h a t Extn(R/(P)g,M)
Extntl(R/P,M)
+
...
... -,Ext"(R/(P)
9
m , M A
+ Extn+l(R/(P)g,M)
"'a
. .. = EXTn(R/(P)9, M) i s a qraded R/(P)q-module,
hence E x t n ( R / ( P ) ,M i s a f r e e araded R / ( P ) q module. Consequently 9 Extn(R/(P) ,M) --$ Extn(R(P) ,M)is rnonomorphic. ma: 9 9 The l o n g e x a c t sequence o b t a i n e d above y i e l d s a s h o r t e x a c t sequence: m Extn(R(PjQ,M) + Ext"'(R/P,M) 30 0 4 Extn(R/(P)g,M)
2
I n t h i s sequence V = Extn(R(P)q,M) whereas V ' = Ext"'(R/P,M)
i s f r e e graded o f r a n k p,,((P),,M),
i s a v e c t o r space o f
d i m e n s i o n P,,+~(P,M)
I n R/(P) -mod we have t h e e x a c t sequence : 9 v -V' 3 0 kn((P)g'M) we deduce f r o m ( * ) and (*) t h a t Since V 2 (R/(P)g)
o v e r t h e f i e l d R/P. m (**) o + v
a,
,
B. Commutative Graded Rings
202
V'
(R/P)
111.1.5.
Pn ( (P)
.M)
.
On t h e ' o t h e r hand V ' = ( R / P )
Corollary. I n the s i t u a t i o n o f III.1.4.,
Pn+ 1(P 9 M)
P,((P)~,M)
,
therefore
=
0 i f and
o n l y i f F ~ + ~ ( P , M )= 0. 111.1.6.
C o r o l l a r y . L e t P be a q r a d e d p r i m e i d e a l o f R.
The m i n i m a l i n j e c t i v e r e s o l u t i o n o f Eg(R/P) i n R-mod i s t h e f o l l o w i n q :
+ E(R/P)-Q
O+
E(R/P')--+O P' where t h e d i r e c t sum i s o v e r a l l p r i m e i d e a l s P ' # P o f R such t h a t (P')
E~(R/P)
9
= P.
-
P r o o f . The r e s o l u t i o n has t h e f o r m : 0-
~ gR/P) (
where Q, =
-0
E( R/P)-Q~
c
P'tSpec R
E(R/P ' )
111(~ , E ~ ( R / P )
and C o r o l l a r y 111.1.5. we g e t Q,
c i n j . dim.R
__ Proof.
E ( R / P ' ) where ( P ' )
9
= P.
o
c n
t
1
D i r e c t l y f r o m t h e p r o p o s i t i o n 1.4. Remark. I f M i s a maximal i d e a l o f R w h i c h i s graded t h e n
111.1.8. Ei(R/'M)
P'
C o r o l l a r y . I f M t R-qr w i t h g r . i n j . dim.R M = n, t h e n
111.1.7. n
= 8
. A p p l y i n g P r o p o s i t i o n 111.1.4.
=
ER(R/M), e . 9 . i f
R
= k [XI
,..., X n l
and
M=(X1 ,..., X n ) .
(k i s a field). 111.1.9.
Lemma. L e t R be a graded r i n g and suppose P
c
Q are d i s t i n c t
graded p r i m e s w i t h no o t h e r graded p r i m e i d e a l between t h e n . I f M i s a graded
f i n i t e l y g e n e r a t e d module t h e n
Proof.
By P r o p o s i t i o n 111.1.2.4.
a c h(R) such t h a t a t Q, a
P.
a;(P,M)
# 0 i m p l i e s g;+l(Q,M)
# 0.
we may assume R = Q g (R) . Choose R-Q We denote A = R/P and B = A/aA = R/(a,P)
203
111. Local Conditions for Noetherian Graded Rings
C l e a r l y , B 1s a n o b j e c t of f i n i t e l e n g t h i n R - g r and a i s a nonzero The e x a c t sequence : O
d i v i s o r on A .
an e x a c t sequence : ExtR(A,M) n
cpa
d
A -6
A L
+Exti
(A,M)
-+
-0 Ext;+l(B,M).
yields By
h y p o t h e s i s R i s g r - l o c a l w i t h Q as t h e or-maximal i d e a l and a t Q, t h e n
# 0. Now, b y i n d u c t i o n on 1(B) ( 1 d e n o t i n g l e n r j t h ) , and Extntl(B,M) R ( . .M) we o b t a i n t h a t u s i n g t h e r i g h t e x a c t n e s s o f f u n c t o r s Ex’:t (R/Q,
Ex$”
111.1.10.
M) # 0, and hence P:+~(Q,F”)
Corollary.
# 0.
L e t R be a commutative N o e t h e r i a n r i n g w i t h l e f t
l i m i t e d g r a d i n g . I f M t R-gr i s a f i n i t e l y t h e n : g r . i n j . dim. R M = i n j . dimR Proof.
Put n = or. i n j . dim.R M c
g e n e r a t e d qraded module
#.
-.
I f i n j . dimR
# n, t h e n i n j . d i m R
= n+l.
Now pn+l (P,M) f 0 f o r some p r i m e i d e a l P o f R, so by t h e f o r e g o i n g results P i s
maximal. C l e a r l y P i s n o t qraded and by P r o p o s i t i o n 111.1.4.
t h a t p n ( ( P ) .M) # 0 hence P:((P)~,M) 9 i s graded maximal, i n d e e d i f n o t , t h e n (P)
# 0. We c l a i m t h a t
i t follows
(P),
graded t h e n
G:+~(P~,M) # 0 by Lemma I I I . 1 . 9 . ,
c P1, b u t i f Plis g c o n t r a d i c t i o n . Since R
has l e f t l i m i t e d a r a d a t i o n i t f o l l o w s t h a t (P) i . e . P = (P) 111.1.11.
9’
i s a maximal i d e a l ,
c o n t r a d i c t i o n . Consequently, i n j . dim.RM = n .
Corollary.
L e t R be a commutative N o e t h e r i a n r i n g w i t h l e f t
l i m i t e d g r a d i n g . L e t M c R-gr. b e a f i n i t e l y g e n e r a t e d qraded module with gr.inj.dim.
M
=
n, n
c
-.
If V M + E q * E F + 0
... + E i *
0
i s a m i n i m a l i n j e c t i v e r e s o l u t i o n of M i n R-ar t h e n E i i s i n j e c t i v e i n R-mod.
# 0. By Lemma
Proof.
L e t P be a araded p r i m e i d e a l such t h a t F:(P,F”)
111.1.9.
we o b t a i n t h a t P i s a qr-maximal i d e a l . Because R has l e f t
l i m i t e d g r a d i n g P i s a maximal i d e a l . By Remark 111.1.8. t h a t E i ( R / P ) i s an i n j e c t i v e an R-module.
R-module.
i t follows
T h e r e f o r e E,“ i s i n j e c t i v e as
204
8. Commutative Graded Rings
I n e x t e n d i n g f r o m R t o R [ X I we need t h e f o l l o w i n g lemma where R has t r i v i a l g r a d a t i o n and R IX ] i s graded as u s u a l . 111.1.12. Lemma.
I f P i s a graded p r i m e i d e a l o f R [ X I t h e n e i t h e r
P = pR [ X I o r P = pR [ X I Proof. S i n c e P
+
(X),
where p = P
1/
pR [ X 1 and R [ X
3
n
R.
pR [ X I = (R/P)[ X I we may assume p = 0,
i . e . R i s a domain. L e t K be t h e f i e l d o f f r a c t i o n s o f R . S i n c e P fl R = 0, PK
I X I i s a graded p r i m e i d e a l o f K [ X I i . e . PK [ X I = 0 o r P K
[ X
1
=
X.
o
111.1.13. P r o p o s i t i o n . L e t R be a N o e t h e r i a n commutative r i n g and t a k e Spec R P u t E = ER(R/P). C o n s i d e r t h e r i n g R [ X I
P C
Denote M = E [ X I
t o t h e degree i n X.
=
E
rn R [ X I R
graded c o r r e s p o n d i n g
.
The m i n i m a l i n j e c t i v e r e s o l u t i o n o f M i n t h e c a t e o o r y R [ X I - q r i s : O-M-Ei(R[X]
(R[Xl/PR[XI+
/PR[X])-+Eg
(X))+O
and t h e m i n i m a l i n j e c t i v e r e s o l u t i o n o f M i n R [XI-mod i s :
t h e d i r e c t sum b e i n g t a k e n P'
n
over t h e prime i d e a l s P' o f
R [ X I such t h a t
R = P.
Proof.
Consider t h e minimal i n j e c t i v e r e s o l u t i o n o f
0-M
-1:
+
Proposition III.1.3.,
'i+1 (PR [ X I
+
(R
= Eq
[ X]/PR [ X I ) . Taking i n t o account
we o b t a i n :
(X),M)
[ X I + (X), E [ X I /
= wi(PR
S i n c e p . ( P . E ) = 1 i f i = 0 and pi(P,E) 1
q(PR [ X I
=
1
+
(X),M)
= 1 and pi(PR
P ' F Supp. M and t h e r e f o r e P '
n
+
[ X I
L e t P ' be a p r i m e i d e a l o f R [ X ]
By Lemma 111.1.12.
R [ X I - gr :
....
If +I;-
S i n c e A s s M = { PR [ X I }, 1:
M in
0 if i
0, we have
= 0 f o r a l l i > 1.
(X),M)
such t h a t +(P',M) R
3
P. I f P '
i t follows that P'
=
X E ( X ) ) = pi(P3E)
n
z
0. Then n e c e s s a r i l y
R # P, p u t P '
n
Q R [ X I or P' = Q R [ X I
R = Q.
+ (X).
205
111. Local Conditions for Noetherian Graded Rings
I n case P '
Q R [X]
=
p i ( P ' , M ) = P.1 ( Q R
we have :
[ X l , E [XI ) = d i m k ( p ) ( E x t A I X l( R [Xl/QR [ X I , E [ X I )
However,
Exti
( R [ X I /QR [ X I , E [ X I )
=
E x t i (R/Q,E)
a n d as Q # P we obtain E x t i ( R / Q , E )
Thus
p.
(P',M) = 0 for a l l i
In case P '
Q R [XI
=
0 R -
[
+
2
-
=
R 1x1
@
R
0 for a l l i
2
.
0.
0.
( X ) , we have an exact sequence mX
X ]/Q R [ X I
R [ X l/QR [ X I d
R [ X ]/PI -0
Yielding a long exact sequence :
... j E x t i ( R
mX
[ X ] / Q R [ X I , M) -Exti
( R [ X I , Q R [ X I , M)
-
( R [X]/P',M) 4 E i + l ( R [ X l / Q R [XI, M) &'Exti+'(R
+ Exti"
[ X]/QR
( a l l E x t a r e E x t R [XI ) ' From the above we deduce : Ext
1
( R [ X]/P',M) = 0 f o r a l l i 2 1.
Finally t h i s amounts t o 1: = Eg ( R [Xl/PR [XI + ( X ) ) a n d Igk = 0 f o r all k
P
1.
For the second statement of the proposition, l e t the minimal i n j e c t i v e in R [ XI-mod be qiven by
resolution f o r
II-J
M -Io+
0-
...
12-
I t i s c l e a r t h a t I. = E R
:
( R [X]/PR [XI). From Proposition 111.1.2.
( R [ X]/P') where P ' R [XI = PR[X] o r varies i n the s e t of prime ideals of R [ X I such t h a t
and 111.1.4. i t follows t h a t I 1 contains
P'
=
PR [ X I
Indeed i f P '
+
n
( X ) . Therefore ( P ' )
R
=
9 P and P ' # ( P ' )
# PR [XI then h t ( P ' ) h t ( P ' ) 2 2 + h t ( P ) . On the PI
n R
=
P, h t ( P ' )
5
g
2
=
9
@
PI
E
PR [XI i f and only i f P ' n R = P . then h t ( P ' )
1 + h t ( P R [XI)
=
=
1 + h t ( P ' ) q , thus i f
1 + h t ( P ) . Thus
other h a n d , since R i s Noetherian and
1 + h t ( P ) , contradiction. Since the converse i s
obvious, t h i s proves the statement.
[XI, M)+
6. Commutative Graded Rings
206
3
L e t P l b e a p r i m e i d e a l o f R [ X I w h i c h i s n o t graded and such t h a t P1 il R f P Then (Pl)g
PR [ X I
would i m p l y
P1
n
# PR [ X I + (X) because(Pl)q
and (Pl)q
R = (P )
l g
From P r o p o s i t i o n 111.1.4.
n
PR [ X I + (X)
R = P.
we r e t a i n t h a t ui((P
) ,M) = pitl(Pl,M). 1 q ,M) = 0 f o r a l l i z 0, hence
already established pi((P.) ' g = 0 f o r a l l i 2 1. T h i s y i e l d s t h a t I1 =
However we have p . (P1,M)
=
8
P'
ER [Xl(R
[Xl/P')
as s t a t e d . I f I 2 were nonzero t h e n t h e r e e x i s t s a p r i m e
where P ' i s
i d e a l P2 o f R [ X I
such t h a t p2(P2,M) # 0.
I f P 2 i s graded t h e n from P r o p o s i t i o n 111.1.2. we deduce $(P2,M)
and t h u s I!
# 0
# 0, c o n t r a d i c t i o n .
I f P2 i s n o t graded, t h e n we deduce f r o m P r o p o s i t i o n 111.1.4. t h a t
p2(P2,M) = PR [ X I +
=
u ~ ( ( P ~ ) ~ , fM 0. ) I n t h i s case i t i s necessary t h a t (P2) ( X ) , hence P2 fl R = ( P 2 )
fl R = P. S i n c e P2
q
=
# (P2)g, h t ( P 2 ) =
h t ( ( P ) ) = 2 + ht(PR [ X I ) = 2 + h t ( P ) . B u t f r o m P2 n R = 2 9 = P, h t ( P 2 ) 5 1 t h t ( P ) f o l l o w s hence we r e a c h a c o n t r a d i c t i o n . = 1
+
T h e r e f o r e p2(P2,M)
=
0 and I p = 0 f o l l o w s .
111.1.14.
C o r o l l a r y . L e t R be a commutative N o e t h e r i a n
i n R-mod,
then :
g r . i n j . dim.
R [XI
E [XI
+
1 = i n j . dim.R
ring,
E [ X I .
E injective
207
111. Local Conditions for Noetherian Graded Rings
111.2. R e g u l a r , G o r e n s t e i n and Cohen-Macaulay R i n g s . r i n g and M a nonzero f i n i t e l y a e n e r a t e d
L e t R be a commutative N o e t h e r i a n
R-module. We p u t : V(M) = supp.(M) = V(AnnRF1) = { P C
M,
Recall t h a t t h e K r u l l dimension o f
Spec R, AnnRM
=
P
1.
denoted by K.dimRM i s d e f i n e d t o
be t h e supremum of t h e l e n g t h s o f c h a i n s o f p r i m e i d e a l s o f V(M) i f t h i s supremum e x i s t s
,
and
-
i f n o t ( t h e r e a d e r may v e r i f y t h a t K.dimRM
c o i n c i d e s w i t h t h e K r u l l dimension o f M d e f i n e d i n s e c t i o n 1 1 . 5 . ) . We have htMP = K.dim QR-p(R) (QR-P(F)). I f I i s an i d e a l o f R such t h a t IM # M t h e n t h e l e a s t r f o r which
# 0 w i l l be c a l l e d t h e grade o f I on P, denoted by : grade (1,M);
Extk(R/I,M) grade
(1,M)
i s e x a c t l y t h e common l e n p t h o f a l l maximal sequences
c o n t a i n e d i n I (see Lemma 111.3.3. o f I f R i s a l o c a l r i n g w i t h maxima!
9
3). s a i d t o be a
i d e a l Q , then M i s
Cohen-Macaulay module o r a C.M.-module
i f grade (si,M)
=
K.dimRM. A n o t
n e c e s s a r i l y l o c a l r i n g i s s a i d t o be a Cohen-Macaulay r i n g o r C.M-ring i f QR-p(R) i s a Cohen-Macaulay QR-p(R)-module and i n t h a t case P i s a C.M-module i f QR-p(M) i s a C.M.-QR-P(R) module. I t i s easy t o v e r i f y u s i n g P r o p o s i t i o n 111.1.2.
t h a t M i s a C.M.-module
each maximal i d e a l Q c V(M), ki(S1,M)
i f and o n l y i f f o r
= 0 whenever i < h t M
(a).We
M i s a Gorenstein-module i f f o r e v e r y maximal i d e a l 9 c V(M), if and o n l y i f i # h t M ( Q ) . P r o p o s i t i o n 1 1 1 . 1 . 2 . e n t a i l s t h a t
say t h a t
wi(Q,M)
= 0
M is a
Gorenstein-module i f and o n l y i f QR-p(M) i s a G o r e n s t e i n QR-p(R)-module for a l l P
c
V(M).
A G o r e n s t e i n r i n q R i s a r i n g w h i c h i s a Gorenstein-module when c o n s i d e r e d as a module o v e r i t s e l f . F o r a d e t a i l e d s t u d y o f we r e f e r t o Bass, graded case.
[ 5 ] . F i r s t we g i v e some l o c a l
these classes o f r i n g
-
g l o b a l theorems i n t h e
B. Commutative Graded Rings
208 111.2.1.
Theorem. L e t R be a commutative N o e t h e r i a n graded r i n g and l e t
M c R-gr be f i n i t e l y g e n e r a t e d . The f o l l o w i n q s t a t e m e n t s a r e e q u i v a l e n t . 1. M i s a Cohen - Macaulay module. 2. F o r any graded p r i m e i d e a l P o f V(M), QR-p(M) i s a Cohen-Pacaulay QR-p( R)-module. ~
P r o o f . 1 = 2 i s t r i v i a l . To p r o v e t h e converse l e t us assume f i r s t t h a t QR-p(M) i s a C.M.
QR-p(R)-module f o r e v e r y graded P
c
V(M). L e t P ' c V ( P ) .
If ( P ' ) g = P ' t h e n we a r e done. So suppose t h a t P ' # we have t h a t h t H P ' = h t M ( P ' ) a
+
I n t h i s case
1. Choose i < h t M ( P L ) .
hence p . ( ( P ' ) ) = 0. By P r o p o s i t i o n 111.1.4. 1-1 g,M pi(P',M)
= 0 and t h e r e f o r e M i s a C.M.-module.
111.2.2.
Theorem.
Then i - 1 c h t M ( ( P ' ) q ) ,
i t follows then t h a t
L e t R be a graded commutative N o e t h e r i a n r i n p and l e t
M C R - g r be f i n i t e l y generated. The f o l l o w i n a s t a t e m e n t s a r e e q u i v a l e n t : 1. M i s a Gorenstein-module 2. F o r each graded P c V(M), QR-p(H) i s a G o r e n s t e i n QR-p(R) module. P r o o f . 1 = 2. Easy. F o r t h e converse, suppose t h a t QR-p(M)
i s a Gorenstein
QR-p(R)-module f o r each qraded P 6 V(M). L e t P ' be a r b i t r a r y i n V(M). I f
P'
(P')
t h e n t h e a s s e r t i o n i s c l e a r . Supposinq t h a t P ' # ( P ' )
and 9 i # h t M ( P ' ) , we o b t a i n i - 1 # h t M ( ( P ' ) ) . S i n c e Q R - ( p , ) ( P ) i s a G o r e n s t e i n g 9 Q R - ( P l ) (R)-module i t f o l l o w s t h a t : 9 =
9
~ i _ l ( ( p ' ) ~ M= )p i - 1 ( Q R - ( p l ) ( ( P ' ) a ) , Q R - ( p l ) ( H ) ) = 0 9 9 From P r o p o s i t i o n 111.1.4. we deduce t h a t pi(P',M) = 0. I n a s i m i l a r way i t may be deduced f r o m v i ( P ' , M )
G o r e n s t e i n module
.
= 0 that i
+ htM(P')
hence
M
is a
0
A l o c a l r i n g w i t h maximal i d e a l Q i s s a i d t o be r e g u l a r i f q l . dim R < o r e q u i v a l e n t l y , ifK.dim R = dim
R/Q
2
(Q/Q ). I f 9 i s not a local rinq,
209
Ill. Local Conditions for Noetherian Graded Rings
t h e n R i s c a l l e d a r e g u l a r r i n g i f f o r e v e r y maximal i d e a l Q,QRTS i s regular.
111.2.3. C o r o l l a r y . L e t R be a commutative N o e t h e r i a n r i n a and M a f i n i t e l y g e n e r a t e d R-module.
I f M i s a Cohen-Macaulay
-
t h e n M [ X I i s a Cohen __ Proof. L e t P
c
n
R t h e n p c V(M).
P = p R [ X] or P = p pi(P,M)
and
M [XI c P
R
[X]
+
if P = pR [ X I
By Lemma 111.1.11. we have
( X ) . I f P = p R [ X I we have vi t
(P,M [ X I ) =
(X) we have by P r o p o s i t i o n 111.1.3. t h a t
'L~+~(P,M [ X I ).
v~(P.M) Moreover h t P = pR [ X I
111.2.2.
Macaulay ( r e s p . F o r e n s t e i n ) R-module.
V(M [ X I ) be a graded p r i m e i d e a l ; t h e n AnnR
I f we w r i t e P = P
=
( r e s p . G o r e n s t e i n ) module
M [ X I
+
(X).
P = htMP i f P = p [ X ] and h t
M [XI
P = 1 t htMP i f
Now t h e c o r o l l a r y f o l l o w i n q f r o m Theorem 1 1 1 . 2 . 1 . and
finishes t h e proof.
:
1 1 1 . 2 . 4 . Theorem. L e t R be a graded commutative N o e t h e r i a n r i n a t h e n t h e f o l l o w i n g statements are equivalent :
1. R i s a r e g u l a r r i n g . 2 . QR-p(R) i s r e g u l a r f o r e v e r y graded p r i m e i d e a l P o f R. Proof. __
2
=
The i m p l i c a t i o n 1 = 2 i s o b v i o u s .
1. If P ' i s an a r b i r a r y p r i m e i d e a l o f R t h e n t h e r i n p Q R - ( p , ) (R) i s q
r e g u l a r i . e . g l . d i m . Q R - ( p , ) (R) < -. S i n c e Q R - p , ( R ) i s t h e l o c a l i z a t i o n 9 of S = QR-(pb) (R) a t t h e p r i m e i d e a l g e n e r a t e d by P i i n i t , i t f o l l o w s 9 t h a t we may assume t h a t we have chosen R t o be g r . l o c a 1 w i t h gr.maximal By h y p o t h e s i s g l . dim S = n c -. C o n s i d e r a f i n i t e l y g' g e n e r a t e d R-module M. By P r o p o s i t i o n 111.1.2. we have t h a t ideal (P')
i n j . dimS Q R - ( p , ) (M) 5 n and hence g r . i n j . dim R M 5 n. T h e r e f o r e 4 3 r . g l . d i m R c m and C o r o l l a r y 111.1.10. y i e l d s q l . d i m . R c m , hence g l . - d i m QR-p,(R)
<
-.
210
6. Commutative Graded Rings
O u r following r e s u l t shows t h a t r e g u l a r and p o s i t i v e l y graded r i n g s a r e c l o s e t o beinq polynomial r i n g s o v e r
regular rings.
111.2.5. P r o p o s i t i o n . Let R be a p o s i t i v e l y graded r e p u l a r r i n g with a
unique graded maximal i d e a l S , then R 0 i s a r e q u l a r l o c a l rinq and
R
- Ro =
t h e X i , i = 1, ..., k being indeterminates over Ro, which
...,X k ] ,
[X1,
a r e taken t o be homogeneous elements o f p o s i t i v e d e a r e e .
Proof. W e
proceed by induction on n
=
K.dim R. If n
i s a field. If n
>
degree i n Q -52'
( i f there i s no such x then R
P' =
R'/ ( X I ,
K.dim R '
=
=
then R
Ro
=
Rofollows). P u t
Q / ( x ) . Now R' i s a qraded r e q u l a r r i n a and
Ro
= =
(R')o [Y1
t h a t R = Ro [ U1,..
,..., Yk]
-
with deq Y i > 0 , i
0 i t follows t h a t ( R ' ) o
d then d
t
i n R f o r t h e images o f Y 1 . . . Y k
.
.,Uk+l]
homomorphism v! : Ro [ X1
k
=
=
i s an isomorphism.
=
1,..., k .
RO'
n-1. Choose r e p r e s e n t a t i v e s U1,...,Uk i n R ' , and
put:Uk+l
=
x . One e a s i l y checks
Therefore we have an epimorphic graded r i n q
,..., X k t l ]
+
Ro [ U 1
,...,Uk+l]
t h e f a c t t h a t K.dim R = k + d t 1 = K.dim Ro [ X1,...,Xk+ll 'v
=
The induction hypothesis y i e l d s : ( R ' ) o i s a r e q u l a r
l o c a l r i n g and R'
So i f K.dim Ro
0
0 , choose a nonzero homoaeneous element x of p o s i t i v e
$2'
n-1.
Since ( x ) f7
=
=
R . However, yields that
o
111.2.6. Remarks.
1. I f R i s a p o s i t i v e l y graded r e g u l a r
r i n a , then Ro i s r e g u l a r . Indeed,
i f w i s a maximal i d e a l i n Ro then Q = W O R+ i s a maximal graded i d e a l . Since t h e l o c a l i z a t i o n QR-Q(R) i s r e g u l a r and a l s o p o s i t i v e l y sraded (note t h a t i t coincides w i t h the
qraded r i n g of q u o t i e n t s Q:-Q(R))
i t follows from t h e foreqoing t h a t ( Q R - Q ( R ) ) o (QR-Q(R))o
=
QR JRo). 0
i s r e g u l a r . However
I t f o l l o w s t h a t Ro i s r e g u l a r t o o .
111. Local Conditions for Noetherian Graded Rings
2 . See a l s o J . M a t i j e v i c [
7 1 . If
R
21 1
0 Riis C-orenstein, hence Cohenizo Macaulay, t h e n i t i s n o t n e c e s s a r i l y t r u e t h a t Ro i s G o r e n s t e i n o r C . V . .
2
/ (X , X Y ) , where R i s any f i e l d . P u t Q1
Indeed, p u t T = k [ X , Y ]
Q2 = ( X ) ,
=
t h e n t h e image of Q ,
n
o f degree 1 and p u t S = T [ W ] , Q ;
Q, i n = Q,S,
T
=
(Y),
i s z e r o . L e t W , V be v a r i a b l e s
QE = Q 2 S +WS and I = Qf fl Q E .
2
P u t U = S / I , R = U [ V ] / ( x V + yV,V ) where x, y denote t h e imaaes o f X, Y r e s p . T h i s c o n s t r u c t i o n y i e l d s a qraded r i n a R w i t h K.dim R = 1 and Ro = T. However
T
is
n o t a C.C-rina.
I f 8 denotes t h e u n i q u e maximal graded i d e a l Gorens t e i n.
o f R then Qq-Q(R)
is
212
5. Commutative Graded Rings
111.3. Graded Rings and M-sequences. R i s a commutative graded r i n a throuclhout t h i s s e c t i o n . Such a r i n a R
i s s a i d t o be c o m p l e t e l y p r o j e c t i v e o r i s s a i d t o have p r o p e r t y C.P. i f f o r any graded i d e a l I and f o r any f i n i t e s e t o f araded p r i m e i d e a l s
PI,
. . . ,Pn
w i t h h ( 1 ) c P1 U . . . U
a t l e a s t one Pi.
Pn i t f o l l o w s t h a t I i s c o n t a i n e d i n
It i s n o t hard t o v e r i f y t h a t , i f R i s completely
p r o j e c t i v e t h e n so i s S-’R
f o r every m u l t i o l i c a t i v e l y closed subset
S o f h ( R ) . I t i s a l s o s t r a i g h t f o r w a r d t o check t h a t e p i m o r p h i c images
o f a c o m p l e t e l y p r o j e c t i v e r i n g have p r o p e r t y C.P. 111.3.1.
too.
L e t R be a commutative r i n g such t h a t a l l graded p r i m e
Lemma.
i d e a l s o f R a r e i n P r o j R, t h e n R i s c o m p l e t e l y p r o j e c t i v e .
4
P r o o f . Assume I
1 5 i 5 n and assume t h e pi a r e n o t comparable.
Pi,
I f n = 1 then h(1)
Q
PI f o l l o w s . We p r o c e e d by i n d u c t i o n on n. The
t h e r e i s an ai
,
c
h(I
n
Pi)
n
Pi)
Q
P 1 5 i 5 n hence j’ such t h a t ai f P . f o r each j # i. L e t J
induction hypothesis y i e l d s h ( I
U j+i
r! a . . C l e a r l y xi c I and xi c P . f o r j # i b u t xi 4, Pi. S i n c e j#i J J no graded p r i m e i d e a l o f R c o n t a i n s R, i t i s easy t o see t h a t we may
x. =
choose deg xi > 0 f o r each i, 1 5 i 5 n. P u t d 1. = dea x . and d = d l . . . . .dn, y,
=
P’di
x.
Then deg yi = d and i t i s o b v i o u s t h a t y = yl+ ...+yn i s
a homogeneous element such t h a t y c I and y f PIU 111.3.2.
... U
Pn.
Examples.
1. I f R i s a p o s i t i v e l y graded r i n g such t h a t Ro i s a f i e l d t h e n R i s completely p r o j e c t i v e . 2 . L e t R be a g r - l o c a l r i n g w i t h qr-maximal
i d e a l Q such t h a t R / Q i s a
non t r i v i a l graded g r - d i v i s i o n r i n g . Then R has p r o p e r t y C.P. Indeed, i f P
.
i s a graded p r i m e i d e a l t h e n P c Q. S i n c e R / Q i s non-
t r i v i a l l y graded, t h e r e i s a homoqeneous t o f p o s i t i v e deqree, t f Q , hence t t P.
213
Ill. Local Conditions for Noetherian Graded Rings 2. L e t O v be a d i s c r e t e v a l u a t i o n r i n g w i t h maximal i d e a l av g e n e r a t e d
i n R we have t x = 0.
by t say. P u t R = O v [ X ] / ( t X ) i . e . (t,x)
i s a graded i d e a l and e v e r y homogeneous e l e m e n t o f ( t , x )
z e r o - d i v i s o r ; however t
+
x i s n o t a z e r o - d i v i s o r . I f P1,...,Pn
t h e prime i d e a l s associated t o R , then these h ( 1 ) c PIU
...U
Pn b u t I
Q
Pi
Then is a are
a r e araded and c l e a r l y
f o r each i, 1 5 i 5
n.
So O,[
X I
does n o t
.
have p r o p e r t y C.P.
L e t R be a commutative N o e t h e r i a n r i n g and M a f i n i t e l y g e n e r a t e d Rmodule. A sequence al,
. . . ,n,
and f o r each i = 1,
,..., ai-l)M.
M/(al
...,an ai
i n R i s s a i d t o be an M-sequence i f (al,
. . . ,an)M#M
i s n o t an a n n i h i l a t o r f o r t h e module
I f I i s a n i d e a l o f R such t h a t I M # M , t h e n we s h a l l
denote by grade (1,M)
t h e common l e n g t h o f a l l maximal Wsequences
contained i n I .
111.3.3. Lemma.
L e t R be a commutative N o e t h e r i a n r i n q and l e t M be a
f i n i t e l y g e n e r a t e d R-module. I M # M. Then grade (1,M)
Proof.
Suppose t h a t I i s an i d e a l o f R such t h a t
i s t h e s m a l l e s t number r such t h a t Extk(R/I,M)#O.
F i r s t we show by i n d u c t i o n on r t h a t i f al,
sequence c o n t a i n e d i n I t h e n E x t i ( R / I , E x t L ( R/ I .M)
=
Hom( R/ I ,M/ ( a l , . . . ,ar) M)
-
a*,
...,a,
i s an M-
M) = 0 for 0 5 i 5 r, and
.
Indeed, from t h e e x a c t sequence
v
0-
C
a1
I M
M/alM
-0
We d e r i v e t h e e x a c t sequence : E x t r - l ( R / I ,M)
'a 4 Ext'-l(R/I
--jr Extr(R/I,M)
'a1 M
By t h e t h e i n d u c t i o n
pal
=
0. Hence E x t r - '
, W ) +xtr-'(R/1
...
Extr(R/I,M)-
h y p o t h e s i s : E x t r - l (R/I,M) ( R / I , M/alM)
,M/alM)
h
E x t r (R/I,M).
=
0 and t h e homomorphism
214
6. Commutative Graded Rings
-
BY i n d u c t i o n : Extr-' Extr(R/I,M) M" = i x
( R / I , M/alM)
5
Hom (R/I,P/(a l...ar)F.l)
Hom(R/I,M/(a l...ar)M).
I f Hom ( R / I , N ' )
I x = 0 } i s not zero. Therefore
C MI,
En' such t h a t I c
t h e r e e x i s t P F Ass M" c Ass
. . . ,an
sequence 1 al,
and t h e r e f o r e
# 0 t h e n t h e module
Ass M" # gj
i.e.
P, and t h e r e f o r e t h e
} i s an M-sequence o f maximal l e n q t h .
C o r o l l a r y . L e t R be a commutative N o e t h e r i a n r i n a w h i c h i s
111.3.4.
graded and c o m p l e t e l y p r o j e c t i v e . L e t F.1 F R-qr be f i n i t e l y g e n e r a t e d and I a graded i d e a l o f R such t h a t I M # M . I f grade (1,M) = m, t h e n t h e r e e x i s t s i n I an M-sequence o f l e n g t h m, c o n s i s t i n o o f homogeneous elements. Moreover any M-sequence formed by homooeneous elements o f I has t h e same l e n g t h grade (1,M).
Proof.
For m = 0 the assertion i s clear. Let m
>
1 and l e t P1'....pS
be graded p r i m e i d e a l s a s s o c i a t e d t o M. S i n c e qrade (1,M)
I4
.U
PIU..
Ps and because o f Lemma 111.3.1. h ( I )
t h e r e e x i s t s fl c h ( 1 ) such t h a t fl n o n - a n n i h i l d t o r o f M. L e t grade (I,M)
Q
P1 U . . . U
Q
0,
P1 U . . . U P,.
P s , t h e r e f o r e fl i s a
M1 be t h e qraded R-module M/flM,
= m - 1 and we a p p l y
Hence
then
t h e i n d u c t i o n h y p o t h e s i s . The second
s t a t e m e n t may a l s o be p r o v e d u s i n g a s i m i l a r i n d u c t i o n aroument. 111.3.5.
C o r o l l a r y . L e t R be a commutative N o e t h e r i a n araded r i n a h a v i n q
p r o p e r t y C.P. L e t P be a p r i m e i d e a l o f R h a v i n g h e i g h t n. Then t h e r e e x i s t homogeneous elements al, Ra
1
t.. .t
Proof.
...,an
i n R such t h a t P i s m i n i m a l o v e r
Ra .
F o r n = 0 t h e r e i s n o t h i n a t o prove, so we may assume n > 0
and proceed by i n d u c t i o n on n . L e t { Q,,
...,Qk
} be t h e s e t o f m i n i m a l
i = 1, . . . ,k
p r i m e i d e a l s o f R and as R i s graded, each one o f t h e Qi
,
i s graded. S i n c e h t ( P ) 3 1, P i s n o t c o n t a i n e d i n any Qi
and t h e r e f o r e k
Lemma 111.3.1. e n t a i l s h(P)
4
QIU
... U
Q,.
P i c k al
F h(P) - U
i=1
Qi.
215
111. Local Conditions for Noetherian Graded Rings
I n R/Ral,P/Ralis
a a r a d e d p r i m e i d e a l w i t h ht(P/Ral)
h y p o t h e s i s y i e l d s t h a t P/Ral i = 2,
ai,
ai
ai f o r
..., n
n-1. The i n d u c t i o n
5
i s m i n i m a l o v e r ( a *+...,an),
a r e homogeneous e l e m e n t s o f R/Ral.
the
where
Choosinq r e p r e s e n t a t i v e s
i n R i t i s o b v i o u s t h a t P i s m i n i m a l o v e r ( a l,...,an).
C o n s i d e r a g r - l o c a l r i n a R w i t h maximal w a d e d i d e a l Q and suppose t h a t R i s c o m p l e t e l y p r o j e c t i v e . L e t M c R-gr have f i n i t e t y p e . The r i n g ?i = R/annRP i s a g r . - l o c a l and
r i n q w i t h maximal q r a d e d i d e a l
= B/annRM
R has p r o p e r t y C.P. W r i t e n f o r t h e K r u l l d i m e n s i o n o f M, t h e n
n = K.dim
R
= h t ( 6 ) . By C o r o l l a r y I I I . 3 . 5 . , t h e r e e x i s t homoqeneous
,..., an c 6
elements
al
i s minimal over
such t h a t
t . . .+anM =
Our h y p o t h e s i s y i e l d s t h a t M/alM
M/alM+..
_-
. .+ Ra i s a g r - A r t i n i a n r i n q , hence M/alMt..
R/Ral+.
(R/Ra,+. =
6
On t h e o;her
..+!%,)-module.
R/annRM
+
R a p ...tRan, t h u s i t f o l l o w s t h a t M/alMt
g r . A r t i n i a n (R/annRM
t
Ral+.
..+Ran)-module;
.+anM.
Furthermore
.+an
i s a qr.Artinian
that
hand we have
- -
+ . . . t R an i n R .
&'Ra,t.
...+anM
. .+Ran
=
is a
consequently, t h e l a t t e r
module a c t u a l l y i s g r . A r t i n i a n i n R-gr. A s y s t e m o f homogeneous p a r a m e t e r s i s a s e t o f e l e m e n t s al,
...,an
A r t i n i a n i n R-gr (hence o f
f o r t h e module M w i t h K.dim M = n
c h ( Q ) such t h a t M/alM+
...+anM
is
f i n i t e lenqth i n R-qr!).
1 1 1 . 3 . 6 . Remarks. 1. As i n t h e ungraded case one may e s t a b l i s h t h a t e v e r y M-system may be i n c l u d e d i n a system o f homoqeneous p a r a m e t e r s f o r #. 2. I f R i s p o s i t i v e l y g r a d e d and such t h a t Ro i s a f i e l d t h e n t h e r e e x i s t s a system o f homogeneous p a r a m e t e r s f o r R+.
However i f Ro i s
n o t a f i e l d b u t a l o c a l r i n g w i t h maximal i d e a l w t h e n t h e g r a d e d i d e a l
B
=
'A
+ Rt may n o t have a system o f homogeneous p a r a m e t e r s , as t h e
f o l l o w i n g example shows.
B. Commutative Graded Rings
216
111.3.7. ideal o
Example. L e t V V
be a d i s c r e t e v a l u a t i o n r i n g w i t h maximal
and u n i f o r m parameter t . L e t R = V [ X I / ( t X ) ;
then R i s
N o e t h e r i a n and p o s i t i v e l y graded. Moreover R i s a g r . l o c a 1 r i n g w i t h K r u l l d i m e n s i o n 1. One e a s i l y v e r i f i e s t h a t t h e r e c a n n o t e x i s t a homogeneous al
c (%+ R + ) / ( t X ) such t h a t {al}
i s a system o f homoqeneous
parameters. I n t h e l i g h t o f t h e f o r e g o i n g o b s e r v a t i o n s w e now suppose t h a t t h e qraded N o e t h e r i a n r i n g R has t h e f o r m Ro 8 R1 8
...,
where Ro i s a f i e l d . Then
R i s g r . l o c a 1 w i t h g r . maximal i d e a l R+.
111.3.8. C
Proposition. L e t M
c
R-gr have f i n i t e t y p e and l e t { a l,....aR}
h(R+). A n e c e s s a r y and s u f f i c i e n t c o n d i t i o n f o r al,
...,an
t o be a
system o f homogeneous parameters f o r M i s t h a t M has f i n i t e t y p e as an o b j e c t o f Ro [al, Proof.
I f al,.
..., a n ]
. . ,an
- g r w h i l e n i s m i n i m a l as such.
i s a system o f parameters f o r M t h e n
i s A r t i n i a n i n R-gr. S i n c e On t h e o t h e r hand s i n c e graded, t h e g r a d i n g o f
k M
k
M
=
V/alM
+. . .+anM
has f i n i t e t y p e i t s g r a d e d i s l e f t l i m i t e d .
i s A r t i n i a n i n R-gr and s i n c e R i s p o s i t i v e l y
i s a l s o r i g h t l i m i t e d . Consequently
M
has t o
be a f i n i t e d i m e n s i o n a l R - v e c t o r space. Nakayama's lemma t h e n y i e l d s 0
that M i s finitely
g e n e r a t e d as a n Ro [ al,
..., a n ] -
module. Conversely,
i f we s t a r t f r o m t h e assumption t h a t M i s a f i n i t e l y g e n e r a t e d Ro [al, module, t h e n M/alM
+.. .+ anM
. . . ,a n ] -
i s a f i n i t e d i m e n s i o n a l R o - v e c t o r space
t h u s A r t i n i a n i n R-gr. N e x t l e t us check m i n i m a l i t y o f n as such. I n d e e d i f { b l,...,bm}
i s a s e t o f homogeneous elements such t h a t M i s a f i n i t e l y
g e n e r a t e d Ro [ bl,
..., b],
-module t h e n M/blM
of f i n i t e l e n g t h t o o . B u t t h e n K.dim M mcn.
5
i s A r t i n i a n i n R-gr,
i.e.
n excludes the p o s s i b i l i t y
217
111. Local Conditions for Noetherian Graded Rings
IV.3.9.
P r o p o s i t i o n . L e t M c R-gr have f i n i t e t y p e . I f al , . . . , an i s a
...,an a r e
system o f homogeneous parameters f o r M t h e n t h e elements al, a l g e b r a i c a l l y i n d e p e n d e n t o v e r R,. ~
...,an
P r o o f . L e t t i n g al,
... , X n ] w i t h g r a d a t i o n 1, ..., n . S p e c i a l i z i n a Xi
r i n g S = Ro [ X1, =
i
deg ai,
=
homomorphism o f degree 0, We have t h a t K.dim Ro [ al,.. then
o f parameters, c o n s i d e r t h e
be such a system
'0 : Ro [
. ,an]
X1,. =
d e f i n e d by p u t t i n g deg Xi +
=
ai d e f i n e s a s u r j e c t i v e r i n g
..,Xn]
+
. . ,an 1
Ro [ al,.
.
K.dim M = n. However i f Ker '0 # 0
t h e K r u l l dimension s h o u l d have dropped, so '0 i s an isomorphism.
111.3.10.
Proposition.
Let M
c
R - g r be f i n i t e l y qenerated. The f o l l o w i n g
statements a r e equivalent. 1. M i s a Cohen-Macaulay module. 2. I f al,
...,an
i s a system o f homogeneous parameters f o r M t h e n M i s
a f r e e ( g r a d e d f r e e ) Ro [ a l,...,an]
-module.
3. E v e r y system of homogeneous p a r a m e t e r s f o r M i s an M-sequence. Proof. 1 = 2. P u t S = Ro [ a l
,...,a n ] .
By P r o p o s i t i o n 111.3.9..
S i s a regular
r i n g w i t h K.dim S = n. S i n c e M i s a Cohen-Macaulay module, grade (R+,V) = K.dim M = n and
bl,...,bn =
M/blM
by
C o r o l l a r y 111.3.3.
=
t h e r e e x i s t s an M-sequence
= h(R) i = 1, . . . , n. F o r i = 1, . . . , n ~ u Mt (i) +...tbiM and p u t M = M. C o n s i d e r i n a t h e f a c t t h a t S i s a
with b
c
(0)
r e g u l a r r i n g , we may d e r i v e f r o m t h e e x a c t n e s s o f m. ltl O+ M M(i) M ( i t l ) +'> (i) (where mitl denotes mu1 i p l i c a t i o n by bi+l) t h a t D.dimSM(i+l)
'
*
= l+p.dim M
s
= M/blM+ ...+ bn M i s an S-module o f f i n i t e l e n g t h . (n) i s a f i n t e l y d i m e n s i o n a l R o - v e c t o r space. Because S i s
On t h e o t h e r hand, M Therefore M
(n) g r - l o c a l we have t h a t p dimSM(,,)=
n = p.dim M
s
(0)'
Now p.dim M
s
(n)
=
n
(i)
218
8. Commutative Graded Rings
y i e l d s p.dimSM(o) = p.dimst4 = 0, c o n s e q u e n t l y M i s a f r e e S-module. 2 = 3, as w e l l as 3 = 1 a r e r a t h e r easy.
o
Supplementay B i b l i o g r a p h y . 1. S. Goto and K. Watanabe, On graded r i n g s I, J. Math. SOC. Japan, 30 (1978) 172-213. 2 . S. Goto and K. Watanabe, On graded r i n g s I1 (Zn-graded r i n g s ) Tokyo J. Math., 1, n r . 2 ( 1 9 7 8 ) . 3. R. Fossum The S t r u c t u r e o f
ndecomposable I n j e c t i v e modules ( p r e p r i n t . )
Exercises 1. L e t R be a graded r i n g o f t y p e Z . Prove t h a t i f i d e a l PI QR-p(R)
f o r any araded p r i m e
i s N o e t h e r i a n t h e n f o r any p r i m e i d e a l Q,QR-Q(R)
i s Noetharian. 2. Suppose t h a t R i s graded N o e t h e r i a n r i n g and
2
i s a araded i d e a l i n
r R. F o r each n C Z we d e f i n e ( R g r ( a ) ) n = l i m ( R / a ) n and we denote by r Rgr(a) = 0 ( R g r ( a ) ) n .Rsr(a) i s a graded r i n g ; t h i s r i n q i s c a l l e d nFZ t h e graded a - a d i c c o m p l e t i o n o f R. Prove t h a t : 1) The r i n g
Rgr(a) i s
Noetherian
2) Rgr(a) i s f l a t R-module. 3 ) I f we denote by i : R
i( a ) Rgr
+
Rgr
(3the
c a n o n i c a l homomorphism t h e n
i s c o n t a i n e d i n t h e Graded Jacobson r a d i c a l o f
4) Ifa i s gr-maximal i d e a l t h e n
Rgr(a) i s
Rgr(a)
g r - l o c a l with i(a)Rgr
t h e gr-maximal i d e a l . 5 ) If2 i s c o n t a i n e d i n t h e graded Jacobson r a d i c a l of R t h e n t h e c a n o n i c a l homomorphism i : R
--f
Rgr(d)
i s t h e 2-adic completion o f R then R a
,gr@)
- c o m p l e t i o n of Rqr@)
i s i n j e c t i v e . Moreover i f c
Rgr@)
c
6 and
i s the
(a) as
219
111. Local Conditions for Noetherian Graded Rings
3. Let R be a graded NoetherIan r i n g and P a graded prime i d e a l . Prove
i s isomorphic t o t h e graded completion
t h a t t h e r i n g FNDR(E:(R/P))
Qi-p(R)
of t h e r i n g 4. Let R be a
a t i t s graded maximal i d e a l .
local graded r i n g with M the gr-maximal i d e a l . Suppose
t h a t R i s graded M-completed ( i . e . R = R g r ( M ) ) .
E
=
E;(R/M).
Prove t h a t :
1) I f X i s gr-Noetherian R module then H O M R ( X , E )
if Y
Write :
i s g r - A r t i n i a n and
i s a g r - A r t i n i a n R-module then H O M R ( Y , E ) i s gr-Noetherian
R-module.
2 ) The f u n c t o r H O M R ( . , E )
i s a d u a l i t y between t h e category
of g r -
Noethbrian modules and t h e category o f g r - A r t i n i a n modules. 5 . Let R = 8 R i be a graded r i n g of type Z and i CZ
T i s an indeterminate
with deg T = 1. Then we can form t h e graded ( T ) - a d i c completion o f
R [ T I ; we denote by gr R [ [ T I ] t h e ( T ) - a d i c completion o f R[ T I . Prove t h a t : 1) For every n C Z , ( g r R [ [ T I ] ) , = {
2 ) I f Ri
=
-?
i=n
a i T”’
,a c
Ri)
0 f o r every i < 0 , then
gr R “ T I 1
=
.
R [TI
3 ) I f R i s Noetherian then gr R [ [ T I ]
i s Noetherian and gr R [ [ T I 1
i s a f l a t R-module. 6 . Let R be a graded Noetherian r i n g . T h e following s t a t e m e n t s a r e
equivalent : 1) The r i n g R i s Gorenstein 2 ) inj.dim Q R - p ( R ) 3 ) i n j . dim
<
-
f o r a l l graded prime i d e a l s P .
Qi-p(R) f o r a l l c
m
graded prime i d e a l s P.
7. Let R be a g r - l o c a l Noetherian ring with gr-maxim1 i d e a l m.
Prove t h a t :
220
6. Commutative Graded Rings
1) E v e r y graded f i n i t e l y g e n e r a t e d R-module M
=
8
iC Z
Mi has a m i n i m a l
resolution, i . e . a resolution :
. . . j L2
-% L1
_ dl j Lo
do > M -0
L2, . . . a r e qraded f r e e modules and K e r di
where Lo, L1,
c
Li
f o r a l l i 3 0. 2 ) TorL(R/m,M) R
s
Li/m Li
3 ) I f f o r a l l i < 0 Ri
=
0 and Mi
=
0 then prove t h a t t h e p - t h
R component o f t h e graded v e c t o r space Tori (Rtm,M) 4 ) p.dim M = n R
8. L e t R
Ln # 0 and Li
Ro 8 R1 8 R2 8
=
1 ) Prove
...
=
=
0 for all i > n
be a graded N o e t h e r i a n r i n g .
that there exists
with ki al,
-
i s z e r o f o r p c i.
homogeneous elements al,...,aj
C R
deg ai such t h a t R i s g e n e r a t e d as an Ro-algebra by
a2,, . . ,as.
2 ) Suppose t h a t R o i s an A r t i n i a n r i n g . I f M = Mo @MI
8
... i s
a
graded f i n i t e l y g e n e r a t e d R-module t h e n f o r a l l i 2 0, Mi i s an Po-module o f f i n i t e l e n g t h . 3 ) I f we denote by l ( M n ) t h e l e n g t h o f Mn i n Ro-mod,
we can c o n s i d e r
t h e power s e r i e s P(M,T)
=
,z0
l ( M n ) Tn
c
ZlITll.
Prove t h a t P(M,T) i s a r a t i o n a l f u n c t i o n i n T s k. f ( T ) / 11 (1-T ’ ) , where f ( T ) c Z [ T I . i=1 4)
I f m r e o v e r k. = 1, i = 1, 1
...,s,
o f t h e form
p r o v e t h a t for a l l s u f f i c i e n t l y
l a r g e n, I(Mn) i s a p o l y n o m i a l i n n. 9. L e t R be a graded N o e t h e r i a n r i n q of t y p e Zn(n 2 1 ) . I f M i s
a
f i n i t e l y g e n e r a t e d graded R-module t h e n t h e f o l l o w i n g s t a t e m e n t s are equivalent :
221
111. Local Conditions for Noetherian Graded Rings
1) M i s Cohen-Macaulay ( r es p . Gorenstein) module. 2 ) For any graded prime ideal P of V ( M ) , QR-p(M)
i s Cohen-Macaulay
(resp. Gorenstei n) QR-p( R) -module 10. Let R = Ro
Q
R1
...
Q
m = f i e l d . We p u t -
be a graded Noetherian ring where Ro = k i s a
R i . Let M 0 Mi be a graded R-module. We define i CZ M* = HOMR (M,k) and cal l i t the graded R-dual of M. Prove t h a t : Q
iz1
1) M* i s a graded R-module with
2 ) I1 =
F1**
{
i f and only i f dimkMi
<
}ncz as i t s grading.
Hornk (M-,.k) m
for all i
C
Z.
3) I f we denote by A k the subcategory of a l l graded R-modules M = 8 Mi such t h a t dimk(Mi) c - ' f o r a l l i F Z ; prove t h a t the
i FZ contravariant functor (.)* = HOMk(.,k)
i s a dua lity over t h i s
category. 4 ) I f M i s graded Noetherian ( r es p . Artinian) R-module then
r1*
is
graded Artinian ( r es p . Noetherian) R-module. 5) I f L i s graded f r e e R-module and f i n i t e l y generated t h e n L* i s
an i n j e c t i v e R-modul e . 6) If M i s graded f i n i t e l y generated R-module and i f
...
L2
d2
4
L1 --Lo-
dl
11 - 0
i s f r e e minimal resolution of M i n R-gr (
o
-+
M*
-+ L*-0
resolution o f M*.
L*-L* 1
2
-+ ...
see exercises 7 ) then
i s an i n j e c t i v e minimal
222
CHAPTER C: STRUCTURE THEORY FOR GRADED RINGS OF TYPE Z
I : Non-Commutative Graded Rings 1.1. Rings o f F r a c t i o n s and G o l d i e ' s Theorems. Throughout R s t a n d s f o r a non-commutative r i n g , w h i c h i s graded o f t y p e Z. F o r i n f o r m a t i o n a b o u t r i n g s o f f r a c t i o n s we r e f e r t o S e c t i o n A . I . 6 . , f o r f u l l d e t a i l on graded r i n g s o f q u o t i e n t s and l o c a l i z a t i o n , see Section A.II.9.
and consequent.
A graded r i n g R h a v i n g f i n i t e G o l d i e d i m e n s i o n i n R-gr and s a t i s f y i n g t h e a s c e n d i n g c h a i n c o n d i t i o n on graded l e f t a n n i h i l a t o r s i s c a l l e d a graded G o l d i e r i n g (sometimes w r i t t e n as g r - G o l d i e r i n g ) . I f R i s t r i v i a l l y graded t h e n t h i s reduces t o t h e d e f i n i t i o n o f a G o l d i e r i n g as i t i s used e.g.
i n [ 49
, I . Now a theorem known as " G o l d i e ' s theorem"
a s s e r t s t h a t a r i n g R has a s e m i s i m p l e A r t i n i a n ( r e s p . s i m p l e A r t i n i a n ) c l a s s i c a l r i n g o f f r a c t i o n s i f and o n l y i f R i s a
semisimple (resp.
p r i m e ) G o l d i e r i n g . Now, e x a c t l y as i n t h e ungraded case one p r o v e s t h a t i f R has a graded r i n g
o f f r a c t i o n s w h i c h i s g r - A r t i n i a n and g r -
s i m p l e t h e n R i s a p r i m e g r - G o l d i e r i n g . The c o n v e r s e does n o t h o l d , as t h e f o l l o w i n g example shows : 1.1.1.
Example. L e t k be a f i e l d and l e t R be t h e p o l y n o m i a l r i n g k [ X,Y
s u b j e c t e d t o t h e r e l a t i o n XY = Y X = 0. P u t Rn = kXn i f n
?
1
0 and Rm = kYm
ifm 5 0. C l e a r l y , each x # 0 i n h(R) w i t h deg x # 0 i s n o n - r e g u l a r . Moreo v e r t h e i d e a l (X,Y)
i s e s s e n t i a l i n R b u t i t does n o t c o n t a i n a r e g u l a r
homogeneous e l e m e n t . T h i s shows t h a t , a l t h o u g h R i s a semisimple q r - G o l d i e r i n g , i t does n o t a l l o w a g r - s e m i s i m p l e g r - A r t i n i a n r i n g o f f r a c t i o n s . 1.1.2.
Lemma. L e t R be a graded r i n g s a t i s f y i n g t h e a s c e n d i n g c h a i n
c o n d i t i o n f o r graded l e f t a n n i h l a t o r s ,
Q i s nilpotent.
then t h e l e f t s i n g u l a r r a d i c a l
Of
223
1. Non-Commutative Graded Rings
J f o r t h e l e f t s i n g u l a r r a d i c a l o f R; note t h a t J i s a
Proof. W r i t e
graded i d e a l . I f S i s a subset o f R l e t 1 ( S ) be t h e l e f t a n n i h i l a t o r 2 of S i n R. Since t h e r e i s an ascending c h a i n 1 ( J ) c l ( J ) c
the hypotheses i m p l y t h a t l ( J n ) 5"'
=
l(Jn+')
... c
happens f o r some n 6
-...,
N. I f
i s nonzero t h e n a Jn # 0 f o r some a c h(R) and we may choose a
such t h a t l ( a ) i s maximal w i t h r e s p e c t t o t h a t p r o p e r t y . I f b F 3 then l ( b )
n
n
h(R
Ra # 0 because l ( b ) i s an e s s e n t i a l l e f t i d e a l o f R. Thus
c
there e x i s t s c
h(R) such t h a t ca # 0 and cab = 0. So l ( a )
2
l(ab);
t h e hypothesis on a e n t a i l s t h a t ab Jn = 0. Consequently, s i n c e = 0 i.e.
graded, a 5"' 5"'
l(Jn)
a F l(J
n+l
J
is
) = l ( J n ) ) , a c o n t r a d i c t i o n . Therefore
= 0 follows.
1.1.3.
Theorem.
L e t R be a s e m i s i p l e graded r i n g .
The f o l l o w i n g statements a r e e q u i v a l e n t . 1. R i s a gr-Goldie r i n g .
2. R i s a Goldie r i n g . Proof. 2. = 1. Easy. 1. = 2 From Lemma 1.1.2.
i t follows t h a t the l e f t singular radical,
Z(R), i s zero. Consider t h e graded maximal q u o t i e n t r i n g
, Qiax(R),
of R and t h e maximal q u o t i e n t r i n g , Qmax(R), o f R. Since Z(R) = 0 we obtain : R c Q R (,;)
c QmaX(R). I t i s n o t h a r d t o see t h a t Q i a X ( R ) i s a g r - r e g u l a r
r i n g i n Von Neumann's sense (see S e c t i o n 1.5. f o r more d e t a i l on these r i n g s ) whereas Qmax(R) i s a r e g u l a r r i n g . Since R has f i n i t e Goldie dimension i n
R-gr, Q(R ,;)
has f i n i t e Goldie dimension i n Q;,,(R)-gr.
Therefore ( s e e a l s o r e s u l t s o f S e c t i o n
A. ) Q;~,(R)
i s a gr-semisimple
g r - A r t i n i a n r i n g and i n p a r t i c u l a r Q i a x ( R ) i s l e f t Noetherian. Since
224
C. Structure Theory for Graded Rings of Type Z
Qmax(R) i s an e s s e n t i a l extension o f R i n R-mod, i t i s a l s o an e s s e n t i a l extension o f Q i a x ( R ) i n Qiax(R)-mod. Now Q i a x ( R ) i s l e f t Noetherian, hence i t has f i n i t e Go1 d i e dimension, t h e r e f o r e Qmax(R) has f i n i t e Goldie dimension t o o . That R has f i n i t e G o l d i e dimension i n R-mod follows. 1.1.4.
Lemma. L e t R be a semisimple graded r i n g s a t i s f y i n g t h e ascending
c h a i n c o n d i t i o n s on graded l e f t a n n i h i l a t o r s . I f I i s a graded l e f t ( o r r i g h t ) i d e a l such t h a t every element o f h ( 1 ) i s n i l p o t e n t , then I = 0. Proof. For a E h(R) t h e r e i s equivalence between t h e statements :
1" every element o f h(Ra) i s n i l p o t e n t . 2" every element o f h(aR) i s n i l p o t e n t .
L e t us prove
t h e case where I i s a r i g h t i d e a l .
I f I # 0 t h e n h ( 1 ) # 0 and we may choose a E h ( 1 )
an$(,)
i s maximal
a h
# 0. By assumption, t h e r e
e x i s t s a t > 0 such t h a t ( a x ) t = 0 and c annR 1 (ax) t - 1 , w i t h ( a x annR(a) 1
1
0, such t h a t
amongst l e f t a n n i h i l a t o r s o f elements of h ( 1 ) which
are nonzero. Take X E h(R) such t h a t
= annR
,a #
Therefore
c
# 0. From t h e i n c l u s i o n
= h I ) , i t f o l l o w s t h a t annR(a) 1
a k a = 0. Thus aRa = 0 b u t t h i s c o n t r a d i c t s
t h e f a c t t h a t R i s a semiprime r i n g . 1.1.5.
C o r o l l a r y . L e t R be a semiprime graded r i n g s a t i s f y i n g t h e
ascending c h a i n c o n d i t i o n on graded l e f t a n n i h i l a t o r s . I f I i s a l e f t ( o r r i g h t ) n i l - i d e a l t h e n I = 0. Proof.
If I
x = x1
+ ...
# 0 t h e n I- # 0. Ifa # O i s i n h(I-) then t h e r e i s an xn i n I such t h a t xn = a, deg x1 <
... <
deg xn. Since
x i s n i l p o t e n t , a i s n i l p o t e n t . So 1- i s a l e f t n i l - i d e a l and i t i s graded, so t h e lemma e n t a i l s 1- = 0, c o n t r a d i c t i o n .
225
1. Non-Commutative Graded Rings
1.1.6.
Theorem. L e t R be a semiprime g r - G o l d i e r i n g s a t i s f y i n g one ( o r
more) of t h e f o l l o w i n g p r o p e r t i e s : 1. R has
r e g u l a r c e n t r a l homogeneous elements o f p o s i t i v e degree.
2 . R i s p o s i t i v e l y graded and m i n i m a l p r i m e i d e a l s o f
3.
R is
p o s i t i v e l y graded and t h e r e e x i s t
R
do n o t c o n t a i n R+
r e g u l a r homogeneous elements
o f p o s i t i ve degree.
4. R i s n i l g r a d e d i . e . e v e r y homogeneous element o f nonzero degree i s n i 1p o t e n t . Then
R
has a g r - s e m i s i m p l e g r - A r t i n i a n r i n g o f f r a c t i o n s .
Proof. __-
L e t L be a nonzero graded l e f t i d e a l o f R. We c l a i m t h a t t h e
h y p o t h e s i s 1.,2.,3.,
l e a d t o t h e e x i s t e n c e o f a n element o f p o s i t i v e
degree i n h ( L ) , w h i c h i s n o t a n i l p o t e n t element. Indeed, suppose 1. By t h e f o r e g o i n g lemma, t h e r e e x < s t s an a C h ( L ) such t h a t a k = 1,2..
..
k
# 0,
P i c k m such t h a t : m deg s + deg a > 0. Then b = sm a F h ( L )
i s n o t n i l p o t e n t and deg b > 0. Now s u p p o s e 2 . I f L i s n o t as c l a i m e d , t h e n t n e graded l e f t i d e a l L ' = L1 lemma we have L ' = 0 and
L2
8
...i s
a n i l l e f t i d e a l . By t h e
t h e n L = Lo. I n t h a t case R+L = 0 and t h e r e f o r e
L i s c o n t a i n e d i n e v e r y m i n i m a l p r i m e i d e a l . Hence L = 0, c o n t r a d i c t i o n . Now suppose 3. Then we have :
0 # SL
c
Lt
@
Lt+l
@
...,
where t = deg s > 0. T h i s l e a d s t o t h e e x i s t e n c e o f a homogeneous element o f p o s i t i v e degree w h i c h i s n o n - n i l p o t e n t i n Lt 8 Lt+l
8
...
The h y p o t h e s i s 4. Leads t o t h e e x i s t e n c e o f a n o n - n i l p o t e n t e l e m e n t o f h ( L ) w h i c h has degree zero. The e s s e n t i a l p a r t o f t h e p r o o f i s now t o e s t a b l i s h t h a t a graded e s s e n t i a l l e f t i d e a l o f R c o n t a i n s a r e g u l a r homogeneous element. Under t h e assumptions l.,Z., i s a n al
c h ( L ) w i t h deg al
0 and a;
# 0
v =
3., we f i n d t h a t t h e r e 1,...,...
.
F o r some
C. Structure Theory for Graded Rings of Type Z
226 n C
N : Ann(ay)
= Ann(a;+l)
Ann ( b l ) = Ann(by).
b2 C L
n
Ann(bl)
=
_. .
P u t bl = a;,
t h e n deg bl > 0 and
If Ann ( b l ) # 0 t h e n t h e r e i s an homogeneous
w i t h deq b 2 > 0
and Ann(b2) = Ann(bg)
,v
...
P r o c e e d i n g i n t h i s way we o b t a i n a d i r e c t sum Rbl 8 Rbp 8 graded l e f t e x i s t s an r Ann(bl)
i d e a l s and s i n c e t h e G o l d i e dimension
cN
1
2, ...
.
o f nonzero
o f R i s f i n i t e , there
such t h a t :
n... n
L e t d . = deg bi
= 1,
Ann(br) = 0 0, 1 5 i 5 r and d = dl...dr.
d/dl Then c = bl
i s a homogeneous element o f L w i t h deg c = d. However Ann c =
d/dr
+...+br n
i=l
Ann(bi
d/di
=
) =
nrAnn(bi) = 0; so Rc i s an e s s e n t i a l graded l e f t i d e a l , c i s a r e g u L a r i=1 element and dea c > 0. Under t h e assumption 4, L c o n t a i n s a r e a u l a r homoqeneous element o f degree 0. From h e r e on t h e p r o o f w i l l be s i m i l a r t o t h e p r o o f o f G o l d i e ' s theorem i n t h e ungraded case. 1.1.7.
C o r o l l a r y . A p r i m e p o s i t i v e l y graded g r - G o l d i e r i n a
R
admits a
gr-simple g r - A r t i n i a n r i n g o f f r a c t i o n s . 1.1.8.
C o r o l l a r y . L e t R be a s e m i s i m p l e gr-(;oldie
ring.
If R i s l e f t -
and r i g h t - l i m i t e d t h e n R a d m i t s a g r - s e m i s i m p l e g r - A r t i n i a n r i n g o f fractions.
1.1.9.
Corollary Let
R
be a l e f t and r i g h t N o e t h e r i a n graded r i n g w h i c h
i s e i t h e r p o s i t i v e l y graded o r commutative. L e t P be any p r i m e i d e a l of R then e i t h e r : ht(P) = h t ( ( P ) ) o r ht(P) = ht((P)J+ 9 Proof.
1.
Suppose t h a t Q i s a p r i m e i d e a l o f R such t h a t (P),
Up t o p a s s i n g t o R / ( P )
4
we may assume t h a t (P),
c Q
c P.
= 0 and hence t h a t R
i s a prime r i n g . Let S
be t h e s e t o f r e g u l a r homogeneous elements o f R. C l e a r l y , R b e i n g
N o e t h e r i a n p r i m e and graded, R i s a graded G o l d i e r i n g hence, t h e
227
1. Non-CornrnutativeGraded Rings
f o r e g o i n g p r o p o s i t i o n i m p l i e s t h a t S - l R e x i s t s and t h a t i t i s a g r - s i m p l e g r - A r t i n i a n r i n g . From o u r r e s u l t s on t h e K r u l l dimension we r e t a i n : K.dim S - l R of S - l R ,
1. On t h e o t h e r hand S - l Q and S - l P a r e nonzero p r i m e i d e a l s
5
o f the r i g h t
t h e r e f o r e S - l Q = S - l P . T h i s means, because
N o e t h e r i a n p r o p e r t y t h a t SP c Q f o r some s C S, s # o hence P = Q o r s E Q;
c
= 0 t h e l a t t e r i s i m p o s s i b l e hence P = Q i s g t h e o n l y r e m a i n i n g p o s s i b i l i t y . Now i f P = P t h e n h t ( P ) = h t ( P ) so q 9 suppose t h a t P # P and assume t h a t h t ( P ) = n < m . F o r n = 1, F o r n = 1, 9 P i s a m i n i m a l p r i m e i d e a l , i . e . h t P = 0. We p r o c e e d by i n d u c t i o n 9 9 on n. L e t Po c P1 c... c Pn = P be an ascending c h a i n o f l e n q t h n o f p r i m e
s
w o u l d i m p l y s E (P)
Q
i d e a l s c o n t a i n e d i n P. By t h e i n d u c t i o n h y p o t h e s i s we have ht(Pn-l) i f Pn-l
= (Pn-l)g
pg # (PnJg Pn-l
3
# (Pn-l)g
s i n c e o t h e r w i s e Pn-l
so h t (P,)
Pg,
= Pn-l and a l s o h t ( P ) = h t ( P )+l. 9 9 t h e n ht(Pn-l) = ht((Pn-l)g) + 1. Now
b u t i n t h i s case P
On t h e o t h e r hand if Pn-l
would be p r o p e r l y c o n t a i n e d i n P and Thus h t ( P ) 9
ht((Pn-l)g).
>
= n-1
?
n - 1 and h t ( P ) 5 h t ( P )+l. 9
L e t R be a N o e t h e r i a n commutative rim. I f P i s a graded
1.1.1O.Corollary.
ppime i d e a l w i t h h t ( P ) = n, t h e n t h e r e e x i s t s a c h a i n of graded p r i m e i d e a l s = Po Proof.
2
P1
2 ... 9
P,
= p.
The s t a t e m e n t i s t r u e f o r n = 1. I f h t ( P ) = n > l t h e n t h e r e i s a
c h a i n o f d i s t i n c t p r i m e i d e a l s : Qo
2
Q, >... Qn
= P.
If Qn-l i s graded t h e n a p p l i c a t i o n of t h e i n d u c t i o n h y p o t h e s i s f i n i s h e s the proof.
Suppose t h a t Qn-l i s n o t graded and t h e n t h e f o r e g o i n g
c o r o l l a r y y i e l d s t h a t ht((Qn-l)g)
n-2. The i n d u c t i o n h y p o t h e s i s a s s e r t s
=
t h a t we may f i n d a c h a i n o f d i s t i n c t graded p r i m e i d e a l s Po c P1 c
...
cPn-2 = ( Q
)
n-1 g
.
Consider t h e graded r i n g R/(Qn-l)g
and denote by P,On-,
t h e image of P, Qnm1
homogeneous
a
element
of
P,
hence
a
E
in
on-1.
R.
=
R
Choose a nonzero
L e t Pn-l
be a m i n i m a l p r i m e
228
C. Structure Theory for Graded Rings of Type Z
ideal containing ht(P) =
Ci.. Then Pn_l
i s graded and
# P because o t h e r w i s e
Pn-l
1 f o l l o w s f r o m t h e p r i n c i p a l i d e a l theorem. P u t t i n g Pn-l
5
Pn-l/(Qn-l)g
we f i n d a c h a i n o f d i s t i n c t p r i m e i d e a l s :
.. .
Po c P1 c
=
c Pn-l
1.1.11.Corollary.
c Pn = P .
0
L e t R be a N o e t h e r i a n commutative praded r i n g and l e t
I be an i d e a l o f R, t h e n : ht(1) 5 ht(I-), Proof.
ht(1)
5
ht(1-).
S i n c e h t ( 1 ) = h t ( r a d ( 1 ) ) and ( r a d ( I ) ) - c
rad(1-)
we may reduce t h e p r o o f t o t h e case where I i s a semiprime i d e a l , I = P1
n... n
Py.Py
...
ht(I-)
=
P s , Pi p r i m e i d e a l s o f R. Then P1...Ps
c Iyields
P y c I-. L e t Q be a p r i m e i d e a l c o n t a i n i n g I-, such t h a t
h t ( Q ) . F o r some i, P y c
Q. B u t I
c Pi
hence we may assume t h a t
I = P i s a p r i m e i d e a l i n p r o v i n a t h e s t a t e m e n t . I f P i s qraded t h e n I f P i s n o t graded t h e n we have : h t ( P ) = h t ( P ) t 1. C l e a r l y q P c P- and Po # P-. I t r e s u l t s f r o m t h i s t h a t h t ( P ) < ht(P-) and 9 9 t h e r e f o r e h t ( P ) 5 ht(P-). I n a s i m i l a r way, h t ( 1 ) 5 ht(1-) may be P = P-.
es t a b 1 ished. I f one i s i n t e r e s t e d i n h a v i n g a n i c e r i n g o f q u o t i e n t s i . e . n o t b e i n g p a r t i c u l a r about i t n o t being a
c l a s s i c a l r i n g o f f r a c t i o n s , t h e n one
may a v o i d e x t r a hypotheses as i n P r o p o s i t i o n 1.1.4.
and d e r i v e a n o t h e r
graded v e r s i o n o f G o l d i e ' s theorem. W r i t e E g ( R ) f o r t h e i n j e c t i v e h u l l o f R i n R-gr. 1.1.12.Proposition.
L e t R be a graded p r i m e G o l d i e r i n g t h e n Eg(R) i s
a g r - s i m p l e g r - A r t i n i a n r i n g such t h a t R
+
t h e c a n o n i c a l r i n g morphism
Eg(R) i s a l e f t f l a t r i n g epimorphism.
229
1. NonCommutative Graded Rings
Proof.
_c_
the i n j e c t i v e hull E ( R ) of R in R-mod i s a
By Theorem 1.1.3.
simple Artinian ring a n d one e a s i l y v e r i f i e s (using Goldie's theorems) t h a t t h i s makes E 9 ( R ) a Goldie ring. I n j e c t i v i t y of E q ( R ) in R-gr e n t a i l s i s a graded regular ring ( i n Von Neumann's sense). The proof
that Eg(R)
t h a t E g ( R ) i s gr-simple gr-Artinian i s now s i m i l a r t o the ungraded case. However, E g ( R )
i s the l o c a l i z a t i o n of R a t the graded torsion theory
corresponding t o the f i l t e r generated by the graded e s s e n t i a l l e f t ideals of R ( s e e a l s o A.II.9. e t c . ) . The f a c t t h a t E 9 ( R ) i s gr-simple Artinian then e n t a i l s t h a t t h i s torsion theory i s graded perfect ( i . e . corresponds t o a graded kernel functor s a t i s f y i n g property T, see A.II.lO.).
The l a s t statement of the proposition then follows by general
1oca 1 i za t i on theory .
1.1.13.Corollary. Let h ( R ) r be the m u l t i p l i c a t i v e l y closed s e t of regular elements i n h(R). P u t E ( R ) = Q , Eg(R)
=
Qg and l e t Qh be the graded ring
of f r a c t i o n s obtained by inverting the elements of h(R)'. ring homomorphisms : R
+
Qh
-t
Qg
+
We have canonical
Q . Each l o c a l i z a t i o n involved
is a perfect l o c a l i z a t i o n (property T ) . We have t h a t Q, = Q9 i f a n d only i f graded e s s e n t i a l l e f t i d e a l s of R contain homogeneous regular elements. Proof.
R s a t i s f i e s the l e f t Ore condition with respect t o the s e t of
regular elements of R because R i s a Goldie ring. Because of Proposition
...,
A.II.lO.
i t follows t h a t R a l s o s a t i s f i e s the l e f t Ore conditions
with respect t o h ( R ) r .
All a s s e r t i o n s of the corollary follow from t h i s
and the foregoing r e s u l t s .
1.1.14.
Theorem.
Let R be a o o s i t i v e l y graded l e f t Noetherian ring and
l e t P be a graded prime ideal of R. Then
K ;
=
K
~
(
where ~ )
K
h ( p ) i s the
r i g i d kernel functor on R-gr associated t o the m u l t i p l i c a t i v e system h(G(P))
=
h ( R ) 11G ( P ) . I f R s a t i s f i e s the l e f t Ore conditions
with respect
230
C. Structure Theory for Graded Rings of Type Z
t o G(P) t h e n i t a l s o s a t i s f i e s Proof.
these conditions w i t h respect t o h(G(P)).
From C h a p t e r A, P r o p o s i t i o n 11.11.6.
, we r e t a i n t h a t - C ( K ; )
c o n s i s t s o f t h e g r a d e d l e f t i d e a l s J o f R such t h a t t h e image o f J i n
By Theorem 1.1.6. i t f o l l o w s t h a t
R / P i s an e s s e n t i a l l e f t i d e a l of R / P .
J n h(G/P)) # 6 ; the f i r s t statement f o l l o w s . Consider s c h(G(P)), r C h ( R ) and l o o k a t (Rs : r ) . and p u t L =
icy
(Rs : r i )
W r i t e r = rlt
c ( R s : r ) . Then
L i s i n - C ( K ~ because ) R s a n d (Rs:ri) Since
rl
L
i s graded, L F - C ( K ; ) .
C R such t h a t
slr
=
L
...t r n w i t h
deg rl< ...c d e g rn,
i s a g r a d e d l e f t i d e a l , and
a r e i n - C ( K ~ ) f o r a l l 1 5 i 5 n.
T h e r e f o r e t h e r e e x i s t s1
c
h ( G ( P ) ) and
rls.
Since R i s l e f t Noetherian t h i s e s t a b l i s h e s t h a t R Ore c o n d i t i o n s w i t h r e s p e c t t o h ( G ( P ) ) .
s a t i s f i e s the l e f t
231
1. Non-Commutative Graded Rings
1.2. Graded Rings S a t i s f y i n g P o l y n o m i a l I d e n t i t i e s . F o r t h e t h e o r y of r i n g s s a t i s f y i n g p o l y n o m i a l i d e n t i t i e s o u r b a s i c r e f e r e n c e s a r e t h e f o l l o w i n g books : N. Jacobson [ 53
1 % C.
Procesi
1 ; L. Rowen [ 96 1 . L e t us j u s t m e n t i o n h e r e some r e s u l t s t h a t
[ 90
a p p l y most d i r e c t l y t o t h i s s e c t i o n . L e t C be a commutative r i n g and l e t R be a C-alqebra, L e t C { X s } be t h e f r e e C-algebra i n u n c o u n t a b l y many v a r i a b l e s . I f f ( X s ) 6 C { X s }
then
o n l y f i n i t e l y many v a r i a b l e s appear i n f ( X S ) and by "abus de l a n g a g e " we say t h a t f ( X s ) i s a p o l y n o m i a l . We say t h a t f ( X s ) i s a _polynomial i d e n t i t y f o r R i f any s u b s t i t u t i o n o f rl,
...,
rn C R f o r t h e v a r i a b l e s
o f f ( X s ) makes t h e p o l y n o m i a l z e r o . The C-algebra R i s s a i d t o be a
P . I . algebra
i f i t s a t i s f i e s a polynomial i d e n t i t y f ( X s ) which i s n o t
c o n t a i n e d i n I { X s } f o r some i d e a l I o f C. The s t a n d a r d p o l y n o m i a l of deqree n i s g i v e n by :
sn
(XI
,..., Xn)
=
c
E ( 0 )
X,(l)-Xo(n)
I
OCS,
where Sn i s t h e symmetric group on n elements and of
U.
O b v i o u s l y S2(x1,
~ ( 5 i )s
the signature
x2) = x x -x x y i e l d s a p o l y n o m i a l i d e n t i t y 1 2 2 1
s a t i s f i e d by a l l commutative r i n g s . However, n o t e v e r y P . I .
algebra
s a t i s f i e s a standard i d e n t i t y , b u t :
1.2.1. Theorem. (S. A m i t s u r )
I f R i s a P.I.
s a t i s f i e s t h e p o l y n o m i a l i d e n t i t y Sn(X l,...,Xn)n, N o t e t h a t R s a t i s f i e s t h u s an The s t r u c t u r e o f p r i m e
algebra over C then R f o r s u i t a b l e n and m.
i d e n t i t y w i t h c o e f f i c i e n t s +1 o r -1.
P. I - a l g e b r a s may now be d e s c r i b e d a c c u r a t e l y
i n the following :
1.2.2.
Theorem. ( E . P o s n e r ) . I f R i s a p r i m e P . I . a l g e b r a t h e n :
C. Structure Theory for Graded Rings of Type Z
232
1. R has a c l a s s i c a l r i n g o f f r a c t i o n s , Q(R) say. 2. Q ( R ) i s a c e n t r a l s i m p l e a l g e b r a 3. Q ( R ) i s a c e n t r a l e x t e n s i o n o f R, i . e . Q ( R ) = R Z Q ( R ) ( R ) .
4. Q(R) s a t i s f i e s a l l p o l y n o m i a l i d e n t i t i e s s a t i s f i e d by R. I n t h e above theorem one i n t u i t i v e l y hopes t h a t t h e
c e n t e r o f Q(R)
i s t h e f i e l d o f f r a c t i o n s o f t h e c e n t e r o f R. T h i s i s a c t u a l l y t h e case and i t f o l l o w s f r o m a r e s u l t o f E. Formanek on c e n t r a l p o l y n o m i a l s . F i r s t n o t e t h a t P o s n e r ' s theorem e n a b l e s us t o embed p r i m e P . I . a l g e b r a s ir. m a t r i x
r i n g s o v e r commutative f i e l d s , j u s t by s p l i t t i n g Q(R) ( s e e
Section A . I . l . ) .
This f a c t h i g h l i g h t s t h e importance o f t h e study o f
p o l y n o m i a l i d e n t i t i e s s a t i s f i e d by n x n m a t r i c e s o v e r a commutative, p r e f e r a b l e i n f i n i t e , f i e l d . A p o l y n o m i a l a ( X s ) i n C{Xs} i s s a i d t o be a c e n t r a l polynomial f o r n x n matrices w i t h c o e f f i c i e n t s i n C i f every e v a l u a t i o n o f g(Xs) i n some Mn(A), where A i s any commutative C-algebra y i e l d s c e n t r a l elements. B o t h E. Formanek and explicit
I.
Razmyslov c o n s t r u c t e d
c e n t r a l p o l y n o m i a l s (Razmyslov's p o l y n o m i a l i s even m u l t i l i n e a r )
A s a consequence one may deduce : 1.2.3.
Proposition.
L e t R be a s e m i s i m p l e P . I . a l g e b r a .
Then nonzero i d e a l s of R i n t e r s e c t t h e c e n t e r n o n - t r i v i a l l y . 1 . 2 . 4 . C o r o l l a r y . If R i s a p r i m e P . I . 1.2.5.
a l g e b r a then, Z ( Q ( R ) ) = Q ( Z ( R ) ) .
Convention. If C i s n o t i m p o r t a n t o r more p r e c i s e l y , i f we c o n s i d e r
Z - a l g e b r a s we w i l l u s u a l l y speak a b o u t
P.I.
r i n g s (not : algebras).
The Formanek c e n t e r F(R) o f a r i n g R s a t i s f y i n g t h e i d e n t i t i e s o f
n x n
m a t r i c e s i s t h e s u b r i n g o f Z(R) o b t a i n e d by e v a l u a t i n a a l l c e n t r a l p o l y n o m i a l s f o r R w i t h o u t c o n s t a n t t e r m . T h i s F may be viewed as a r i g h t semi e x a c t f u n c t o r and R i s an Azumaya a l q e b r a o f c o n s t a n t
rank
233
1. Non-Commutative Graded Rings
n2 o v e r
Z(R) i f and o n l y i f R F(R) = R h o l d s .
The p . i .
degree o f a p r i m e P . I . a l g e b r a R w i l l be
fl where
N = dim
z ( Q (R ) ) Q (R,
If P i s a p r i m e i d e a l o f R t h e n we d e f i n e p . i . deg P = p . i . deg R / P . 1 . 2 . 6 . Theorem (M. A r t i n ) . I f R s a t i s f i e s t h e i d e n t i t i e s o f n x n m a t r i c e s t h e n R i s an Azumaya a l g e b r a of c o n s t a n t r a n k n2 o v e r i t s c e n t e r i f and o n l y i f f o r e v e r y p r i m e i d e a l P o f R, p . i . deq P = n .
... o f I , ... .
For extensions, applications e t c . 3
above we r e f e r t o [
I , [ 90
t h e v e r y i n t e r e s t i n g theorem
L e t us now r e t u r n t o t h e s i t u a t i o n where R i s g r a d e d o f t y p e Z and c o n t i n u e w i t h n o t a t i o n s as i n 1.1.13.. 1.2.7.
I f R i s a p r i m e graded P . I . r i n g t h e n Qh = Qg i . e
Proposition.
Qg i s a g r - s i m p l e g r - A r t i n i a n ' r i n g o f f r a c t i o n s o f R .
Put C = Z(R).
Proof,
I f t h e r e e x i s t s a c F h(C) w i t h deg c > o t h e n
1. a p p l i e s , s i n c e c i s r e o u l a r . I f t h e r e i s a c c h(C) 2 o t h e n t h e graded r i n g o f f r a c t i o n s o f R a t I l , c , c , . . . I ,
Proposition I.1.4., w i t h deg c <
Q:
say, embeds i n Qg. So we have Qh
R + Q : +
+
Qg
+
r i n g homomorphisms :
Q.
C l e a r l y , Qg i s a g a i n a p r i m e graded G o l d i e r i n g and c C
i s i n t h e c e n t e r o f .:Q
Q:
A g a i n by P r o p o s i t i o n 1.1.4.,
-1
w i t h deg c - l > o
1, i t f o l l o w s t h a t
has a g r - s e m i s i m p l e g r - A r t i n i a n r i n g o f f r a c t i o n s w h i c h o b v i o u s l y
has t o c o i n c i d e w i t h Qh = Q g . So we a r e l e f t w i t h t h e p r o b l e m C = Co. BY t h e graded v e r s i o n o f Wedderburn's theorem ( s e e A . I . 5 . ) , Qg 2 Mn(A)(cJ
Posner's
f o r some graded d i v i s i o n r i n g , n c N and d
c
Zn.
By
theorem, Q i s f i n i t e d i m e n s i o n a l o v e r t h e f i e l d o f f r a c t i o n s
of C and t h u s i t f o l l o w s t h a t f o r e v e r y x c h(D) we have an e q u a t i o n c 0 xm
+
... +
Cm-l
x + cm = 0 w i t h ci
C, i = 0, ...,rn.
B u t C = Co i m p l i e s
C. Structure Theory for Graded Rings of Type Z
234
deg x
=
0 hence D
=
DO, and t h e l a t t e r e n t a i l s t h a t P i s a s k e w f i e l d ,
thus Q g = Q f o l l o w s . I n case D i s a n o n - t r i v i a l l y graded d i v i s i o n a l g e b r a and a l s o a P . I . r i n g then t h e r e s u l t s of A.I.4. e n t a i l t h a t D = Do [ X , X - l , q ~ ] where skewfield and
'p
i s an automorphism of
Do some power of which i s an inner
automorphism ( c f . a l s o 6 . Cauchon [ 17 ] ) . P u t m k = Z(Do)"
the f i x e d f i e l d of
k [T,T-l]
where T = X Xe,
9
in Z(Do).
some e
C
Do i s a
N, X
2
=
[Do
: Z(Do) ]
The c e n t e r Z ( D ) e q u a l s Since D i s an Ore domain
F .D :
i t has a skewfield of f r a c t i o n s Q ( D ) = DO(X,,?)
having k ( T ) f o r i t s c e n t e r .
Furthermore one e a s i l y v e r i f i e s t h a t : [ Do(X,*o):k(T)] = [ D : k [T,T-'Il
1.2.8.
=
Theorem. (Graded v e r s i o n of P o s n e r ' s theorem).>
A graded prime r i n g R i s a P . I . r i n g i f and only i f R i s a graded o r d e r
of M n ( K ) ( ~ ) f o r some g r - f i e l d K , and some Proof.
4
C
Zn, n
C l e a r l y , each (graded) o r d e r o f M n ( K ) ( < )
C
N.
i s a P . I . r i n g . Conversely,
i f R i s a prime graded P . I . r i n g then by P r o p o s i t i o n 1 . 2 . 7 . i t follows t h a t Q h = Qg i s a gr-simple g r - A r t i n i a n r i n g o f f r a c t i o n s of R . By t h e graded v e r s i o n of Wedderburns's theorem ( c f . A . I . 5 . ) ,
Qg = M n , ( D ) ( d ' )
f o r some n ' F N, d ' c Z n ' . I f D = Do then t h e theorem follows from Qg = Q ( R ) s i n c e the l a t t e r can be s p l i t by some t r i v i a l l y graded commutative f i e l d . Now assume D # Do i s a n o n - t r i v i a l l y graded d i v i s i o n a l g e b r a . The remarks preceding the theorem imply t h a t
Consider a maximal commutative graded s u b r i n g of D , L s a y . I t i s n o t hard t o v e r i f y t h a t L i s a g r - f i e l d and t h a t i s a l s o maximal a s a commutative subring of 0.
235
1. NonCornrnutative Graded Rings
W r i t e Q(L) = k ( T )
@
k [ T,T-'
1
L ; t h i s i s a commutative s u b r i n g o f Q ( D ) .
Ify c Q(D) commutes w i t h Q ( L ) t h e n f o r some s 6 k [ T,T-'],sy
c L because
L i s a maximal commutative s u b r i n g o f D. Consequently, Q ( L ) i s a
maximal commutative s u b f i e l d of Q f D ) , hence a s p l i t t i n g f i e l d . Since
Now, D
D
@
k [ J,J-l]
@
k [ T,T-l]
L i s a p r i m e r i n g , we g e t :
L i s a gr-c.s.
a. w i t h c e n t e r L hence i t i s i s o m o r p h i c
t o some M k ( A ) ( d ) f o r some graded d i v i s i o n a l a e b r a A, k c
N,
d
k c Z .
C e n t r a l l o c a l i z a t i o n a t t h e p r i m e i d e a l 0 y i e l d s = Mk(Q(A)) 2' Mn(Q(L)), s o i t f o l l o w s i m m e d i a t e l y t h a t k = n and Q(A) = Q ( L ) hence A i s commutative o r A = L. F i n a l l y we have e s t a b l i s h e d t h a t Mn(L) 2
D
60
k [ T,T-l]
L
becomes an isomorphism o f graded r i n g s i f t h e g r a d a t i o n chosen on Mn(L) i s d e s c r i b e d by d c Zn.
1.2.9.
Remark. P o s n e r ' s theorem i m p l i e s t h a t e v e r y (graded) P . I .
ring
can be embedded as an o r d e r i n a f u l l m a t r i x r i n g Mn(K) o v e r a commutative f i e l d K. There e x i s t s an i n f i n i t e number o f n o n - i s o m o r p h i c Z - g r a d a t i o n s on M,(K)
e x t e n d i n g t h e t r i v i a l g r a d a t i o n o f K. Unless R i s i t s e l f t r i v i a l l y
graded none o f t h e s e g r a d a t i o n s w i l l e x t e n d t h e q r a d a t i o n o f R. The above graded v e r s i o n o f P o s n e r ' s theorem may t h u s be t h o u g h t o f as a s t a t e m e n t a b o u t t h e e x t e n s i o n o f t h e a r a d a t i o n of R t o some " n i c e " s u b r i n g o f M,(K). 1.2.10. = P.i.
Proof. n 6
N
P r o p o s i t i o n . I f R i s a graded P . I . deq R/P
4
r i n g t h e n p . i . deq R/P =
f a r e v e r y P c Spec R.
T h a t p . i . deg R / P 5 p . i . deq R / P
i s o b v i o u s . Now i f f o r some 9 t h e s t a n d a r d i d e n t i t y Sen vanishes on R/P, t h e n t h e i d e a l g e n e r a t e d
C. Strucrure Theory for Graded Rings of Type Z
236
be t h e e v a l u a t i o n s o f S2n i n h(R) i s a graded i d e a l o f R because Szn i s m u l t i l i n e a r , thus i f t h i s i d e a l i s i n P then i t i s i n P a l s o vanishes on R/P p.i.
deg R/P
1.2.11.
9
5
p.i.
g
a'
T h e r e f o r e Szn
and t h e r e f o r e : deg R/P.
C o r o l l a r y . L e t R be a P . I .
r i n g satisfying the identities o f
n x n m a t r i c e s and l e t R be graded o f t y p e Z. The r a d i c a l r a d ( I n ) o f ( c f . [ 96 1 , f o r more d e t a i l s , In = i d e a l generated t h e c e n t r a l k e r n e l In by t h e e v a l u a t i o n s i n R o f a m u l t i l i n e a r c e n t r a l p o l y n o m i a l f o r n x n m a t r i c e s ) i s a graded i d e a l o f R. Proof.
Prime i d e a l s c o n t a i n i n g Ina r e e x a c t l y t h o s e h a v i n a p . i .
degree < n ( c f . [ 90
I).
Hence P
3
In i f and o n l y i f P g
3
In, follows
from the foregoing p r o p o s i t i o n . 1.2.12.
Corollary.
(Graded v e r s i o n o f
r i n g R i s an Azumaya a l g e b r a o f
M.
c o n s t a n t r a n k o v e r Z(R),
e v e r y graded p r i m e i d e a l has p . i .
i f and o n l y i f
degree equal t o p . i . deg R, and i f
and o n l y i f f o r e v e r y maximal i d e a l M o f p.i.
A r t i n ' s Theorem).A graded P . I .
R t h e p . i . degree o f
M
9
equals
deg R.
P r o o f . S t r a i g h t f o r w a r d f r o m 1.2.10.
and 1.2.6.
.
Note t h a t i n g e n e r a l graded maximal i d e a l s , maximal graded i d e a l s and t h e graded p a r t s Mg o f maximal i d e a l s M a r e
very
poorly
r e l a t e d . L e t us
g i v e a t r i v i a l example t o show t h a t C o r o l l a r y 1.2.12 does a l l o w t o economize a l o t
when c h e c k i n g p . i . d e g r e e ' s o f p r i m e i d e a l s i n o r d e r t o
v e r i f y w h e t h e r a graded P . I . r a n k o r n o t . Take R = D [ 0
X,,p]
r i n g R i s an Azumaya a l q e b r a o f c o n s t a n t ,fp
an automorphism o f t h e s k e w f i e l d Do;
t h e n R i s Azumaya if and o n l y i f p . i .
deg(X) = p . i . deg R and i t i s
known t h a t t h i s happens o n l y i f f o r some 1 c U ( D o ) ,
X X i s c e n t r a l i n R.
1. NonCommutative Graded Rings
1.2.13.
237
Proposition. If R i s a graded P . I . ring such t h a t i t s center i s
a g r - f i e l d then R i s an Azumaya algebra of constant rank. Proof. Let f be a m u l t i l i n e a r central polynomial f o r R . Clearly, f
cannot vanish f o r every homogeneous s u b s t i t u t i o n i . e . there
e x i s t u1 ,..., u k
F h ( R ) such t h a t 0 # f ( u l ,..., u k ) 6
h ( C ) where C = Z ( R ) .
Since homogeneous elements o f C are i n v e r t i b l e by hypothesis, i t follows
t h a t f ( u l , ..., u k ) of rank n 1.2.14.
2
.
Corollary. Let G a f i n i t e abe i a n group of automorphisms of the
f i n i t e dimensional skewfield Do. The terated twisted polynomial ring -1 Do [ Xo,Xo , IS C G I i s an Azymaya algebra. Proof.
A r a t h e r straightforward
modification of the techniques used
in foregoing r e s u l t s . Let us conclude t h i s section
with a r e s u l t concernina s p l i t t i n q f i e l d s
and s p l i t t i n g g r - f i e l d s , f i r s t an example : 1.2.15. Example. Consider D = C[ X,X-’,q] where ‘p i s complex conjugation. 2 -2 2 -2 Then Z ( D ) = 9 X , X ] and [ D:Z(D)] = 4 . Clearly L = E [ X , X ]and K=R[ X,X-’1 M2(L)(d) f o r c e r t a i n are s p l i t t i n g f i e l d s for D in the sense t h a t D rn L 2 Z(D) d C Z determined by the n a t u r a l gradation of the tensor product ( s i m i l a r
2
f o r K ) . Also, t(X ) andR(X) a r e s p l i t t i n g f i e l d s of C(X,cp) and both a r e 2
graded r e a l i z a b l e f i e l d s ( i n the sense of B . I . 3 . ) over R(X ) . On the other handR(Xti) i s a maximal commutative subfields of t ( X , ( p ) which i s n o t graded r e a l i z a b l e . 1.2.16. Corollary.
Let D be a graded division P . I . algebra then Q(D)
has graded r e a l i z a b l e s p l i t t i n g f i e l d s . Proof.
Write D = D o [ X , X - l , q ] ,
Q(0) = D o ( X , , ~ ) . I t i s e a s i l y checked
t h a t a maximal graded commutative subring of D , L say, y i e l d s a r e a l i s a b l e S p l i t t i n g f i e l d Q ( L ) of Q ( D ) .
238
C. Structure Theory for Graded Rings of Type Z
1.3. F u l l y Bounded Graded Rings.
Throughout t h i s s e c t i o n R i s a graded r i n g o f t y o e Z which i s suDposed t o be l e f t N o e t h e r i a n . I n t h i s and t h e f o l l o w i n g s e c t i o n we i n t r o d u c e c l a s s e s o f graded r i n g s g e n e r a l i z i n g i n d i f f e r e n t d i r e c t i o n s t h e c l a s s o f graded P . I .
. We use r e s u l t s and
r i n g s , s t u d i e d i n S e c t i o n 1.2.
r e s u l t s and n o t a t i o n s from A . I I . 7 . ,
A.II.9.,
A.II.ll
.,...
L e t Zg(R) be t h e s e t o f e q u i v a l e n c e c l a s s e s uo t o s h i f t s and isomorDhisms
of t h e indecomposable i n j e c t i v e s i n R-gr. L e t +g : Zg(R)
E
t h e map d e f i n e d by a s s o c i a t i n g t o a c l a s s Ass(E) 1.3.1.
,
where E r e p r e s e n t s
Spec ( R ) be g
t h e graded p r i m e i d e a l
5.
Lemma.
I f P t Spec ( R ) t h e n Eg(R/P) 9 t h e map ag i s s u r j e c t i v e . Proof.
+
i s P-cotertiary.
I n particular,
Decompose Eg(R/P) as a d i r e c t sum o f indecomposable i n j e c t i v e s i n
R-gr, say Eg(R/P) = El @...@
En.
From P r o o o s i t i o n 1.1.8.
Eg(R/P) i s g r - s i m o l e l e f t g r - A r t i n i a n ,
hence E4(R/P) =
we r a t a i n t h a t
L1 Q...QLr
where
t h e Li a r e m i n i m a l graded l e f t i d e a l s o f Eg(R/P) such t h a t f o r each i, j t 11, ...,r ) t h e r e i s an nii,j
c
Z such t h a t Li
5
Krull-Remak-Schmidt theorem we o b t a i n : r = n and Ei
L . ( n . . ) . By t h e J 1J = Ln(i), f o r some
p e r m u t a t i o n n o f { l ,..., n } . T h e r e f o r e we have : Ass(Ei)
= Ass(L
u ( i ) ) = ASS(Lv(j)(nv(i),v(j) T h i s y i e l d s t h a t Eg(R/P) i s P - c o t e r t i a r y .
) ) = Ass(L
u(j)
= Ass(Ej)
R e c a l l t h a t a p r i m e r i n g i s bounded i f e v e r y e s s e n t i a l l e f t i d e a l c o n t a i n s a nonzero i d e a l . A r i n g i s f u l l y bounded i f each o f i t s p r i m e f a c t o r r i n g s i s bounded. A p r i m e graded r i n g R i s s a i d t o be graded bounded i f each gr-essential
l e f t i d e a l c o n t a i n s a nonzero graded i d e a l . A graded r i n g
R i s g r a d e d f u l l y bounded i f e v e r y
R/P,
P F Spec (R), i s graded bounded. g
239
1. Non-Commutative Graded Rings
1.3.2.
Theorem.
L e t R be a l e f t N o e t h e r i a n g r a d e d r i n q .
The f o l l o w i n g p r o p e r t i e s a r e e q u i v a l e n t : 1.
c
Spec ( R ) , R/P i s graded bounded.
9
3. Each c o t e r t i a r y g r a d e d R-module M i s i s o t y p i c i n R - g r i . e . Eg(M) i s a d i r e c t sum o f e q u i v a l e n t indecomposable i n j e c t i v e s i n R - g r .
Proof. 1
3
3 If M
c
R-gr i s c o t e r t i a r y and E4(M) =
Q
i
Ei where t h e Ei
indecomposable i n j e c t i v e i n R-gr t h e n Ass(M) = Ass(Ei) Ei
are
implies t h a t a l l
a r e e q u i v a l e n t ( u p t o s h i f t and isomorphism) by 1.
3 = 1. If E , E ' a r e indecomposable i n j e c t i v e s i n R-gr such t h a t P = =
Ass(E) = A s s ( E ' ) t h e q E Q E;' i s P - c o t e r t i a r y and by 3. t h e n E and E '
are equivalent. 1
2 . F i r s t n o t e t h a t i f +g i s b i j e c t i v e f o r R t h e n t h e
map +g(R) i s b i j e c t i v e f o r each g r a d e d e p i m o r p h i c image we may assume t h a t R i s p r i m e
corresponding o f R . Hence
and P = 0. I f t h e i m p l i c a t i o n 1 = 2
f a i l s , l e t L be a g r - e s s e n t i a l l e f t i d e a l o f R the property o f not containing a
maximal w i t h r e s p e c t t o
nonzero graded i d e a l . O b v i o u s l y L has
t o be g r - i r r e d u c i b l e and t h e r e f o r e Ass(R/L) c o n s i s t s o f a s i n g l e e l e m e n t w h i c h c l e a r l y has t o be t h e z e r o i d e a l . I t w i l l now s u f f i c e t o c o n s t r u c t an indecomposable i n j e c t i v e E d i f f e r e n t f r o m Eq(R/L) i n R-gr, Ass(E) = {O}. I f we decompose Eg(R) as El Q...@ En, where Ei indecomposable i n j e c t i v e i n R-gr f o r each i = 1, t h a t Ass(Ei)
= 0
. . . ,n,
f o r each i . S i n c e R i s n o n - s i n g u l a r
such t h a t is
then i t i s c l e a r no e l e m e n t o f Eq(R)
240
C. Structure Theory for Graded Rings of Type Z
i s annihilated by L whereas E g ( R / L )
c e r t a i n l y contains such nonzero
elements. So, putting E = El, E and E g ( R / L )
cannot be equivalent.
2 = 1.By TheoremA.II.7.3. we obtain t h a t P = A s s ( E ) , where E i s a n
indecomposable i n j e c t i v e i n R-gr, i s equal t o annR(Rx) f o r some x # 0 in h ( E ) . I f we p u t L
=
a n n R x then L i s a gr-irreducible araded l e f t
ideal containing P . Now i f L/P i s a gr-essential l e f t ideal of R/P then L / P contains a graded ideal J/P # 0 of R / P . However J 2P and JRx c Lx = 0 contradicts a n n R ( R x ) = P , therefore L/P i s n o t g r - e s s e n t i a l . The l a t t e r leads t o the existence of a graded l e f t ideal K
2
P such
t h a t P = L n K. Writing K as an i n t e r s e c t i o n of gr-irreducible graded l e f t ideals K = K1 E
n... n
Kr, we obtain :
Eg(Rx) 2 E g ( ( R / L ) ( n ) )
=
8 . . . @E q ( R / K r )
c Eg(R/K1)
=
E9(R/P).
Consequently, we have a n i n j e c t i v e graded morphism of degree zero :
E(n)
--f
E g ( R / P ) f o r some of n f Z . From Lemma 1.3.1. i t t h u s follows
t h a t E i s equivalent t o one of the indecomposable i n j e c t i v e s in the decomposition of E g ( R / P )
.
A theorem of G. Cauchon s t a t e s t h a t a l e f t Noetherian ring i s f u l l y
l e f t bounded i f a n d only i f i t s a t i s f i e s P . G a b r i e l ' s condition H . W e now proceed t o 1.3.3. Lemma.
prove the graded version of t h i s r e s u l t .
For a l e f t Noetherian graded ring R the followinq statements
are equivalent :
1. I f L i s a graded l e f t ideal of R then there e x i s t s a f i n i t e s e t 0
I Xl , . . . , ~ n }c h ( R ) such t h a t L
=
(L:xl)
n...n
0
( L : x n ) , where L denotes
the l a r g e s t graded ideal of R contained in L.
2:For
every f i n i t e l y generated M c R-gr there e x i s t s a f i n i t e s e t
I m l , . . . , m r j c h ( M ) such t h a t :
annR(M) = a n n R( m 1 ) n...nann R( m r ) .
24 1
1. NonCommutative Graded Rings
Proof. 0
1 - 2. L e t gl, L = {xl
S
n annR(gi). i=l
,..., x n }
F h(M) g e n e r a t e
...,g,
=
By 1 we have 0
n.
c h ( R ) . Thus L =
i 3J
M, t h e n :annR(M) = L where
On ( fls annR(gi):x.) j=1 i = l J
f o r some
annR ( x . 9 . ) and t h e a s s e r t i o n f o l l o w s . J '
0
0
2 = 1. S i n c e ann (R/L) = L i t f o l l o w s f r o m 2 t h a t L = annR(x,) R f o r some
xi
where xi
represents
F h(R/L).
Now we have
xi,
t h a t annR(xi)
= (L:xi)
f l . . . flannR(xn)
f o r a l l i,
and so 1. f o l l o w s .
The ungraded v e r s i o n o f t h e e q u i v a l e n t c o n d i t i o n s i n t h e above lemma y i e l d s P. G a b r i e l ' s c o n d i t i o n ( H ) . I f a graded l e f t N o e t h e r i a n r i n g R s a t i s f i e s c o n d i t i o n (H) t h e n i t a l s o s a t i s f i e s t h e c o n d i t i o n s mentioned w h i c h we w i l l r e f e r t o as c o n d i t i o n ( H ) g .
i n Lemma 1 . 3 . 3 . , 1.3.4.
L e t R be a g r a d e d l e f t N o e t h e r i a n r i n g s a t i s f y i n g
Proposition.
c o n d i t i o n ( H ) g t h e n R i s graded f u l l y bounded. Proof.
Let
II.7.3.,
E
P = annR(Rm) f o r some m c h ( E ) . I f (H)'
P = ann (Rm) = annR(xl) R
R/P
R X ~ ( V a. ~ ). . Q
--f
Eg(R/P) i C
--t
Ass(E). By Theorem A.
r e p r e s e n t a c l a s s o f Eg(R) w i t h P =
Qn
i=l
{I, ...,n } .
Ei,
n. ..n RX,(V~)
where Ei
i s s a t i s f i e d then
ann ( x ) and i t f o l l o w s t h a t R n f o r some (vl
= Eg(Rxi)(vi)
,...,Vn)
F
zn.
Consequently :
i s equivalent t o
E
f o r each
F o r e g o i n g r e s u l t s y i e l d t h a t P d e t e r m i n e s E up t o
s h i f t s and isomorphisms i . e . +g i s b i j e c t i v e o r R i s graded
fully left
bounded. Note t h a t a
l e f t N o e t h e r i a n graded f u l l y bounded r i n g need n o t be f u l l y
bounded. F o r example, c o n s i d e r R = A [ X,,D]
where 'D i s an automorphism
of t h e s k e w f i e l d A , X a v a r i a b l e and m u l t i p l i c a t i o n g i v e n by Xa = a"X. In A [
X,,p]
e v e r y graded l e f t i d e a l i s t w o - s i d e d b u t i t i s f u l l y l e f t
bounded i f and o n l y i f some power o f 'D i s i n n e r i n A .
242
C. Structure Theory for Graded Rings of Type Z
1.3.5.
Remarks.
1. I f R i s a p o s i t i v e l y graded f u l l y l e f t bounded l e f t N o e t h e r i a n r i n g , t h e n Ro i s l e f t N o e t h e r i a n f u l l y l e f t bounded. 2. I f R i s s t r o n g l y graded t h e n i t i s graded f u l l y l e f t bounded l e f t N o e t h e r i a n i f and o n l y i f Ro i s f u l l y bounded l e f t N o e t h e r i a n .
3. A l e f t N o e t h e r i a n graded r i n g R i s l e f t g r - A r t i n i a n i f and o n l y i f R i s graded f u l l y bounded and a l l graded p r i m e i d e a l s a r e gr-maximal.
I f R i s p o s i t i v e l y graded t h e n R i s l e f t g r - A r t i n i a n i f and o n l y i f Ro i s l e f t A r t i n i a n and R,
i s nilpotent.
Now we t u r n t o a t o r s i o n t h e o r e t i c d e s c r i p t i o n o f graded f u l l y l e f t bounded r i n g s and i t s a p p l i c a t i o n s . 1.3.6.
Lemma.
L e t R be a l e f t N o e t h e r i a n qraded r i n g and l e t
K
be a
r i g i d k e r n e l f u n c t o r on R-qr. L e t C 9 ( ~ )be t h e s e t o f graded l e f t i d e a l s o f R w h i c h a r e maximal amongst graded l e f t i d e a l s o f R n o t i n - P ( K ) . Consider Then
=
K
= inf{KKII,I
K~
cC~(K)}.
1'
K~
P r o o f . I f L 6 C g ( ~ )t h e n K ( R / L ) = 0 t h u s Conversely i f
11.9.7.).
graded l e f t i d e a l
K~
2
K
K
i s such t h a t
and
5 K~ K ;
#
K
5
K~
r
(see a l s o A.
then there e x i s t s a
K
J o f R, J c L ( K ~ -) L ( K ) . S i n c e R i s l e f t N o e t h e r i a n
t h e r e i s a J1 C C g ( ~ )such t h a t J c J1. Now : K ~ ( R / J ~= ) R/J1 i m p l i e s that
2
K~
1.3.7.
K
~
1
Theorem.
hence / ~
K~
3 K:.
It follows t h a t
K
=
K;.
A g r a d e d l e f t N o e t h e r i a n r i n a R i s qraded f u l l y l e f t
bounded i f and o n l y if e v e r y r i g i d k e r n e l f u n c t o r on R-gr i s symmetric. Suppose R i s graded f u l l y l e f t bounded and l e t
Proof.
k e r n e l f u n c t o r on R-gr. By t h e lemma i t that
K
r
~
K
be
d
rigid
w i l l suffice t o establish
i s/ symmetric, ~ f o r each L c C ~ ( K ) .Now L
i s gr-irreducible
243
1. Non-Commutative Graded Rings
by d e f i n i t i o n , hence b i j e c t i v i t y o f ag e n t a i l s t h a t
R / I - “;/P
K~
for
some P C Spec ( R ) . By P r o p o s i t i o n A.II.11.6.(2.) and i n v i e w o f 9 Theorem 1.3.2. i t i s c l e a r t h a t graded f u l l y boundedness o f R i m p l i e s that
K;/~
i s symmetric f o r e v e r y P c Spec,(R).
Conversely, suppose now t h a t I f L C P(K;)
then
L
K~
R/ P
=
K;
i s symmetric f o r e v e r y P
c
Spec ( R ) . 9
mod P i s an e s s e n t i a l graded l e f t i d e a l o f R / P and
e v e r y graded l e f t i d e a l o f R w i t h t h e l a t t e r p r o p e r t y i s i n L ( K ; ) . The symmetry o f
t h e r e f o r e i m p l i e s t h a t each graded e s s e n t i a l l e f t i d e a l
K;
I o f R/P c o n t a i n s a graded i d e a l o f R/P which i s nonzero i . e . each Spec (R) i s graded bounded. Then Theorem 1.3.2. e n t a i l s t h a t 9 R i s graded f u l l y l e f t bounded. CI R/P,
P
1.3.8. C o r o l l a r y .
L e t R be a l e f t N o e t h e r i a n p o s i t i v e l y graded r i n g ,
w h i c h i s graded f u l l y l e f t . bounded, t h e n f o r e v e r y P c Spec (R) we 9 have : K; = K ~ and K~h ( p ) = K ~ ( ~ ( -n o~t a )t i o n o f A.II.9., and 1.1.). __ Proof.
A s t r a i g h t f o r w a r d consequence o f t h e above, combined w i t h
Thsorem I . 1.10.
1.3.9. P r o p o s i t i o n . L e t R be a l e f t N o e t h e r i a n graded f u l l y bounded r i n g , then t h e f o l l o w i n g p r o p e r t i e s h o l d : 1. If
M c R - g r i s K - t o r s i o n f o r some r i g i d k e r n e l f u n c t o r on R-gr, t h e n
Ass(M) c
L(K).
2. Every graded f i l t e r t h e form L = Po,...
,Pn
c
{L,
P(K)
L ( K ) o f a r i g i d kernel functor
graded l e f t i d e a l o f R
n
K
on R-gr has
such t h a t t h e r e e x i s t
Spec (R) such t h a t Pn.Pn-l...Po 9
c
L}.
Proof. 1. Easy. 2. S i n c e Pn.Pn-l...
Po
c
L ( K ) , L ( K ) J L i s obvious.
F o r t h e converse, f i r s t
because ) R is n o t e t h a t we may r e s t r i c t t o f i n i t e l y g e n e r a t e d L C l ( ~
244
C. Structure Theory for Graded Rings of Type Z
l e f t N o e t h e r i a n . Now f o r any f i n i t e l y g e n e r a t e d M F R-gr t h e r e e x i s t s a chain 0 = M
0
c M
1
c...cMn
graded submodule o f Mi+l
= M i n R-gr such t h a t each Mi
and Mi+l/Mi
i s a tertiary
i s a n n i h i l a t e d by i t s a s s o c i a t e d
graded p r i m e i d e a l ( t o check t h i s s t a t e m e n t one may f o l l o w t h e l i n e s o f t h e p r o o f o f t h e unqraded a n a l o g e ) . So i f we s t a r t w i t h some f i n i t e l y g e n e r a t e d graded l e f t i d e a l L i n L ( K ) , we f i n d a sequence o f graded l e f t i d e a l s o f R,Lo = L c L1 c . . . c L n = R such t h a t Lr/Lr-l a n n i h i l a t e d by a graded p r i m e i d e a l Pr i . e . Lr/Lr-l
L
S i n c e each
Po...Px.
3
is
i s an e p i m o r p h i c image o f a graded submodule o f R/L,
f o l l o w s that(KLr/Lr-l) t h e n t h a t Pr-l 1.3.10.
f o r r = 1, ...,n+1. From 1 i t f o l l o w s
= Lr/Lr-l
F L ( K ) f o r r = 1,
Proposition.
it
...
n
+
1.
L e t R be a l e f t N o e t h e r i a n
a symmetric r i g i d k e r n e l f u n c t o r
qraded r i n q . Consider
on R-gr h a v i n g p r o p e r t y
K
(T) and
such t h a t i d e a l s I o f R e x t e n d t o i d e a l s Q;(R)
j , ( I ) o f Q:
R i s graded f u l l y bounded i f and o n l y i f Q:(R)
i s araded f u l l y
(R). Then
bounded.
Proof. 1.3.11.
S t r a i g h t f o r w a r d , n o t e t h a t Q:(R) _
= Q,(R) _ -
by C o r o l l a r y A . I I . 1 0 . 1 2 .
P r o p o s i t i o n . A graded r i n g i s l e f t N o e t h e r i a n graded f u l l y
bounded i f and o n l y i f Mn(R)(4) i s l e f t N o e t h e r i a n graded f u l l y bounded f o r some (hence a l l ) n
Proof.
A.I.5.
c N,
d
6 Zn.
The g r a d a t i o n on Mn(R)(d) - d e s c r i b e d by d E Zn
. That
E i s l e f t N o e t h e r i a n i f and o n l y i f t h e
Mn(R) i s l e f t N o e t h e r i a n i s w e l l known. Assume t h a t Mn f u l l y bounded f o r some n F
N, 4
s as i n S e c t i o n ungraded) r i n g
R)(d) i s
graded
and c o n s i d e r P F Spec (R) and 9 a g r - e s s e n t i a l graded l e f t i d e a l L o f R/P. O b v i o u s l y , M n ( L ) ( ~ )i s a F Zn,
g r - e s s e n t i a l graded l e f t i d e a l o f Mn ( R / P ) ( d ) and t h e r e f o r e i t c o n t a i n s a nonzero graded i d e a l M n ( I ) ( G ) of Mn(R/P)(d). Thus L 31 # 0,
245
1. Non-Cornrnutative Graded Rings
where I i s a graded i d e a l o f R/P. Conversely, assume t h a t R i s graded f u l l y bounded. A graded p r i m e i d e a l Q o f Mn(R)(d) i s n e c e s s a r i l y o f t h e form M,(P)(?)
f o r some graded p r i m e i d e a l P o f R. S i n c e Mn(R)(<)/Mn(P)(4)
'Z Mn(R/P)@)
we o n l y have t o show t h a t Mn(R/P)(_d_) i s bounded. I f we a p p l y
P r o p o s i t i o n 1.1.6.
t o R/P we o b t a i n t h a t Qg = Eq(R/P) i s o b t a i n e d b y
l o c a l i z i n g R/P a t t h e r i g i d k e r n e l f u n c t o r
K
o n R/P-gr where
has graded
K
f i l t e r J ( K ) g e n e r a t e d b y t h e g r - e s s e n t i a l graded l e f t i d e a l s o f R/P. Now
i s symmetric b y Theorem 1.3.7.
K
( c f . 1.1.6.).
I t i s e a s i l y seen t h a t Mn(Qq)(d) = Mn(R/P)(d)
Q:(Mn(R/P)(d)
=
i s a l s o o b t a i n e d by l o c a l i z i n q
Mn(R/P)(cJ)-gr
where
K '
However Mn(Qg)
'
(4)
K
was symmetric. So i f L ' i s a g r - e s s e n t i a l
6 jK, ( L ' ) f o r some graded i d e a l
jKI
S
t h a t J ' = Mn(J)($ Mn(R/P)($)
t h e n Q;,(S)
jK ( L ' ) = Q:,(S)
f o r some qraded i d e a l J o f
K
entails
i s graded bounded.
commutes w i t h R, i s c o n s i d e r e d as a graded r i n g o f t y p e Z
1.3.12.
'
R/P and J ' cL'. Hence
L e t R be an a r b i t r a r y graded r i n g , t h e p o l y n o m i a l r i n q R [ T I
RITln=
and t h u s
J ' F J ( K ' ) . Since S i n prime,
(S) i s i n j e c t i v e ; t h e r e f o r e t h e d e f i n i t i o n o f
+
in
i s a r - s i m p l e l e d t q r - A r t i n i a n hence graded bounded and
graded l e f t i d e a l of S = Mn(R/P)@) J I . 1
I
H F J(K).
c l e a r l y symmetric s i n c e
JS
Mn(R/P)(A)
Qg =
60
R/P at K
i s t h e r i g i d k e r n e l f u n c t o r w i t h graded f i l t e r
1 : ( ~' ) g i v e n by t h e Mn(H)(<),
K
and Q g i s q r - s i m p l e l e f t g r - A r t i n i a n
Z
i+j=n
, where
via :
RiTJ.
Theorem. L e t R be a graded r i n g . The f o l l o w i n g s t a t e m e n t s a r e
equivalent : 1. R i s f u l l y l e f t bounded 2 . R [ T I i s graded f u l l y l e f t bounded.
T
246
C. Structure Theory for Graded Rings o f Type Z
Proof.
P then
( T ) where p F Spec (R), P = R n R. On t h e o t h e r hand , i f T 1 P 4 ( see A . I I . 8 . ) i s a p r o p e r p r i m e i d e a l o f R . Moreover we may
+
P = p
c
I f P i s a graded p r i m e i d e a l o f R [ T ] such t h a t T
t h e n P,
assume t h e r i n g s R and R [ T I t o be l e f t N o e t h e r i a n and p r i m e and i t w i l l s u f f i c e t o e s t a b l i s h t h e boundedness c o n d i t i o n . 1 - 2. C o n s i d e r P C Spec ( R [ T I ) . I f T C P t h e n R [ T ] / P = R/P where 9 p = P n R. A graded e s s e n t i a l l e f t i d e a l L o f R [ T I /P w i l l t h e r e f o r e c o n t a i n an i d e a l I o f R [ T I /P.
S i n c e L i s graded i t w i l l a l s o c o n t a i n
t h e graded i d e a l o f R [ T I / P g e n e r a t e d by t h e homoqeneous p a r t s o f elements o f I, i . e . R [ T ] / P i s graded bounded. I n t h e o t h e r T f P, L,
i s an e s s e n t i a l l e f t i d e a l o f R/P,
as i s e a s i l y checked.
P u t J = 11 C R [ T I /P,
By 1. L* c o n t a i n s an i d e a l I o f R/P,.
situation
1,
C I}.
t h e p r o p e r t i e s o f ( - ) * and ( - ) * y i e l d - t h a t J i s an i d e a l af R [ T I / P such t h a t J, (L,)*
i . e . J c (J,)*
L,
c
c (L,)*.
Since R [ T I i s l e f t Noetherian.
i s f i n i t e l y g e n e r a t e d hence t h e r e e x i s t s an n
cN
c L. So Tn J c L, b u t TnJ i s an i d e a l o f R [ T I
Tn(L,)*
such t h a t
,
therefore
R [ TI, i s qraded bounded. 2
3
1. Take
that T
p C Spec(R),
p*.
Let
L
p be a
2
i d e a l o f R/p. Thus L* R [ T I then N
J
* #
2 p*
p implies L
S i n c e , f o r some n c f o l l o w s t h a t L* Q N
N 2
#
w h i c h i s graded and J R[T]
t h e n p* i s a qraded p r i m e i d e a l o f R [ T I such
4
and i f N
n N,
2
p
2 p* or
p*.
By 2.,
Q pz
L*
By 2.,
i s a qraded l e f t i d e a l o f
(L n
we have : Tn(L* fl (N,)*)
N),* c L
*f
p” .
fl N w i t h T 1 p*.
has t o c o n t a i n a n i d e a l J o f R I T ]
L*
has t o c o n t a i n an i d e a l J o f
w h i c h i s graded and J 4 p*.
Finally this implies that L
J,
l e f t i d e a l o f R mapping t o an e s s e n t i a l l e f t
)
J, where J, i s a n i d e a l o f R such t h a t
p and t h u s R/p i s bounded.
it
247
1. Non-Commutative Graded Rings
1.3.13.
Proposition.
Let R be a p o s i t i v e l y graded l e f t Noetherian qraded
f u l l y l e f t bounded r i n g . The following
properties a r e equivalent :
1. R i s f u l l y l e f t bounded
2. For every P c
Proj(R), Qg(R/P)
=
Mn(a)(4)where
graded skewfield such t h a t e i t h e r A 'Q
i s an automorphism of A.
= A.
or A
$ c Zn and
A is a
[ X,X-',,D]
where
= A.
such t h a t ,oe i s i n n e r f o r
and where A, s a t i s f i e s c o n d i t i o n Co ;Co:
some e
c N,
For every m c N, Mm(ao)i s
a l g e b r a i c over i t s c e n t e r . Proof.
S t r a i g h t f o r w a r d , usinq the f a c t t h a t A. [ X,,D]
bounded i f and only i f Co (a
toe
i s i n n e r f o r some e
r e s u l t o f G. Cauchon).
F
M and
i s fully left A.
satisfies
C. Structure Theory for Graded Rings of Type Z
248
1.4. Birational Graded Algebras. Birational extensions of r i n g s , i n p a r t i c u l a r Zariski central rings, have been introduced and studied in [
113 and I 114 I ,
[ 115 I
.
We w i l l r e c a l l b r i e f l y some of the d e f i n i t i o n s and basic properties and afterwards specify t o the Z-graded case. I n view of the conventions introduced in Section A.II.9. we c a l l pre-kernel functor exact subfuvctor
K
,
any l e f t
of the i d e n t i t y on R-mod ( i t d i f f e r s from a kernel
functor because i t may be non-idempotent). To f i l t e r of l e f t i d e a l s of R , Z ( K ) say, and
K
K
there corresponds a
i s a kernel functor exactly
then when L ( K ) i s an idempotent f i l t e r i . e . i f L i s a s u b - l e f t ideal of I , I c L ( K ) , such t h a t I / L i s torsion then L 6 L ( K ) . A pre-kernel functor i s b i l a t e r a l i f Z ( K ) allows a cofinal s e t of i d e a l s . Consequently, a b i l a t e r a l kernel functor i s j u s t a symmetric kernel functor i n t h e Sense of A . I I . 9 .
I f R i s l e f t Noetherian then a b i l a t e r a l pre-kernel functor
i s a symmetric kernel functor i f and only i f Z ( K ) i s m u l t i p l i c a t i v e l y
closed. I n general a pre-kernel functor
K
on R-mod i s said t o be radical
i f a n d only i f i t i s b i l a t e r a l and r a d ( J ) c E ( K ) implies J any ideal J of R . Again,
L ( K ) for
i t i s e a s i l y checked t h a t , i f R i s a l e f t
Noetherian r i n g , a pre-kernel functor i s radical exactly then when i t i s a symmetric kernel functor. I n the absence of the Noetherian condition however, r a d i c a l i t y does not imply symmetry and vice versa. I n the l e f t Noetherian case a symmetric kernel functor
K
i s completely determined
by the s e t { P c Spec(R), P ,t L ( K ) and P i s maximal with respect t o t h i s property} . If R i s n o t l e f t Noetherian a s i m i l a r statement f a i l s , moreover, the l a t t i c e of
supremum of
b i l a t e r a l pre-kernel functors ( i n the
pre-kernel functors on R-mod) need n o t be b i l a t e r a l . We
aim t o avoid these problems by r e s t r i c t i n q a t t e n t i o n t o radical prekernel functors.
Here a r e the two basic examples of radical pre-kernel
249
1. NonCornrnutative Graded Rings
f u n c t o r s we w i l l use i n t h e sequel : 1" C o n s i d e r
K
~
I an i d e a l o f
,
such t h a t r a d ( J ) 2" C o n s i d e r
3
R, g i v e n by f ( ~ , )
= {L l e f t ideal o f
I}.
P i s a p r i m e i d e a l o f R, g i v e n by t h e f i l t e r
K ~ - where ~ ,
l ( ~ = I L l e~ f t idea l o f~ R , L c) o n t a i n s an i d e a l
J
R
:
J o f R such t h a t
4 PI.
I n case
R i s a graded r i n g o f t y p e Z ( o r any G) a l l d e f i n i t i o n s g i v e n
above have n a t u r a l graded e q u i v a l e n t s . We l e a v e t o t h e r e a d e r t h e t a s k o f f i n d i n g f o r m u l a t i o n s o f t h e s e concepts i n terms o f r i g i d k e r n e l f u n c t o r s on R-gr, e t c ... Here we w i l l c o n s i d e r b i r a t i o n a l e x t e n s i o n s o f
commutative r i n q s o n l y ,
f o r a s l i g h t l y more g e n e r a l s e t up c f . [ 1 1 5 ] . L e t C be a commutative r i n g and l e t A be a C-algebra.; 1.4.1.
D e f i n i t i o n . We say t h a t A i s a b i r a t i o n a l C-algebra i f t h e r e
e x i s t non-empty open s e t s
U
c Spec(A), V c %ec(C)
such t h a t t h e
following conditions hold :
i. I f P
c
Spec(A) i s such t h a t
2. The correspondence P
--f
P
f-l(P),
fl C c V t h e n
where f : C
d e f i n e s a t o p o l o g i c a l homomorphism
U
--f
P < U. A defines t h e algebra s t r u c t u r e
V.
We say t h a t A i s a g l o b a l l y b i r a t i o n a l C-algebra i f U = Spec(A). Note t h a t any semiprime P . I .
a l g e b r a A i s a b i r a t i o n a l Z ( A ) - a l g e b r a where
may be t a k e n t o be t h e open s e t c o n s i s t i n q o f p r i m e i d e a l s o f A o f maximal p . i .
degree, ( s e e 1.2. f o r more d e t a i l ) .
Of course, Azumaya a l g e b r a s p r e s e n t a n i c e c l a s s of examples o f
g l o b a l l y b i r a t i o n a l alqebras. L e t us f i x n o t a t i o n s as f o l l o w s : Y = Spec(A), X = Spec(C), f o r some i d e a l I o f A, V = X ( 1 ' )
U
=
Y(1)
f o r some i d e a l I ' o f C . We summarize
Some e l e m e n t a r y p r o p e r t i e s i n t h e f o l l o w i n q :
U
C. Structure Theory for Graded Rings of Type Z
250
1.4.2. Lemma.
Let A be a birational C-alqebra.
1. We have U = Y(A1') i . e . rad(1) = rad(A1'). 2 . For any ideal J of A there e x i s t s an ideal J ' of C such t h a t Y ( I J ) = =
Y(1) n Y(J) = X ( 1 ' ) = X ( 1 ' J ' ) and thus rad(1J) = r a d ( A 1 ' J ' ) =
=
rad(A1')
n rad(AJ')
= rad(1)
n rad(AJ')
=
rad(1J').
3. I f H c I i s an ideal of A then there e x i s t s a n ideal H ' such t h a t rad(H)
=
c
I ' of C
rad(BH'); Y ( H ) = X ( H ' ) .
4. I f A i s a globally birational C-algebra then AI'
=
A.
Note t h a t the homeomorphism Y(1) = X(1') i s in f a c t given by
Y ( H ) = X ( ( H n C ) I ' ) f o r a l l H c I ; t h i s i s due t o the commutativity of C here, in general one has t o distinguish between birational extensions a n d Zariski extensions ( t h e l a t t e r s a t i s f y i n q the forementioned property).
The following theorem enables us t o l o c a l i z e a t c e r t a i n radical prekernel functors even in the absence of the Noetherian condition. 1.4.3. Theorem.Let A be a birational C-algebra, and l e t pre-kernel functor on A-mod such t h a t I
c
K
be a radical
P ( K ) . Let E ( K ' ) be the f i l t e r
generated by the s e t { H ' , H ' an ideal of C such t h a t X ( H ' ) = Y ( H n I ) f o r some ideal H of B , H c . E ( K ) } . I f
K'
i s idempotent then
K
is
idempotent. 1.4.4. Corollary. I f C i s Noetherian and
A-mod such t h a t I c P ( K ) t h e n
K
i s a ore-kernel functor on
K
will be symmetric i f i t i s r a d i c a l .
Let us now focus on special pre-kernel functors. 1.4.5. Proposition.
Let A be a birational C-alaebra.
1. I f H c I i s a n ideal o f A then
from
K~
( K ~ ) '
=
K
~
where ,
( K ~ ) '
i s obtained
a s indicated in 1 . 4 . 3 . , and H ' i s associated t o H as in
I .4.2. ( 3 ) .
251
J. Non-CommutativeGraded Rings
2. If P c Y ( 1 ) then
=
K
1. Since C i s commutative and
K
where p
c-P
=
A n P.
I . 4 . 6 . Corol lary. are r a d i c a l , they a r e also C-P symnetric kernel functors. Therefore, i f H i s an ideal of A contained ~
and ,
K
in I o r i f P i s a prime ideal of A in Y ( I ) - t h e n K~
and
K
~
a r- e
~
idempotent i . e . kernel functors. 2 . If A i s a globally birational C-algebra then t o every radical pre-
kernel functor functor
K
on A-mod there corresponds
K
' on C-mod. If
K
' i s idempotent then
a radical pre-kernel K
i s idempotent.
All the above r e s u l t s a r e special cases of the r e s u l t s i n [ 115 ]
an a r b i t r a r y ring homomorphism homomorphism
: S-ker
'p
we correspond
.O(K')
M c R-mod i s
--f
R
To
S there corresponds a l a t t i c e
--f
R-ker where t o a kernel functor
K '
on S-mod
on R-mod gi'ven by i t s torsion c l a s s as follows
' ) - t o r s i o n i f ,op*(M) i s
K
:
'-torsion.
Proposition. Let the C-algebra A contains C i n i t s center.
1.4.7.
Consider a kernel functor 1. I f
,p :
.
K
' on C-mod a n d p u t ;= ? ( K
K
' has property T then Q; a n d
QK
I
I ) .
coincide on A-mod and ;has
property T . 2. ,For a r b i t r a r y
Proof. 1.4.8.
K ' ,
Q;(A)
2' Q,, ( A ) .
Special case of Proposition 2 . 7 . i n [I15 1 . Corollary.
Let A be a birational C-algebra. Let
pre-kernel functor such t h a t I c L ( K ) a n d l e t pre-kernel functor on C-mod.
If
K
K
K
be a radical
' be the corresponding
' i s idempotent then ?(K ' )
= K
.
The following proposition sums up some f u r t h e r r e s u l t s of [ 115 ]
.
C. Structure Theory for Graded Rings of Type Z
252
1.4.9.
Proposition.
Let A be a birational C-algebra.
1. Any radical pre-kernel functor
K
on A-mod such t h a t I c f ( ~ )i s
idempotent hence symmetric. For a l l P E Y ( I ) ,
K
~
i s- a ~ symmetric kernel
functor having property T. If A i s globally birational then Spec(A) has
a basis f o r the Zariski-topology consisting of T-sets. i . e . open s e t s Y(J) such t h a t K J has property T.
2 . Let
K
be a radical kernel functor such t h a t I F Z ( K ) a n d suppose
has property T. Then any ideal J of A where jK : A 3. Let K '
K
+
K
extends t o a n ideal Q K ( A ) j K (of J) QK(A),
Q K ( A ) i s the canonical localizing ring morphism.
be a symmetric kernel functor such t h a t I F f ( ~ a n) d such t h a t
has property T , then Q K ( A ) i s a globally birational Q K , ( C ) - a l g e b r a .
From hereon R i s a ring with u n i t a n d C denotes the center of R . The ring R i s said t o be Zariski central i f - t h e Zariski topology of Spec R = X has a basis of open sets. X(S)
= {
P F X, P
p
S } where S varies through the
subsets of C; equivalently : R i s Zariski central i f rad(1) = rad(R(1 n C ) ) f o r every ideal of R . Examples of Z a r i s k i
central rings a r e , semisimple
Artinian rings, Azumaya algebras, rings R = A [ X,,p,6]
i s a simple
where
ring and where the center of R i s n o t a f i e l d : in p a r t i c u l a r A [ X,,p 1 i s Zariski c e n t r a l . 1.4.10.
Lemma.
I f R i s a Zariski central ring then the f i l t e r s :
f ( I ) = iL l e f t ideal o f R , L
2
L ( R - P ) = { L l e f t ideal of R, L
J ideal o f R, r a d ( J ) 3
3
I}
J ideal of R, J Q P ,
are idempotent f i l t e r s f o r every ideal I of R and every prime ideal P of R . Because of the above lemma we are able t o l o c a l i z e a t
K
~ K, ~
-
~ the ,
kernel
functors on R-mod associated t o f ( I ) , l ( R - P ) r e s p . , even i f R i s n o t necessarily l e f t Noetherian. If
K
i s a symmetric kernel functor on R-mod
253
1. Non-Commutative Graded Rings
then l e t L ( K ' )
RI 6
Z(K).
be the f i l t e r generated by the i d e a l s I of C such t h a t
The following statements are equivalent, f o r some M c R-mod :
1. M i s K-torsion 2. C~ i s Kc-torsion
The c l a s s of Zariski central rings i s closed under taking epimorphic images. 1.4.11. Theorem.
Let
K
be a symmetric kernel functor on R-mod, R a
Zariski central r i n g , and suppose t h a t
K'
has property T then has
K
has
property T. Moreover : 1. QK(R)i s a central extension of R and QK(R)and Q c ( R ) a r e isomorphic K
rings
2 . For every M c R-mod t h e C-modules Q, ( M ) and Q c(cFI)
3. I f I i s an ideal of R then QJR) j K :R
+
K
a r e isomorphic.
j K ( I ) i s an ideal of Q , ( R ) ,
where
Q K ( R ) i s the canoncia! morphism.
4. There i s a one-to-one correspondence between prime i d e a l s of QJR) prime ideals of R which a r e not in 1.4.12. Lemma.
and
L)(K).
I f D i s a graded division ring then D i s a Zariski central
ring. Proof.
D = Do [ X,X-l,t~] f o r some skewfield Do and an automorphism
Do, cf.A.Theorem 6.3., unless D the statement i s t r i v i a l .
=
'0
of
Do i s a skewfield and in the l a t t e r case
Now S = Do [X,,D]i s Zariski c e n t r a l , c f . [ 3 1 1
and localizing S a t the Ore s e t { l , X , X ' ,
. . . } y i e l d s t h a t D i s Zariski
central t o o .
Now suppose t h a t C i s a graded commutative ring and l e t R be a graded C-aIgebra. 1.4.13. Definition.
R i s said t o be a graded birational (Gr-biratiqnal)
C-algebra i f i t i s a b i r a t i o n a l C-algebra and the open s e t s "(1) and
C. Structure Theory for Graded Rings of Type Z
254
X ( 1 ' ) a r e given by graded i d e a l s I and I ' of A and C r e s p e c t i v e l y .
Let R be a G r - b i r a t i o n a l C-algebra, l e t
1.4.14. Proposition.
r a d i c a l kernel f u n c t o r on R-mod and l e t on C-mod a s s o c i a t e d t o
K .
Then
K
K
K
be a
' be t h e r a d i c a l kernel f u n c t o r
i s graded ( r i g i d ) i f and only i f
K
' is
graded ( r i g i d ) . Proof.
Suppose
i s graded. I f L
K
F Z(K)
t h e n L c o n t a i n s a graded l e f t
i d e a l J c X ( K ) . Then J n C c d : ( ~ ' )i s a graded i d e a l of C , hence, s i n c e t h e i d e a l s J r! C with J F Z ( K ) from a b a s i s f o r d : ( ~ ' ) i t follows t h a t K '
i s graded. Conversely suppose t h a t
Then X ( L r! C ) J'
F Z(K
I).
ir
Y(L) y i e l d s L
From L
The r a d i c a l i t y of
K
3
n C
2
K '
i s graded. Let L c Z ( K ) .
J ' f o r some graded i d e a l J' of C ,
A J ' w i t h J ' c L ( K ' ) i t follows t h a t r a d ( A J ' ) c
Z(K).
e n t a i 1 s AJ ' F 6: ( K ) and so L c o n t a i n s a graded i d e a l
which i s again i n Z ( K )
i.e.
K
i s graded. The statements about r i g i d i t y
a r e e a s i l y checked. We leave to t h e r e a d e r t h e t a s k of r e f o r m u l a t i n g t h e p r o p e r t i e s of b i r a t i o n a l a l g e b r a s mentioned before i n
the graded s i t u a t i o n . The
proofs of t h e s e graded v e r s i o n s a r e v e r y s i m i l a r t o t h e proofs of t h e r e s u l t s i n [ 115 ]
r e f e r r e d t o , keeping
i n mind t h a t the r a d i c a l of a
graded i d e a l i s again graded. We s k i p t h e s e t e c h n i c a l d e t a i l s and turn
t o Z a r i s k i c e n t r a l graded r i n g s . A graded r i n g R i s s a i d t o be a GZ-ring i f i t i s a Z a r i s k i c e n t r a
ring.
A graded r i n g R i s s a i d t o be a ZG r i n g i f f o r every graded i d e a l I of R we have t h a t rad I ) = rad(R(1 n C ) ) . A GZ-ring i s a ZG-ring; we w i l l deal with GZ-rings here, most r e s u l t s may a l s o be obtained f o r t h e l a r g e r class of
ZG-ring.
255
1. Non-Commutative Graded Rings
Let R be a graded Z a r i s k i c e n t r a l r i n g and l e t P
1.4.15. Theorem.
be a
graded prime i d e a l of R . Then :
1.
K
c o- i n c~ i d e s on R-gr w i t h
~
f o r every M F R-gr.
K
c-P
Qi-p(M)
where p = P n C i . e . K ~ - ~ ( M =)
K
~
(
M) -
~
2' Q:-p(M) f o r a l l M c R-gr. R- P has property T and 3. I f J i s a graded i d e a l of R then j R - p ( J ) i s a graded i d e a l 2.
K
of Q;-,(R)
(here jR-p: R
of degree
4 . There
+
Qi-p(R)
Q;-,(R)
i s t h e canonical
graded morphisrn
0).
i s a one-to-one correspondence between graded prime i d e a l s P
o f R such t h a t P F X ( K ~ - ~ ) and , proper graded prime i d e a l s of Q:-,(R). Consequently, i f I i s a graded i d e a l of R (hence r a d ( 1 ) i s a graded i d e a l of R t o o ) , then : b;-p(R)JR-p(rad(I))
= rad ( Q i - p ( R ) J R - p ( I ) ) .
Proof.
1. Since
K
i s- induced ~ on R-gr by E ~ and - ~s i n c e K ~ and - ~cc-pc o i n c i d e
~
mR-mod i t follows t h a t
K ~ - ,
and
K
C -P
c o i n c i d e on R-gr. Consequently,
may be regarded a s being the kernel f u n c t o r on R-gr a s s o c i a t e d t o t h e
KR-P
c e n t r a l (Ore) s e t h ( C - p ) . 2 . From 1 i t follows t h a t , I c f ( ~ ~ i-f ~ and ) only i f t h e r e i s an
x
F
I f' h ( C - p ) ,
(Rx1x-l c R w i t h jR-, : R
-cK
Q 9R - ~ ( M )
Q;-,(R)
=
i s the canonical epimorphism of degree 0. For M c R-gr,
Q c-p ( M-) follows from Z a r i s k i c e n t r a l i t y o f R . Since both F ~ - ~
QR-p(!)s and
-f
Qi-p(R) follows from x - l c Q R - p ( R ) and Rx c L ( K ~ - ~ )T .h u s Qi-p(R) j R - p ( I ) Q;-,(R), where
but then x - l c
p
a r e graded and of f i n i t e type i t follows t h a t QR-p($ 2 QC-p(M)
Q:-p(M)
3. Let J be any graded i d e a l o f R and t a k e x F Q i - p ( R ) j R - p ( J ) Q i - p ( R ) .
Write x = z ' q . j . q ! w i t h q i , q i 6 h(Q!-,(R)), 1 1 1
j i F h ( J ) . We may f i x an
C. Structure Theory for Graded Rings of Type Z
256
e l e m e n t c 6 h(C-p) such t h a t c q ' i n Q;-,(R),
we g e t :
CX C
c
f o r e v e r y i. S i n c e c i s c e n t r a l
jR-p(R)
Q;-p(R)jR-p(J)
Qi-p(R)jR-p(J)
Or X 6
by l e f t
c Q;-,(R).
m u l t i p l i c a t i o n w i t h c-'
4. F o r a l l graded p r i m e i d e a l s Q o f R we have t h a t Qe = Q:-p(R)jR-p(Q) an i d e a l o f
w h i c h i s non-proper i f and o n l y i f Q c f ( R - P ) .
Q;-,(R),
i s e a s i l y checked t h a t Qe i s a graded i d e a l and t h a t Qe = j R - p ( Q ) i f Q d, f ( R - P ) .
then I
n
i s an i d e a l of j R - p ( R )
jR-p(R)
n
Q;-,(R)(I
I f I i s an i d e a l o f Q:-,(R)
K
T h e r e f o r e , i f I and J a r e i d e a l s i n Q;-,(R)
I n jR-p(R)
or J
n
c Qe
n jR-p(R)
jR-p(R)
It
=
w h i c h i s n o t i n Qe
w h i c h i s n o t i n j R m P ( Q ) because
j R - p ( R ) ) = I by p r o p e r t y T f o r
( I n jR-p(R))(J n j R - p ( R ) )
n
is
~
-
~
.
such t h a t
IJ
c Qe t h e n
= jR-p(Q) and t h i s e n t a i l s t h a t
j R - p ( R ) i s c o n t a i n e d i n jR-p(Q) since the l a t t e r i s a
I o r J i s c o n t a i n e d i n Qe. Combination o f t h e f o r e g o i n g
prime i d e a l . This
properties y i e l d s t h a t every prime i d e a l o f
a;-,
graded p r i m e i d e a l Q o f R. The second s t a t e m e n t
R) e q u a l s Qe f o r some
n 4. i s
a trivial
consequence o f t h e f i r s t . 1.4.16.
Remarks.
1. The same h o l d s f o r ZG-rings. 2. I n t h e same way s i m i l a r p r o p e r t i e s may be e s t a b l i s h e d f o r any k e r n e l
f u n c t o r a s s o c i a t e d t o a c e n t r a l m u l t i p l i c a t i v e s e t . I n p a r t i c u l a r , such p r o p e r t i e s h o l d o n a b a s i s f o r t h e Z a r i s k i t o p o l o g y o f Spec R, i n d e e d one may c o n s i d e r
K
~
c ~# 0, i n C, w h i c h i s a s s o c i a t e d t o 11, c,c
2
,... 1 .
3. From t h e g e n e r a l t h e o r y o f graded l o c a l i z a t i o n s i t f o l l o w s t h a t : i f R i s prime if
K
t h e n Q:(R)
i s p r i m e f o r any r i g i d k e r n e l f u n c t o r on R-gr,
has p r o p e r t y T t h e n Q:(R)
1.4.17.
i s l e f t Noetherian i f R i s Noetherian.
P r o p o s i t i o n . L e t R be a p o s i t i v e l y graded ZG-ring. Then Ro i s
a Z a r i s k i c e n t r a l algebra.
257
1. Non-Commutative Graded Rings
L e t I.
Proof.
be an a r b i t r a r y i d e a l o f Ro. C l e a r l y
C,
i s i n the center
I f Po i s a prime i d e a l o f Ro such t h a t Po xRo(Io
o f R.,
n Co)
then Po
+
R+
i s a prime i d e a l o f R such t h a t :
P 0 + R+
3
n Co)+R+)
rad(Ro(Io
n C
Now, ( R o ( I o fl Co)+R+)
3
= rad(R((Ro(Io
R I o R fl C
n
Co)
+
R+)
n
C))
i s e a s i l y checked by comparing p a r t s
o f a r b i t r a r y degree; t h e r e f o r e P0
+
hence Po
R,
3
2
rad(R(RIoR
n
C)) = rad(RIoR),
Io.
The f i r s t main advantage o f Z a r i s k i c e n t r a l r i n g s i s t h a t , even i n t h e non l e f t Noetherian case, P r o j i s having t h e n i c e p r o p e r t i e s one can hope f o r
,
c f . [ 125
I
.
C. Structure Theory for Graded Rings of Type Z
258
1 . 5 . Graded Von Neumann Regular Rings.
I n t h i s section we study graded rings s a t i s f y i n g Von Neumann's r e g u l a r i t y condition on homogeneous elements. P a r t of the theory i s presented in the general case of graded rings of type G . We do n o t go i n t o
the theory of
Gr-V-rings ( V f o r Villamayor) because the l e s s controlable behaviour of g r - i n j e c t i v e modules makes this theory diverge in many d i f f e r e n t d i r e c t i o n s . A graded ring R of type C: i s said t o be Gr-regular i f f o r every homogeneous
x
C
h ( R ) there e x i s t s y in R
such t h a t x = xyx. Obviously i f x c Ro
then we may replace y by i t s p a r t y - 1 of degree
o-'C
G.
U
1 . 5 . 1 . Proposition.
For a graded ring of type G the following statements
a r e equivalent :
1. R i s Gr-regular. 2 . Every principal l e f t ( r i g h t ) graded ideal i s generated by a homogeneous
idempotent element.
3. Every f i n i t e l y generated l e f t ( r i g h t ) graded ideal i s generated by a homogeneous idempotent element. Proof. 1.5.2.
As in the ungraded case, c f . [
411
Remarks.
1. Homogeneous idempotents have degree e
c G
( e : the neutral element
of G); however a r b i t r a r y idempotents need not be homogenous. 2 . I f f o r some u
c G, Ru # 0 then R - 1 # 0 and R - 1 Ra # 0.
3. The ring Re i s a regular ring,
U
0
'
4 . I n a Gr-regular ring each l e f t ( r i g h t ) graded ideal i s generated as
a l e f t ( r i g h t ) ideal by i t s p a r t o f degree e. 5 . Graded l e f t ( r i g h t ) - i d e a l s are idempotent. Graded i d e a l s a r e semiprime.
259
1. NonCommutative Graded Rings
1.5.3. Corollary. I f R i s a strongly graded ring of type C. then R i s Grregular i f and only i f Re i s regular. Apply Theorem A.I.3.4. a n d Proposition 1.5.1..
Proof.
1 . 5 . 4 . Remarks.
1. I f R i s G r - r e g u l a r and prime while G i s ordered o r abelian, then the center Z ( R ) i s Gr-regular and without zero-divisors, hence a graded f i e l d of type G. I f R i s moreover a P . I . ring then i t will be an Azumaya algebra (use the technique of 1.2.10, 1.2.11.)
2 . I f P i s a f i n i t e l y generated projective graded module over a Gr-reqular ring R of type G , then EndRP i s gr-regular. 3 . A graded ring R of type 6 i s Gr-regular
i f a n d only i f a l l graded l e f t
modules a r e g r - f l a t ( i . e . f l a t ) ,
4 . Let A be a graded f i n i t e l y generated l e f t module over a Gr-regular ring R and l e t B be a graded projective l e f t R-module, f c H O H R ( A , B ) , C be
and l e t
any f i n i t e l y generated graded submodule of B, then f - l ( C ) i s f i n i t e l y
generated and a d i r e c t summand. 5. Let B1,. ..,B,
be f i n i t e l y generated graded submodules of a
graded
projective module A , then B1 n...n Bm i s f i n i t e l y generated. For the proof of 4. a n d 5. see Lemma's 2 . 1 . and 2 . 2 . in 1.5.5.
Proposition.
[ 41
1.
Let J be a graded ideal o f the Gr-regular ring R of
type G . Let f l , f 2 be orthogonal homogeneous idempotent of R/J, then there exist el
a n d e 2 , orthogonal
e l mod J = f l , e 2 mod J such t h a t e l + e2 Proof.
=
=
homogeneous idempotents in R such t h a t
f 2 . If f l
i.f 2
= 1 then el a n d e2 may be chosen
1.
This statement only a f f e c t s degree e , and since Re i s regular
and (R/J), = Re/Je, t h i s
i s exactly Proposition 2.18 of [ 4 1
I.
C. Structure Theory for Graded Rings of Type Z
260
1.5.6. Definition.
Recall t h a t a r e g u l a r r i n g R i s s a i d t o be a b e l i a n
r e g u l a r i f R has no nonzero n i l p o t e n t elements, o r e q u i v a l e n t l y : i f a l l idempotent elements of R a r e c e n t r a l . Let
US
term t h e graded r i n g R of
type G Gr-abelian r e g u l a r i f a l l homogeneous idempotents a r e c e n t r a l , i . e . i f and only i f R
e i s abelian regular.
Let us r e c a l l the following easy but useful lemma : Let R be any semiprime r i n g . For any idempotent element
1 . 5 . 7 . Lemma.
e of R t h e following s t a t e m e n t s a r e e q u i v a l e n t : 1. e i s c e n t r a l i n R . 2 . e commutes w i t h every idempotent element of R . 3. Re i s an i d e a l of R .
4. (1-e)Re = 0 . 5. e R ( l - e )
=
0.
1 . 5 . 8 . P r o p o s i t i o n . For a G r - r e g u l a r r i n g R t h e following c o n d i t i o n s a r e equivalent : 1. R i s G r - a b e l i a n r e g u l a r . 2 . For every graded prime i d e a l P of R , R/P i s a Gr-division r i n g . 3. R has no nonzero homogeneous n i l p o t e n t elements.
4. All graded l e f t (right) i d e a l s of R a r e i d e a l s . 5. Every nonzero graded l e f t i d e a l c o n t a i n s a nonzero c e n t r a l homogeneous idempotent. Proof.
Easy, along the l i n e s of Theorem 3.2. i n
[41 1 .
In o r d e r t o deduce from t h e Gr-abelian r e g u l a r i t y c o n d i t i o n s t a t e m e n t s about a r b i t r a r y n i l p o t e n t elements we need t h a t C. i s
an ordered group.
F i r s t we note : 1.5.9. P r o p o s i t i o n .
Let G be an ordered group and l e t R be G r - r e g u l a r
of type G . Then R i s a l e f t and r i g h t nonsingular r i n g .
261
1. Non-CommutativeGraded Rings
Proof. Let us f i r s t es t ab l i s h t h a t each x # 0 i n h(R) i s nonsingular. Indeed, suppose Jx = 0 f o r some es s en t i al l e f t ideal J of R. Then JxR
=
0
y ie l d s JeR = 0 f o r some idempotent i c Re such t h a t x R = i R. Hence J c R ( 1 - i ) b u t t h i s contradicts J c Ri f 0. Now take x # 0 a r b i t r a r y in R , say x = x
+...t
+...+ xo
u1 with
n
with u1
>...
>
n . Any j e J may be written
and j x = 0 y i elds t h a t j T x = 0. 1 Ol = 0. Since xo f 0 by assumption, i t will suffic e 1 1 t o show t h a t J" i s es s en t i al as a l e f t ideal of R , (J- i s defined i n
as j
j
Tl <m Consequently J" xu
T~
>...>
T~
Section A.II.2.) Since J i s an es s en t i al l e f t ideal of R, we have tha t f o r any y c h ( R ) there e x i s t s a n r
r
=
r
+. . .t r
Y1
Yt
with y1 >-..
. > yt,
highest degree f o r which r
C
R such t h a t 0 f ry c J . Writing
we obtain r y yi Consequently
E
3- where yi i s the
y # 0. i n t e r s e c t s every "i . graded l e f t ideal of R in a nontrivial way, i . e . J" i s gr-e sse ntia l.
Now Lemma A.I.2.8. y i el d s t h a t J- i s a n es s en t i al l e f t ideal of R , so J- xol = 0 leads t o a contradiction and so R i s l e f t nonsingular A symmetrical argumentation may be used t o derive t h a t R i s a l s o r i g h t
nonsi ngul a r
.
1.5.10. Proposition.
Let G be an ordered group and l e t R be a d r - r e g u l a r
ring of type G . Then the conditions of Proposition 1.5.8. being equivalent t o R i s Gr-abelian regular, ar e also equivalent t o : 6 . R has no
nonzero nilpotent elements.
7. Every nonzero graded l e f t ideal of R contains a nonzero central idempotent. Proof.
That 6 . i s equivalent t o 3 in Proposition 1.5.8. i s obvious when
G i s ordered. Furthermore, condition 5 in Proposition I .5.8. obviously
implies 7. The implication, 7
-
1 follows from Proposition 1.5.9. by the
arguments used in proving Theorem 3.2. in
[
41 1 .
C. Structure Theory for Graded Rings of Type Z
262
C o r o l l a r y . L e t R be G r - r e g u l a r o f t y p e C- where G i s a n o r d e r e d
1.5.11.
group. I f R i s g r - a b e l i a n r e g u l a r t h e n : 1. A l l i d e m p o t e n t elements o f R a r e c e n t r a l . 2 . Every m i n i m a l graded p r i m e i d e a l i s c o m p l e t e l y p r i m e . Proof. 1. I f i i s i d e m p o t e n t i n R t h e n ( 1 - i ) R i c o n s i s t s o f n i l p o t e n t elements so, i f R i s G r - a b e l i a n t h e n ( 1 - i ) R i = 0 by P r o p o s i t i o n 1.5.10. and t h e n
i i s c e n t r a l because R i s semiprime and because o f Lemma 1.5.7. 2. I f P i s a minimal
.
graded p r i m e i d e a l o f R t h e n R/P i s a g r - d i v i s i o n
r i n g because o f P r o p o s i t i o n I . 5 . 8 . ,
2.
.
S i n c e C: i s ordered, g r - d i v i s i o n
r i n g s o f t y p e G c a n n o t have z e r o - d i v i s o r s . The s t r u c t u r e o f G r - a b e l i a n r i n g s may be c o m p l e t e l y d e s c r i b e d i n t h e s t r o n g l y graded case f o r a r b i t r a r y groups. We need : Lemma. L e t A be an a b e l i a n r e g u l a r r i n g . Then e v e r y i n v e r t i b l e
1.5.12.
A-bimodule P i s f r e e as a l e f t ( o r r i g h t ) A-module. Suppose P, Q a r e A-bimodules such t h a t :
Proof.
P ~ ~ Q ~ A ~ Q Q o P A A We t h e n know t h a t P and Q a r e f i n i t e l y g e n e r a t e d p r o j e c t i v e l e f t ( r i g h t ) A-modules. The s t r u c t u r e t h e o r y f o r f i n i t e l y g e n e r a t e d p r o j e c t i v e modules over a r e g u l a r r i n g , c f . [ 4 1 P = RX
1
8
y i e l d s t h a t we may w r i t e
... 8 RX n .
D e f i n e a morphism f : A l e f t A-morphism). Ker f = { a
1,
c A,
+
P by f ( a ) = axl t ax2
+
... +
ax,.
(f i s a
We have t h a t :
On annA(xi). Since A i s a b e l i a n regular, i=l each l e f t i d e a l i s an i d e a l , hence Ra c aR. So i f a x . = 0 t h e n a Rx = 0;
f ( a ) = 0} =
t h e r e f o r e any a 6 K e r f w o u l d a n n i h i l a t e P. B u t P i s
f a i t h f u l since i t
1. Non-Commutative Graded Rings
263
i s i n v e r t i b l e , hence f i s i n j e c t i v e . Since Q i s p r o j e c t i v e a s a r i g h t A-module we have the e x a c t sequence : 0
+
l?f
Q63A
A
Q m P - A A
The map 1 8 f i s l e f t A-linear and so we o b t a i n
0
--f
AQ
2
AA.
NOW
Q i s f i n i t e l y generated a s a l e f t A-module, hence t h e r e exists an
idempotent i i n t h e c e n t e r of A such t h a t AQ
A i . From ( l - i ) A i = 0 i t
follows t h a t ( 1 - i ) Q = 0 but Q i s f a i t h f u l a s a l e f t A-module s i n c e i t i s invertible,
hence 1-i = 0 and A Q 2 A A f o l l o w s . S i m i l a r l y AP
AA
f o l 1ows. 1.5.12. Theorem. Let R be a s t r o n g l y graded r i n g of type G . The following s t a t e m e n t s a r e e q u i v a l e n t : 1. R i s Gr-abelian r e g u l a r . 2 . R i s t h e c r o s s e d product Re [
E
.aR] where Re
i s an a b e l i a n r e g u l a r
r i n g ( f o r t h e s t r u c t u r e o f t h e crossed products s e e S e c t i o n A . I . 3 . ) . Proof. 2 = 1. Follows immediately from Corollary 1 . 5 . 3 . and then from D e f i n i t i o n
1.5.6. 1
2 From t h e lemma above and t h e r e s u l t s o f S e c t i o n A.1.3. i n p a r t i c u l a r
Lemma A. I .3.20. 1.5.13. C o r o l l a r y .
Let R be a s t r o n g l y graded r i n g of type Z then the
following s t a t e m e n t s a r e e q u i v a l e n t : 1. R i s Gr-abelian r e g u l a r . 2. R
=
Ro [ X,X-l,fo] where
'0
i s an automorphism o f t h e a b e l i a n r e g u l a r
r i n g Ro. 1 . 5 . 1 4 . Example. I f R i s graded o f type Z but not s t r o n g l y graded then
t h e r e e x i s t Gr-abelian r e g u l a r r i n g s , which need n o t be o f the form
C. Structure Theory for Graded Rings of Type Z
264
Ro
.
[ X,X-',?]
L e t k be a f i e l d and f o r each n F N consider k [ X
where t h e g r a d a t i o n on k [ X (,), Consider B = of all b
=
(bl,
1 Xin)]
X-' ] (n)' (n) i s d e f i n e d by p u t t i n g deg X t n ) = n.
n k[X X - l ] and l e t A be t h e s u b r i n g o f B c o n s i s t i n g ~ F N (n)' (n) bp,
...,
bn, .. . ) i n B which become e v e n t u a l l y constant
s e r i e s . I t i s c l e a r t h a t A i s Gr-abelian r e g u l a r and A,
i s the regular
subring o f
c o n s i s t i n g o f t h e e v e n t u a l l y constant s e r i e s . C l e a r l y ,
A # A.
f o r any p o s s i b l e X one m i g h t t h i n k o f .
[X,X-l]
L e t R be a graded r i n g o f type G; we d e f i n e :
c
R i R ( l - i ) R , where i runs through t h e s e t o f a l l idempotent i elements o f R ;
N(R) =
Ng(R) = E R j R ( l - j ) R , where j runs through t h e s e t o f idempotent
j
homogeneous elements o f R; S(R) = t h e s u b r i n g ( w i t h o u t i d e n t i t y perhaps) generated by t h e n i l p o t e n t elemenst o f R; Sg(R) = t h e graded s u b r i n g o f R generated by t h e homogeneous n i l p o t e n t elements o f R. 1.5.15.
Theorem. L e t R be graded o f type G where G i s an ordered group
and suppose t h a t R i s a Gr-regular r i n g
.
Then Ng(R) = N(R) = S(R)
=
Sg(R)
and R/Ng(R) i s g r - a b e l i a n r e g u l a r . Proof.
I f f F R/N9(R) i s an idempotent homogeneous element then t h e r e
e x i s t s an idempotent homogeneous j F R which l i f t s f. Since j R ( i - j ) c Ng(R) i t f o l l o w s t h a t 0 = f ( R / N g ( R ) ) ( l - f ) and thus f i s c e n t r a l i . e . R/Ng(R)
i s Gr-abelian r e g u l a r . By P r o p o s i t i o n 1.5.10.,6.,
we o b t a i n
Sg(R) c S(R) c Ng(R) c N(R). I f E i s an idempotent, E F N(R) image i n R/Ng(R) i s c e n t r a l because o f C o r o l l a r y 1.5.10. ER( I-E) c Ng(R), hence N(R) c Ng(R) and N(R) = Ng(R) .
, then i t s
. Therefore
I. @on-CommutativeGraded Rings
Let
US
now proceed t o show t h a t N g ( R )
c
sg(R). Let
265
j be an idemPotent
homogeneous elements of R, then : R j R ( 1 - j ) R = j R j R ( l - j ) R + ( 1 - j ) R j R(1-J) R c j R(l-j)R
+ (1-j)R J R.
sg(R)
Moreover : j R ( 1 - j ) R = j R ( l - j ) ( l - j ) R j + j R(1-j) R(1-j) Similarly
:(l-j)R j R c
Therefore
zR j
j R(l-j)R
s~(R). = Ng(R) c sg(R).
I n the sequel R will be graded of type Z . 1.5.16. Proposition. I f R i s Gr-regular of type 2 then J(R) Proof.
=
0.
In Section A . I . 7 . we established t h a t the Jacobson radical of
a Z-graded ring i s a graded i d e a l . As a graded ideal of a Gr-regular
or e l s e J(R)
ring J(R) then has t o contain a n idempotent element
= 0.
Since the f i r s t i s impossible the l a t t e r i s a f a c t . 1 . 5 . 1 7 . Proposition.
Let R be a Gr-regular ring of type Z such t h a t
f o r each prime ideal p of Ro there e x i s t s a graded prime ideal P of
R such t h a t Po = p . Then the following properties hold : 1. For a l l n 6 Z , Rn R-, 2 . For a l l n
F
Z , RRn
=
=
RR-,
R-n Rn i s an ideal of R .
Proof. 1. I n a Gr-regular ring each l e f t ( r i g h t ) graded ideal i s generated
as a l e f t ( r i g h t ) ideal of R by i t s p a r t of degree zero. Thus RR1 = RIo where I.
R1 =
=
R-l
R1 i s a n ideal of Ro. From RR1 = RR-lR1
RIR-l
R1 = Jo R1 where Jo
RIIo R-l
a n d RIIo = JoR1, R-l
=
=
R1 R-l.
Hence : Jo
Jo = I, R-l.
Since R,
=
i t follows t h a t
RIR-l.
RIR-l
=
i s regular, I 0
and J o a r e semiprime i d e a l s . I f p i s a prime ideal of R, containing I.
C. Structure Theory for Graded Rings of Type Z
266
t h e n t h e hypothesis on R i m p l i e s t h a t t h e r e i s a graded prime i d e a l P o f R such t h a t Po = p i . e . P contains R I. R1 IoR-l
R. Consequently p = Po contains
Jo and thus we o b t a i n J, c I o . S i m i l a r l y , by symnetry, we
=
f i n d I, c
Jo and I.
= Jo. I n f o r m a l l y t h e same way i t can be shown
t h a t Rn R-n = R-n Rn f o r a l l n t
Z.
Consider RR1 Rn f o r some p o s i t i v e n c Z.
2.
Rn = R1 I.
C l e a r l y : I.
R 1: Rn-l
=
R I o Rn-l
=
Rm c
RR1 R n - l .
=
Rn-l
and thus KR1 Rn c RR1 1, Rn-l
R e p e t i t i o n o f t h i s argumentation
f o r a71 m c 0 i n Z. B u t s i n c e RRl
RR-l
= RR-lR1
= RR1 R-l
i t f o l l o w s t h a t RR1 i s an i d e a l o f R. Now note t h a t :
= RR-l
Rn R1 = (RRl)n+l
= (RIR)n+l
holds f o r a l l n 6 Z.
= R1 Rn
To prove t h a t RRn i s an i d e a l we proceed
by i n d u c t i o n
on n (we w r i t e
t h e p r o o f f o r n > 0, the case n < 0 i s s i m i l a r ) . Suppose we know t h a t RRm i s an i d e a l f o r each m, 0 < m < n. I f f o r some m, then RRm i s as r i g h t i d e a l generated by i t s RRm = R-,Rm
R1 Rt RR
n
R
R = R,
R-,,
R
=
R,
=
t
Rt R1 f o r a l l 1 t Z. = RR-,
Rn Rt = RRn R-,
R f o l l o w s . Consider RRn Rt
R t c RRn Rt-n.Repetition where to = t mod n
hence RRnRt c RRn as b e f o r e . I f t <
RRn i s an i d e a l o f R. 1.5.18. C o r o l l a r y .
ideal
f o r some RtR y i e l d s
n then of t h i s
i .e. to <
n,
- n then i n t e r c h a n g i n g t h e r o l e
RRn Rt c RR-,
o f n and - n y i e l d s :
=
If t
Hence RRn Rt c RR,.
Rt c RRn R
argument leads t o RR,
RRm i s an
p a r t o f degree zero i . e .
I f - n < t < n thenRRt i s an i d e a l , hence RRt
t 6 Z.
=
c RR1 f o r a l l n 2 0. A s i m i l a r c a l c u l a t i o n e n t a i l s
y i e l d s t h a t RRIRn t h a t RR-l
c R1 I.
Rn
R-l
= RtO
RR
R
to
c RR,.
Therefore
I f R i s G r - a b e l i a n r e g u l a r then i t s a t i s f i e s t h e
c o n d i t i o n o f P r o p o s i t i o n 1.5.17.
=
267
1. NonCornrnutative Graded Rings
Proof.
I f p i s a prime ideal of Ro then R p i s a graded l e f t ideal of
R hence i t i s an i d e a l . Since ( R P ) ~= p i t i s c l e a r t h a t there e x i s t s a graded prime ideal P of R such t h a t Po = p .
1.5.19. Proposition. I f R i s a l e f t Noetherian Gr-regular ring of type Z , then i t i s a gr-semisimple gr-Artinian ring. Moreover, Ro i s semisimple
Artinian and Ri i s a semi-simple Ro-mdule f o r each i
Proof.
I f R i s n o t gr-semisimple then there i s a graded l e f t ideal of
C
Z.
R which i s not a d i r e c t summand, hence i t cannot be f i n i t e l y generated. The l a t t e r contradicts the l e f t Noetherian hypothesis. The r e s u l t s of e n t a i l t h a t Ri i s a semisimple Ro - module f o r each i c Z .
Section A . I I . 6 .
The decomposition of R = S ( l ) 8 rings y i e l d s t h a t SL1) 8
...
... @ S i n )
@
S ( n ) i n t o gr-simple gr-Artinian
i s a decomposition of Ro i n t o simple
Note also t h a t ' i n t h i s case Ri = SI1)
Artinian rings.
where S i j ) i s a simple
Q
...
8 Sin)
,
Ro-module or zero depending on whether i i s
d i v i s i b l e by e . o r not, where e . i s the lowest p o s i t i v e degree appearing J
J
in the gradation of the gr-simple ring S ( J ) . If M
c R-gr
then M+= 8 Mn (notation of Section A . I I . 2 . ,
nzO
Proposition A.II.2.3.).
in particular
We i n v e s t i g a t e graded prime i d e a l s of R and R+
and a l s o the l e f t Ore conditions with respect t o these prime i d e a l s . 1.5.20.
Let P be a graded prime ideal of t h e Gr-regular ring R
Lemma.
of type Z . I f P Proof.
3
N g ( R ) then P+ i s a graded prime ideal of R'.
Since R/P i s a prime epimorphic image of the Gr-abelian regular
ring R / N g ( R ) skewfield A
i t follows t h a t R/P i s a gr-division ring i . e . e i t h e r a i f R/P i s t r i v i a l l y graded, or e l s e of the form A [ X,X-l,,~]
as mentioned in A.I.4. So R+/P+ i s e i t h e r A or A [ X,,D] ring in e i t h e r case.
i . e . a prime
C. Structure Theory for Graded Rings of
268 1.5.21.
Type Z
I n t h e s i t u a t i o n o f Lemma 1.5.20.
Proposition.
,
= h(G(P))+ = h(G(P)) fl R+
h(G(P+)) =
and R+ s a t i s f i e s t h e l e f t Ore c o n d i t i o n
w i t h r e s p e c t t o h(G(P+)) i f and o n l y i f R s a t i s f i e s t h e l e f t Ore condition w i t h respect t o h(G(P)). Proof.
I f R/P i s t r i v i a l l y graded t h e n a l l s t a t e m e n t s a r e e a s i l y deduced
f r o m t h e f a c t R/P i s a G k e w f i e l d . So now suppose R/P
A [ X,X-l,p].
S i n c e G(P) i s e x a c t l y t h e s e t o f
elements mapping t o l e f t r e g u l a r e l e m e n t o f R/P i t f o l l o w s t h a t G(P) = R-P and t h e f i r s t s t a t e m e n t s a r e e a s i l y v e r i f i e d . Assume
h ( G ( P ) ) = h(R-P),
deg X > 0 and choose a r e p r e s e n t a t i v e Y i n h(R) f o r X. O b v i o u s l y Y
c
h(G(P))+ and f o r any s
c
enough. I f r 6 h(R), s
c
h(G(P)),
c
Ym s
h(G(P))+ f o r some m l a r g e
h ( G ( P ) ) + a r e such t h a t r s = 0 t h e n t h e l e f t
Ore c o n d i t i o n s w i t h r e s p e c t t o h(G(P)) e n t a i l s t h a t s ' r = 0 s'
c
h(G(P)) and t h u s Y m s ' r = 0 f o r some Yms'
f o r given r
c
h(R), s
c
h ( G ( P ) ) + we f i n d r '
c
c
h(G(P))+.
f o r some
Furthermore,
h ( R ) , s ' c h ( G ( P ) ) such
t h a t s ' r = r ' s , hence Y m s ' r = ( Y m r ' ) s . Consequently, i f R s a t i s f i e s t h e l e f t Ore c o n d i t i o n s w i t h r e s p e c t t o h(G(P)) t h e n R+ s a t i s f i e s t h e s e c o n d i t i o n s w i t h r e s p e c t t o h(G(P))+. Conversely, i f r where m,l
c
h(R), s
c
h(G(P)) a r e such t h a t r s = 0 t h e n YmrsY1 = 0
a r e chosen such t h a t mdeg Y
+
deg r > 0,
1 deg Y
The l e f t Ore c o n d i t i o n s w i t h r e s p e c t t o h(G(P))+ y i e l d slym, some s '
c
p u t t i n g s" = s l y m we f o u n d s "
h(G(P))+,
s " r = 0. A l s o , f o r g i v e n s t h a t Y m r c R+, some r '
c
the proof.
Yls
h(R+),
c
c
h(G(P)),
r
c
c h(G(P))
+
deg s > 0. =
0 for
such t h a t
h(R) we f i n d a g a i n m, 1 such
1
h ( G ( P ) ) + . T h e r e f o r e we o b t a i n : s'Ym = r ' Y s, f o r
s ' c h(G(P))+
and s i n c e s l y m
c
h(G(P)) t h i s f i n i s h e s
1. Non-Commutative Graded Rings
1.5.22.
269
C o r o l l a r y . If P i s a graded prime i d e a l o f t h e Gr-regular r i n g R
such t h a t P
3
Ng(R) then R‘
s a t i s f i e s t h e l e f t Ore c o n d i t i o n w i t h r e s p e c t
t o h(G(P+)). Proof.
R s a t i s f i e s t h e l e f t Ore c o n d i t i o n w i t h r e s p e c t t o h(G(P)) i f
and o n l y i f R / K ~ ( R )s a t i s f i e s t h e l e f t Ore c o n d i t i o n w i t h respect t o t h e image o f P, where
K
h(G(P)) (see A . I I . 9 . ) .
P
i s the r i g i d
kernel f u n c t o r associated t o
B u t K ~ ( R )= P; indeed, i f p c h(P) then p = p y p
y i e l d s ( 1 - p y ) p = 0 w i t h I - p y c G(P) ! Furthermore R/K
P
(R) i s of t h e
f o r some s k e w f i e l d A . E v i d e n t l y R/P s a t i s f i e s
form A o r A [ X,X-’,,p]
t h e Ore c o n d i t i o n s w i t h r e s p e c t t o zero. The statement i s t h e n a d i r e c t consequence o f P r o p o s i t i o n 1.5.21. 1.5.23.
Remark.
I n a way f o r m a l l y s i m i l a r t o t h e above we can prove
t h a t a Gr-regular r i n g R o f type Z s a t i s f i e s t h e l e f t and r i g h t Ore c o n d i t i o n s w i t h r e s p e c t t o G(P), and h(G(P)), f o r any graded prime i d e a l P o f R c o n t a i n i n g Ng(R). 1.5.24.
An example : t h e Weyl-algebra.
L e t k be any f i e l d and l e t S = k { x , y } be t h e s p e c i a l i z a t i o n o f t h e f r e e algebra i n two i n d e t e r m i n a t e s d e f i n e d by t h e r e l a t i o n X Y - Y X = 1. Put deg X = 1, deg Y = -1. Then S i s a graded r i n g such t h a t So = k [ X Y ]
.
The t o t a l graded r i n g o f f r a c t i o n s o f S i s then given by Qg(S) = k(YX) [ X , X - ’ , Q ] by ,p(XY) = 1 + XY.
where
I f char k
,Q
i s t h e automorphism o f k(XY) d e f i n e d
# 0. then
‘P
has f i n i t e o r d e r and then Qg(S)
i s a f i n i t e dimensional g r - d i v i s i o n algebra over i t s c e n t e r k(XY)”[ T,T-’l where T = Xn,
some n 6
N.
I f char k = 0 then S i s t h e f i r s t Weyl-algebra
i . e . a simple Ore domain which i s n o t a r e g u l a r r i n g . Therefore we see t h a t Qg(S) i s a simple domain which i s G r - r e g u l a r and even G r - a b e l i a n r e g u l a r b u t c e r t a i n l y n o t a r e g u l a r r i n g . I t i s well-known
that
C. Structure Theory for Graded Rings of Type Z
270 S
k [XI
[y, 6 J
where 6 denotes t h e " d e r i v a t i v e w i t h r e s p e c t t o X " .
I t may be s u r p r i z i n g t o see how t h i s d i f f e r e n t i a l p o l y n o m i a l r i n g has
a graded r i n g o f f r a c t i o n s w h i c h i s i s a c r o s s e d p r o d u c t . The g r a d a t i o n on S shows t h a t t h e p a r t o f degree z e r o o f a s i m p l e and graded domain need n o t be simple ! One can c a l c u l a t e f u r t h e r m o r e t h a t t h e s u b r i n g s k [ XnYm]
w i t h (n,m) = 1 a r e maximal graded commutative s u b r i n g s o f S
and t h e s e a r e a l s o maximal as commutative s u b r i n g s o f S . F i n a l l y l e t us m e n t i o n some s t a t e m e n t s a b o u t t h e i n c o m p a t i b i l i t y of g r a d a t i o n s and commn Von Neumann r e g u l a r i t y . A non
# 0
t r i v i a l l y graded
i f Rn c o n s i s t o f n i l p o t e n t elements f o r
r i n g i s s a i d t o be n i l g r a d e d all n
-
; I f R i s n o t n i l g r a d e d t h e n we say t h a t t h e g r a d a t i o n o f R
i s reduced, o r s i m p l y t h a t R i s reduced.
I f R i s a reduced graded r i n g t h e n R i s r e g u l a r
1.5.25. P r o p o s i t i o n
o n l y i f R i s t r i v i a l l y graded.
Proof.
Suppose 0 # xn
c
Rn i s n o t n i l p o t e n t , say n > 0. I f R i s r e g u l a r
t h e n t h e r e e x i s t s y E R such t h a t (l+x), W r i t e y = yi
+. . .+yi
1
m
where i
P . .
1
y ( l + x n ) = l+xn.
.>i,.Then yi
degree i n t h e l e f t hand s i d e hence yi
m
= 1,
i
m
m =
i s t h e p a r t o f minimal
0 and il>.. .>i = 0 m
follows. Straightforward calculations y i e l d : 2xn+yi
+...+y. +x y . +...+yi
1
lm
Since a l l y.
'k
'1
xn+ 1
...
= l+xn
have p o s i t i v e degree i t f o l l o w s e a s i l y t h a t :
=O i f ik < n,yn = -xn, yn+, = 0 f o r 0 k ( l + x n ) ( 1-xn+xn-xn+. 2 3 . .) ( l + x n ) = l + x n .
yi
S i n c e t h e homogeneous d e c o m p o s i t i o n
<
m < n, yPn = x2 and so on, i . e n
o f y i s f i n i t e i t f o l l o w s t h a t xn
must be n i l p o t e n t , c o n t r a d i c t i o n . The o n l y p o s s i b i l i t y l e f t i s t h a t R
i s t r i v i a l l y graded.
27 1
1. NonCommutative Graded Rings
P r o p o s i t i o n . L e t R be a reduced graded r i n g and c o n s i d e r t h e
1.5.26.
where X commutes w i t h R and t h e g r a d a t i o n
graded r i n g S = R [ X,X-’] o n S i s d e f i n e d by
Sm =
c
iFZ
Ri
?-’. I f
S i s gr-regular then R
i s t r i v i a l l y graded and r e g u l a r . Suppose 0 # xn c Rn i s n o t n i l p o t e n t and c o n s i d e r 1 + x,X-~
Proof.
By r e g u l a r i t y o f S ( 1 + x n ~ - n ) (1
+
0
we o b t a i n :
zix-’+...)(l
+
xn~-n) = 1
+
xnx-’
f o r some zi
c R ~ .
Now f i r s t use t h e degree i n X i n o r d e r t o g e t r i d o f t h e terms i n X with i
p
c So.
i
nZ and t h e n t o d e r i v e t h a t i > 0 f o r a l l a p p e a r i n g terms, Then
u s i n g t h e g r a d a t i o n o f S one e a s i l y c a l c u l a t e s , as i n t h e f o r e g o i n g
z~~ -- x:
p r o o f , t h a t zn = -xn, t o be n i l p o t e n t i . e .
R
=
a.s.0.
l e a d i n g t o t h e f a c t t h a t xn has
Ro = So and R i s r e g u l a r .
L e t us i n c l u d e some o b s e r v a t i o n s a b o u t graded r i n g s o f t y p e Z w i t h Gr-regular center. I f graded ( l e f t ) i d e a l s a r e i d e m p o t e n t t h e n R i s s a i d t o be G r - f u l l y I l e f t ) idempotent.
A G r - f u l l y ( l e f t ) i d e m p o t e n t r i n g has a G r - r e g u l a r
c e n t e r . A graded r i n g R such t h a t each graded p r i n c i p a l i d e a l i s g e n e r a t e d by a homogeneous c e n t r a l i d e m p o t e n t i s s a i d t o be G r - b i r e g u l a r So if R i s G r - b i r e g u l a r t h e n Ro i s b i r e g u l a r , R i s G r - f u l l y l e f t idem-
p o t e n t and Z(R) i s G r - r e g u l a r . The f o l l o w i n g r e s u l t i s an easy a d a p t a t i o n of P r o p o s i t i o n 1.2. and 1.3.
1.5.27.
Theorem.
i n [ 85
I.
L e t R be a graded r i n g w i t h c e n t e r C, t h e n t h e
following conditions are equivalent :
1. C i s G r - r e g u l a r and f o r each the set o f 2. R i s & - f u l l y
M c s?
r-maximal i d e a l s o f R .
9
(R),
M
=
R(P n C) where 8 (R) i s 9
l e f t i d e m p o t e n t and a Z a r i s k i e x t e n s i o n o f C ( c f . 1 . 4 .
f o r Zariski extensions).
C. Structure Theory for Graded Rings of Type Z
272
3. R i s Gr-fully idempotent and a Zariski extension of C .
4. R i s Gr-biregular. 5 . For every m 6 Q (C), Q;-,,(R) i s gr-simple.
4
6. R i s a Zariski extension of C and Qi-,(R)
i s gr-simple f o r a l l
M C Qg(R). I f moreover, R i s f i n i t e l y generated as a C-module then R i s a Zariski extension of C i s and only i f R s a t i s f i e s the l e f t and r i g h t Ore conditions with respect t o G ( P ) f o r a l l graded prime ideals P of R. Let R be a graded ring with regular center C , which
1.5.28. Theorem.
i s f i n i t e l y generated as a C-module then the following statements a r e equivalent : 1. R i s semiprime 2. R i s Gr-regular 3 . R i s Gr-birerjular
4. R i s a Zariski extension of C and a l l graded prime ideals of R a r e idempotent. 5. R i s a n Azumaya algebra over C. Proof.
Easy modification of 2.6. i n
85
1 . Note t h a t already the
weaker homogeneous conditions on C make R i n t o an Azumaya algebra. This i s of course r e l a t e d t o the graded version of the Artin-Procesi theorem
for P . I . rings as expounded i n Section 1.4. 1.5.29 Remark.
The above may be used in describing the "graded Brauer
group" o f a commutative Gr-regular r i n g C . However, the study of graded Azumaya algebras over C reduces e a s i l y t o the study of graded Azumaya algebras over the s t a l k s a t gr-maximal i d e a l s of C . The l a t t e r s t a l k s are graded f i e l d s of the form k[ X,X-
1
]
a n d hence the study of the
graded Brauer groups of C i n the sense of [ 1171 reduces t o the study of t h e graded Brauer groups of graded f i e l d s f o r which we can r e f e r to [ 1 2 1 .
273
1. Non-Comrnutative Graded Rings
1.6.
E x e r c i s e s , Comments, References.
1. E x e r c i s e s .
1. L e t R be a s t r o n g l y graded r i n g o f t y o e G, where G i s a f i n i t e g r o w . Show t h a t : r a d Re = R e fl r a d R. 2. I f R as i n 1 i s s e m i p r i m i t i v e t h e n Re i s s e m i p r i m i t i v e . The converse h o l d s i f t h e o r d e r o f G i s i n v e r t i b l e i n Re. 3. L e t R be s t r o n g l y graded o f t y p e G, i s semi-prime. Show t h a t
iGl= n <
m
, and
assume t h a t Re
R has a u n i q u e maximal n i l n o t e n t
deal N
and Nn = 0. Moreover, i f Re has no n - t o r s i o n t h e n R i s semiprime.
4. L e t R be s t r o n g l y graded by a f i n i t e grouD G o f o r d e r n. The i n j e c t i v e h u l l o f R R i n R-mod i s graded;; a c t u a l l y E(R),
i s t h e i n j e c t i v e h u l l of
R o i n Re-mod. A p ~ l yt h i s t o d e r i v e t h e f o l l o w i n g : a) The maximal q u o t i e n t r i n g Qmax(R) i s s t r o n g l y graded o f t y n e G.
b ) (QmaX(K)Ie = Qmax(Re) c ) I f R i s a c r o s s e d p r o d u c t t h e n Qmax(R) i s a c r o s s e d p r o d u c t . 5. L e t R be as i n 4. I f R i s a semiprime l e f t G o l d i e r i n g t h e n Re i s a semiprime l e f t G o l d i e r i n g . In t h i s case, t h e c l a s s i c a l r i n g o f q u o t i e n t s o f R , Q say, i s s t r o n g l y graded of t y p e G and Qe = Q(Re). 6. L e t R be as i n 4, and assume t h a t Re i s a semiorime l e f t G o l d i e r i n g t h e n Qmax(R) = E(R) i s a q u a s i -Frobenitis r i n g . Moreover Qmax(R) = S - I R where S i s t h e s e t o f a l l non-zero d i v i s o r s o f Re.
7. L e t R be a graded r i n g o f t y o e Z and c o n s i d e r F1 r e s p . s:(M)
be t h e
R-gr. L e t sR(M)
s o c l e r e s p . t h e graded s o c l e o f
M. Show t h a t
274
C. Structure Theory for Graded Rings of Type Z
8. Let R be a graded ring of type Z / D Z , where p i s a arime number, and l e t M be a graded R-module. Suppose t h a t M contains a f i n i t e number of non-isomorphic simple R-submodules N1'.
.. , N r .
P u t D~ = char Di,
where Di = End R( N i ) , i = 1,... , r , and assume t h a t D # p i f o r a l l i . I f x C sR(M) has homogeneous decomDosition x = xi + ... + x . , then 1 'k p xi
j
c sR(M) f o r a l l
j c
(0,
p-l}.
9. Using 8, prove the following r e s u l t : I f R i s graded of tyae Z, M C R-gr, then sR(M) i s a graded submodule of M. 10. Deduce from 9 a new proof f o r the f a c t t h a t i f M c R-gr i s gr-simple
then M i s e i t h e r a simple R-module o r a 1 - c r i t i c a l R-module. 11. Let R be a graded ring of type Z a n d consider a
projective graded R
module P . Show t h a t the following statements are equivalent :
a ) Homomorphic images of P a r e f l a t . b) F i n ; t e l y generated submodules o f P are direct-summands.
c) For every submodule P ' of P and every r i g h t ideal I of R , I P =
n P'
=
I P'.
If one of these conditions holds then P i s said t o be a gr-regular module. 12. With terminology as in 11. The graded orojective R-module P i s regular
i f P i s isomorphic in R-gr t o a d i r e c t sum I 1 ( n l )
nl,
...,n r
6 Z and I . i s
regular, f o r each j
J
=
@...a I r ( n r )
where
a gr-orinciDal l e f t ideal of R, which i s gr-
l,.. . , r .
1. NonCommotative Graded Rings
13. Let R be a reduced graded r i n g , then a
g r - r e g u l a r module P i s
r e g u l a r i f and only i f i t i s t r i v i a l l y graded. 14. I f M
R-gr then ENDR (M) i s & - r e g u l a r (Von Neumann) i f f o r every
F
f C h ( E N D R ( M ) ) , Ker f and Im f a r e d i r e c t summand of M.
15. Let R be a graded r i n g of type Z . The following p r o p e r t i e s a r e equivalent : a) R
is
gr-semisimple g r - A r t i n i a n .
b) There e x i s t s an i n f i n i t e l y generated ? r - f r e e module F such t h a t E N D R ( F ) i s a Gr-regular r i n g .
16. Let P
C
R-gr be a f i n i t e l y generated g r - r e g u l a r R-module, then
E N D R ( P ) i s a Gr-regular r i n g . 17. Let R be a commutative graded r i n g of type Z . I f P i s a graded
p r o j e c t i v e R-module such t h a t ENDR(P) i s a Gr-regular r i n g , then P i s a g r - r e g u l a r module.
18. For a graded R-orogenerator P . t h e following s t a t e m e n t s a r e
equivalent :
a ) R i s a Gr-regular r i n g b) P i s a g r - r e g u l a r R-module. c ) E N D R ( P ) i s a Gr-regular r i n g . d ) P i s a g r - r e g u l a r ENDR(P)-module.
275
276
C. Structure Theory for Graded Rings of Type Z
2 . Comments.
The theory of graded P . I . r i n g s , i n p a r t i c u l a r t h e p r o j e c t i v e spectrum P r o j may be f u r t h e r s t u d i e d i n a geometrical c o n t e x t c f . t h e theory of non-commutative p r o j e c t i v e a l g e b r a i c v a r i e t i e s i n [ 125 1 . The r e s u l t s in Chapter C may a l s o be used in studying (graded) maximal o r d e r s over graded domains i n p a r t i c u l a r graded Krull domains, c f . [ 18 I , [ 64 1 , both i n t h e P . I . case o r i n t h e more general gr-Goldie case.
If G i s
a t o r s i o n f r e e a b e l i a n group s a t i s f y i n g the ascending chain c o n d i t i o n on c y c l i c subgrouos then a gr-maximal o r d e r A i n some c . s . a . 2 i s a maximal o r d e r over i t s c e n t e r which i s a (graded) Krull domain, c f [
64 1 , hence a non-commutative Krull r i n g i n t h e sense of [ 18 1 .
The notion of a g e n e r a l i z e d Rees r i n g may be extended by using the c l a s s group o f a maximal o r d e r over a Krull domain i n s t e a d of t h e Picard group o f a maximal o r d e r over a Dedekind domain. These r e c e n t developments may be found i n : L. Le Bruyn, F. Van Oystaeyen, [ 64
I.
The notions of GZ-rings ( o r ZG-rings) and g r - b i r a t i o n a l i t y may a l s o be a p p l i e d t o non-commutative p r o j e c t i v e geometry, cf [ 125 I , and a l s o the theory of g r - o r d e r s over gr-Dedekind r i n g s because a gr-maximal o r d e r ( o v e r a gr-Dedekind r i n g )
i n some c , s . a .
t u r n s o u t t o be a GZ-ring.
W e did n o t i n c l u d e t h e s e a p p l i c a t i o n s here but we hope t h a t t h e r e f e r e n c e s above e s t a b l i s h c o n t a c t w i t h some o f t h e up-to-date r e s e a r c h in t h i s a r e a .
277
1. Non-Commutative Graded Rings
3. References f o r Chapter C. G . Cauchon [ 17 1 ; [
M. Chamarie
39 I ; I . Herstein [
[ 18
1;
E. Formanek [ 28
I
; A. Goldie
49 1 ; L . Le Bruyn, F. Van Oystaeyen [ 64 1 ;
C. Nxstxsescu, F. Van Oystaeyen [ 82 1,184 I ; C. Procesi [ 90 I ; Y. [
Razmyslov [ 9 1 I ; L. Rowen [ 96 1 ; F. Van Oystaeyen 1110 ] ,[ 115 I , 1181, [ 1191 ; A. Verschoren, F. Van Oystaeyen [ 1241
For Picard groups of o r d e r s we refer t o A.Frohlich [ 311
[
1 2 5 1 , [ 1261.
.
The g e n e r a l i z a t i o n of t h e P i c a r d g r o w introduced by A. Verschoren i n [
1281 allows t h e a p p l i c a t i o n t o o r d e r s over Krull domains mentioned i n
the comments.
CHAPTER D: FILTERED RINGS AND MODULES
I : The Category of Filtered Modules
1.1.
Definition.
rins i f
An a s s o c i a t i v e r i n g R i s sasd t o be a f i l t e r e d
c
t h e r e i s an ascending c h a i n {FnR, n
Z) o f a d d i t i v e subgroups
, o f R s a t i s f y i n g t h e f o l l o w i n g c o n d i t i o n s : 1 t FOR, FnR.FmR c Fn+R f o r any (n,m)
c
Z2. The
f a m i l y o f t h e s e subgroups, denoted by FR, i s
c a l l e d t h e f i l t r a t i o n o f R. Note t h a t t h e d e f i n i t i o n i m p l i e s t h a t FOR i s a s u b r i n g o f R .
I . 2.
Examples.
1" Any r i n g R can be made i n t o a f i l t e r e d r i n g by means o f t h e t r i v i a l filtration
, i.e.
FnR = 0 i f n < 0, FR ,
=
R if n
?
0.
2" L e t I be an i d e a l o f R. The I - a d i c f i l t r a t i o n i s o b t a i n e d by p u t t i n g f o r n < 0. F R = R f o r n 2 0 w h i l e FnR = I-" n
3" L e t R be any r i n g and l e t ,D : R 6 : R
+
+
R be an i n j e c t i v e r i n g homomorphism,
R a s o - d e r i v a t i o n i .e. a group homomorphism f o r t h e a d d i t i v e
s t r u c t u r e o f R such t h a t 6 ( x y ) = f i ( x ) y + x ' ~~ ( y f)o r x,y o f skew p o l y n o m i a l s R [ X , , Q , ~ ] i s o b t a i n e d
from R by a d j o i n i n g a v a r i a b l e
X and d e f i n i n g m u l t i p l i c a t i o n a c c o r i n g t o : Xb = r D X i n such a way t h a t R becomes action o f
a subring o f
F R . The r i n g
+
F(r),
for r
c
R,
R [ X,fop,6 1. (Note t h a t t h e
endomorphisms i s w r i t t e n e x p o n e n t - w i s e ) . The f a c t t h a t
3 0
i s i n j e c t i v e a l l o w s t o i n t r o d u c e t h e t h e degree f u n c t i o n ( d e n o t e d by : deg) i n t h e u s u a l way and so we may d e f i n e t h e degree - f i l t r a t i o n by: FnR [ X , V , ~ ] = { P
I . 3.
c
Definition.
R [X,~,E1 Let R
c a l l e d a f i l t e r e d module
, deg
P
5
n ) hence FOR [ X,,o,h
I = R.
be a - f i l t e r e d r i n q A l e f t R module M i s i f t h e r e e x i s t s an ascending c h a i n {FnM,ncZ}
279
1. The Category of Filtered Modules
of additive subgroups of M such t h a t FnR.FmM c Fn+,,, F" f o r any (n,m) C Z The family {FnFI, n
2
.
c Z} i s the f i l t r a t i o n of M. Obviously, i f R i s a
f i l t e r e d ring RR ( r e s p . R R ) i s a f i l t e r e d l e f t ( r e s p . r i g h t ) R-module. I f FOR = R, then FiR i s a two-sided ideal f o r any i
5
0 and FiM a r e
submodules of M. If M i s a f i l t e r e d l e f t R-module then i t s f i l t r a t i o n F
may have one
of the following properties : ( E ) F i s exhaustive i f M =
FnM.
U
ncZ ( 0 ) F i s d i s c r e t e i f there i s an no c Z such t h a t FiM ( S ) F i s separated
if
n FnM
ncZ
=
=
0 for a l l i
<
no.
0.
I n the f i l t r a t i o n of R i s t r i v i a l and M c R-mod, t h e n any ascending chain of submodules of M defines a
f i l t r a t i o n f o r Fn; i n t h i s case
the t r i v i a l f i l t r a t i o n of M i s 'given by FnM = 0 i f n < 0, FnFn = M i f
n
2
0.
I . 4.
Remarks.
1. I f the f i l t r a t i o n FR of the ring R i s exhaustive then R i s a
topological ring with the family ( F n R ) n c Z
f o r a fundamental system
of neighbourhoods of 0 in R . Moreover, i f the f i l t r a t i o n FM of the module Fn i s exhaustive then M i s a topological R-module with the family (FnMIncZ f o r a fundamental system
of neighbourhoods of 0 in M. 2 . I f the f i l t r a t i o n FR i s exhaustive, then
(ij)
=
n F ~ M
n cZ
i s a submodule o f M.
Indeed, i f h c R and x c
fl FnM, there e x i s t s a p C Z such t h a t X F FpR. nFZ As x c F M then Xx c F R . F M c FnM ( a r b i t r a r y n ) . Therefore n-p P n-P
Xx
c
fi
n FZ
FnM.
280
D. Filtered Rings and Modules
I . 5.
Definition. Let R and S be f i l t e r e d rings. An Y c R-mod-S i s i f there e x i s t s a n ascending chain
said t o be a f i l t e r e d R-S module tFnM, n
<
Z } of additive subgroups such t h a t : FnR.FmM c Fn+,,,M f o r a l l
n,m c Z , and FmM.FtS
c
Fm+tM f o r a l l n , t
<
Z.
Let R be a f i l t e r e d ring, M and N f i l t e r e d l e f t R-modules. An R-homomorphism f c HomR(M,N) i s s a i d t o have degree p i f f(FiM) c Fi+pN f o r Homomorphisms of f i n i t e degree form a s u b g r o u p HOMR(P?,N)
all i c Z.
of HomR(M,N), homomorphisms of degree p form a subgroup FpHOMR(M,N) of H O M R ( M , N ) . One e a s i l y checks : 1. I f p
5
q then : Fq H O M R ( W , N )
2. HOMR(M,N) 3. If f : M
= +
c
Fq HOMR(M,N).
U F HOMR(M,N). pcz N has degree p and g : N
+
P has degree qthen g o f
has degree p t q. These properties allow us t o introduce the category R - f i l t of f i l t e r e d l e f t R-modules where the morphisms a r e the homomorphisms in HOPR(-,-) of degree 0. I f M,N c R - f i l t then FoHOMR(V,N)
HornFR ( M , N ) .
will simply be denoted by
Clearly R - f i l t i s a n additive category and furthermore
i f f c HornFR ( M , N )
then Ker f and Coker f a r e in R - f i l t . Indeed, Ker f
has the induced f i l t r a t i o n Fi Ker f f i l t e r e d by putting Fi Coker f
=
=
Ker f n FiM, whereas Coker f i s
( I m f t FiN)/
Im f . I n p a r t i c u l a r i t
follows t h a t monomorphisms and epimorphisms in R - f i l t coincide with the i n j e c t i v e resp. s u r j e c t i v e morphisms of R - f i l t . Arbitrary d i r e c t sums as well as d i r e c t products e x i s t Indeed, i f ( M i ) i C I
i s a family of objects
in R - f i l t .
of R - f i l t r a t i o n
F ( 8 M i ) = @ F M. i s the d i r e c t sum and the module n Mi with P ic1 ic1 P ic1
’
281
1. The Category of Filtered Modules
the f i l t r a t i o n F ( n Pi) = n F M. i s the d i r e c t product i n t h e P icI icI P
’
category R - f i l t . Moreover, i f ( M i , soij), with i , j in some index s e t I , i s an inductive ( p r o j e c t i v e ) system i n R - f i l t then i t s inductive
9 Mi (lim M i ) ( p r o j e c t i v e ) l i m i t e x i s t s i n R - f i l t ; i t i s the R-module 1 with f i l t r a t i o n F I. 6.
P
li+m Mi = li+m Fp M i ,
( F p lim Mi
=
ljm F M.).
P ’
Remarks.
1. Let {Mi,icI} be a family of objects from R - f i l t such t h a t t h e i r
f i l t r a t i o n s FMi, i c I , a r e exhaustive, then the f i l t r a t i o n of
n
exhaustive. This property f a i l s f o r the d i r e c t product
Mi i s I Mi! The 8
I s i m i l a r r e s u l t however does hold f o r inductive o r projective l i m i t s .
2. R - f i l t i s preabelian ( c f . [ 46 1 ) b u t not abelian. Indeed,
take
M # 0 in R-mod and p u t FiM = 0 f o r a l l i t Z . Let F’M be another f i l t r a t i o n on M and denote
G , resp. H , the f i l t e r e d modules from M using FM resp.
F’M. The i d e n t i t y morphism of M i s in HornFR (G,H)
b u t n o t i n HomFR(H,G)
and thus the i d e n t i t y of M i s a b i j e c t i v e mapping which f a i l s t o be a n isomorphism.
3. I n a s i m i l a r way as before one defines the category R-filt-S f o r a p a i r of f i l t e r e d rings R and S . Before introducing some special functors in R - f i l t l e t us point o u t t h a t , i f R i s a f i l t e r e d ring with f i l t r a t i o n FR, we have a functor D : R-mod
--f
R - f i l t which associates t o M t R-mod the R-module M with
f i l t r a t i o n FnM = FnR.M. filtration.
This f i l t r a t i o n
of M i s c a l l e d the induced
Note t h a t FnD(M) = FnR.M = M f o r any n 2 0 .
Dually, we can define the functor
6
: R-mod
--f
R - f i l t which associates
t o every module M, the module M with f i l t r a t i o n FnM ( n c 2 ) . FnM i s a
=
{xtM,F-nR.x=O}
subgroup and F R.FnM 5 Fp+nM. This f i l t r a t i o n of P
282
0.Filtered Rings and Modules
M i s c a l l e d t h e c o - i n d u c e d f i l t r a t i o n . Note t h a t Fn6(M) = 0 f o r any n 5 0. S i n c e 1 F FOR, t h e c o - i n d u c e d f i l t r a t i o n i s d i s c r e t e .
P a r t i c u l a r case. L e t I be a t w o - s i d e d i d e a l . I f f o r t h e r i n g R we c o n s i d e r t h e I - a d i c f i l t r a t i o n , t h e n F-nD(M) = I n M and Fnb(M) The suspension f u n c t o r . Tn : R - f i l t
R-filt,
+
For n F
=
annMIn f o r any n?o.
Z we d e f i n e t h e n - t h s u s p e n s i o n f u n c t o r
w h i c h a s s o c i a t e s t o M F R - f i l t FM t h e f i l t e r e d
module M(n) w i t h f i l t r a t i o n FiM(n) Suspension i s c h a r a c t e r i z e d by :
= FitnM
Tn o T,
hence FiTn(M) = Tn+m,
= Fitn(M).
To = I d ( t h e i d e n t i t y
p a r t i c u l a r , e v e r y T , i s an e q u i v a l e n c e o f c a t e g o r i e s
functor). I n
w h i c h commutes w i t h d i r e c t
sums, p r o d u c t s , i n d u c t i v e and p r o j e c t i v e
l i m i t s . I t i s e a s i l y v e r i f i e d t h a t t h e f o l l o w i n g h o l d s : f o r p F Z, and f o r M, N
c
F HOMR(M,N) P
R-filt : =
HornFR (M(-p),N)=
The e x h a u s t i o n f u n c t o r .
HornFR (M,N(p)).
L e t R have f i l t r a t i o n FR. I f we p u t R ’ =
t h e n R ’ i s a f i l t e r e d s u b r i n g o f R. I f M F R - f i l t ,
U
ncZ
FnR
then, c l e a r l y
U FnM i s a f i l t e r e d R’-submodule and t h e f i l t r a t i o n FM’ g i v e n by ncZ FnM‘ = FnM f o r a l l n F Z i s an e x h a u s t i v e f i l t r a t i o n . I n t h i s way we
M’ =
obtain a functor : R - f i l t
+
R ’ - f i l t carrying M
M I , which i s c a l l e d t h e exhausting f u n c t o r .
i n t o the corresponding
II. Complete Filtered Modules. The Completion Functor
Let M
c
R-filt.
of elements o f M i s s a i d t o be a
A sequence (mi)iC
Cauchy -sequence i f f o r e v e r y p E mS - mt
c
N t h e r e i s an N(p)
N
such t h a t
F-p M f o r a l l s , t 2 N ( p ) . O b v i o u s l y , i n t h i s d e f i n i t i o n i t
i s s u f f i c i e n t t o have
c
X ~ - X ~ + F-p ~
f o r any s 2 N ( p ) . A sequence
converges t o m 6 M i f f o r e v e r y p
(mi)icN
6
cN
t h e r e e x i s t s N(p) F
M
-
such t h a t mS
m C F- M whenever s 3 N(p). P I f t h e f i l t r a t i o n FM i s s e p a r a t e d t h e n a l l s e q u e n c s c o n v e r g e t o a
unique l i m i t .
11.1.
A f i l t e r e d module M C R - f i l t i s s a i d t o be complete
Definition.
i f FM i s s e p a r a t e d and a l l Cauchy sequences converge i n M.
Remarks.
11.2.
1. O b v i o u s l y M
c R-filt
i s complete i f and o n l y i f TnM i s complete,
n c
Z.
2.
I f t h e f i l t r a t i o n o f M i s d i s c r e t e , t h e n M i s complete.
3. A c c o r d i n g t o d e f i n i t i o n Sum
Of
11.1. one deduces t h a t any f i n i t e d i r e c t
c o m p l e t e m o d u l e s i s a l s o a complete module.
Now l e t M
c
R - f i l t and c o n s i d e r t h e p r o j e c t i v e system
M/F M, p z q . q L e t us p u t M = l i m M/F M. I f n = M --t M/F M a r e t h e c a n o n i c a l morphisms P P P we n o t e F M = K e r n Since n are surjective then the applications P P’ Pq n a r e s u r j e c t i v e . I f up : M --t M/F M a r e t h e c a n o n i c a l morphisms, t h e n P P a l l elements o f M have t h e f o l l o w i n g form : 5 = ( u ( x ) ) p F Z w i t h P P x C M and xp - xp+l c F M f o r a l l p E Z. P P+ 1 Then F fi = 15 = ( u ( x ) ) P p p p e 2 ’ xp FpM). I t f o l l o w s t h a t F M = l j m F M/F,M. P r-p tM/FpM, npq:M/FpM
--t
284 Let
0. Filtered Rings and Modules pM
: M
fi
+
be t h e c a n o n i c a l morphism : D ~ ( x )= ( U ~ ( X ) ) ~ ~ ~ .
O b v i o u s l y , np o oM = u
for all p
The f a m i l y o f subgroups { F
P
h}PCZ
c
2.
i s a f i l t r a t i o n f o r t h e a b e l i a n group
11 .3. P r o p o s i t i o n . The a b e l i a n group
fi
fi.
w i t h f i l t r a t i o n t F p ~ ) p C zhas
the following properties : 1. Ker p
=
n
P a r t i c u l a r l y FM i s s e p a r a t e d i f and o n l y i f oM i s
FiM.
icz
injective. 2. I f t h e f i l t r a t i o n FM i s e x h a u s t i v e t h e n t h e f i l t r a t i o n FM i s exhaus t i ve. 3. F o r a l l p
4.
M
c
Z, F
-
P+l
-
M/ F M = Fp+lM/ P
FpW.
i s a complete a b a l i a n group.
5 . vM(M) i s dense i n M. 6. The module M i s complete i f and o n l y i f
i s an isomorphism.
pM
Proof. 1. Obvious. be an element o f M I f pocZ, f o r t h e element x 2. L e t 5 = ( u ( x ) ) P P PEZ PO c FkM. I f k 5 po t h e n x C F M t h e r e e x i s t s a k c Z such t h a t x PO Po Po and t h e r e f o r e 5
5
c
F
PO
i. I f
po 5 k, we g e t x
k
c
F M and c o n s e q u e n t l y k
C FkM.
3. L e t 5 =
be an element o f
h. T h e r e f o r e x P+ 1 F Fp+lM. P+l L e t us d e f i n e t h e a p p l i c a t i o n
F M, t h e n x c Fp+lM. P P+l P+l P ( u p ) . One may e a s i l y deduce t h a t t h i s FpM + F p + lM/FpM,
Since x - x Fp+@
F
--f
a p p l i c a t i o n i s an isomorphism. be a Cauchy sequence o f M. T h i s n F M = 0. L e t (
4. O b v i o u s l y
II. Complete Filtered Modules
qi+l - qi
c F- 1. M f o r any i
such t h a t 5,
N1
<
N2
- 5, c
... .
<
285
0. Indeed, there e x i s t s a natural N i
2
F-; M
for r,s
Let q i =
sNi;the
Ni.
2
We get the ascending sequence
sequence ( q i ) i 2 1 s a t i s f i e s the
required condition. Therefore i n order t o show t h a t M i s complete i t will be s u f f i c i e n t t o es t ab l i s h t h a t a l l sequences (
- xi c
So xi P+l
P
Fp+l M, v p F Z v i
1. As
2
ci+l
-
ci
F F-i
M, then
i+l- i M, t h e n for p = -i-1 x - ~F F-i M f o r a l l i.1. As x i + l - xi+' c F xTi P+ 1 P+t pi+l i+l i+l we get x - ~ - x - ( ~ + ~ F) F-i M and so x!i - x - ( ~ + ~c ) F-i M. i 1 If we denote 5 = ( y i ) i e z where ybi = u - ~ ( x - ~ ) f o r i z l a n d yi = u ~ ( X - ~ )
f o r j20 i t follows t h a t 5 c M. On the other hand, f o r i z 1, therefore
5 =
ei.
lim
j i -
5 - ci F
F-i M.
Consequently M is complete.
be a n element of M. Let us denote Ei 5 . Let 5 = ( u ( x ) ) P P P@ f o r a l l i21. As n-i (5 - c i ) = 0 one deduces 5 - si F F-i61, 5
=
lim 5.. consequently mM(N) i s dense i n i-t-
'
=
,oM(xYi) therefore
M.
6 . I f M i s complete, then FM i s separated, a n d so ,pM i s i n j e c t i v e . Let
E C 6l; a s coM(M) i s dense i n
b,
xn F M,
-
and
'pM
such t h a t
5
=
lim
n-
i s i n j e c t i v e , then
then there e x i s t s a sequence ( x ~ ) ~ ? ~ , ,pM(xn).
( x ~ ) , , ? ~i s a Cauchy sequence of M . As M i s
complete, there e x i s t s an x = lim
x n . Then 5
n--
'pFI
Since 'oM (FiM) = Fi M n ",,,(M)
= ,pM(x)
and therefore
i s su r j e c t i v e .
Let us assume now t h a t the f i l t r a t i o n of the ring R i s exhaustive. I f M c R - f i l t and FM i s exhaustive, then M i s a topological R-module (Remark 1 . 4 . ) . I n t h i s case M Sense
. Since
R x M
--f
k, ( a , x )
i s the completed of M, --f
in
the topological
a x i s continuous,
i t may be extended t o a continuous application : Rx
i
+
h.
Since
D. Fiftered Rings and Modules
286 F.R. F.M 1
c
3
F . .M, we g e t F. k . F . M c Fi+j J+' I J
M. I f M
=
R
R , then
k
is a
FI i s a l s o a f i l t e r e d R-module
f i l t e r e d r i n g w i t h f i l t r a t i o n {Fik}icz. w i t h f i l t r a t i o n {FiM}. 11.4. Remarks. 1. I f we denote M'=
n F
pcz
p
M, t h e n c l e a r l y M/W'
i s separated f o r t h e q u o t i e n t
f i l t r a t i o n and M=M/M'. L e t us assume now t h a t M has s e p a r a t e d f i l t r a t i o n . A c c o r d i n g t o P r o p o s i t i o n I I . 3 . , t h e completed M may be o b t a i n e d by C a n t o r ' s method, t h a t i s by means o f c l a s s e s o f e q u i v a l e n t Cauchy sequences.
2. I f FOR = R, t h e n t h e m u l t i p l i c a t i o n on R may be d e f i n e d as f o l l o w s : i f 5, q
c R w i t h E,
t h e n E, q = (u (a ) ) and q = ( u ( b ) ) P P Pcz P P PCZ = ( u p ( a p b p ) ) p c Z . The subgroups F R, pcZ, a r e t w o - s i d e d i d e a l s o f k. P Moreover, i f FoM = M, t h e n t h e module s t r u c t u r e o f M i s g i v e n by t h e =
following e q u a l i t y : i f u = (u
E
P
(m ) ) P
be t h e c a t e g o r y of f i l t e r e d modules w i t h e x h a u s t i v e f i l t r a t i o n ;
then t h e a p p l i c a t i o n M
+
M defines a functor c : R ' - f i l t
w h i c h i s s a i d t o be t h e c o m p l e t i o n (If f : M
+
N
N
--f
functor.
i s an isomorphism o f t h e c a t e g o r y R ' - f i l t , c ( f )
i s the f o l l o w i n g morphism:if =
i s an element o f Y , t h e n :
( u (a m ) ) P P P P
a =
Let R ' - f i l t
+(<)
PCZ
(np(f(xp))pcZ).
=
(nMP ( x P ) ) PCZ
i s an element o f
i-filt
-
-
= f :M
then
-
+
N
287
II I: Filtration and Associated Gradation
L e t R be a f i l t e r e d r i n g G(R) =
8
ic Z
FiR/Fi-lR,
,M c
R-filt.
F(M) = .8 1
Consider t h e a b e l i a n groups :
M. I f x c F P t h e n x denotes P P
FiM/Fi-l
cz
= FpP/Fp-lM. I f a c FiR, x C F . P t h e n P J d e f i n e a . x . = ( a x ) . . and e x t e n d i t t o a Z - b i l i n e a r mapping li. : G(R) x G(M) 1 J 1 +J + G(M). T a k i n g M = R, p makes G(R) i n t o a graded r i n g and i n g e n e r a l G(M) i s
denotes t h e image o f x i n G(M)
t h u s made i n t o a graded G(R)-module.
Let f
c
HomFR(M,N) f o r some
M, N t R - f i l t ;
t h e n f i n d u c e s c a n o n i c a l mappings fi
g i v e n by fi(xi)
= (f(x))i
m o r p h i s n o f G(R)-modules. G : R-filt
D.3.1.
--f
c
+
FiN/Fi-lN.
Z. P u t t i n g G(f) =
8 fi d e f i n e s a ic Z A l l o f t h e s e amounts t o t h e s t a t e m e n t t h a t
G(R)-gr i s a f u n c t o r . We have t h e f o l l o w i n g :
Proposition.
The f u n c t o r G 1. I f M
c
2. I f n
c
M c
has t h e f o l l o w i n g p r o p e r t i e s :
R - f i l t and FM i s e x h a u s t i v e and s e p a r a t e d t h e n M
o n l y i f G(M)
3. I f
for x
: FiP/Fi-lP
=
=
0 i f and
0.
2 and M F R - f i l t ,
t h e n G(TnFI) = Tn(G(M)).
R - f i l t and FM i s d i s c r e t e t h e n G(M) i s l e f t l i m i t e d .
4. I f M C R - f i l t and b o t h FR and FM a r e e x h a u s t i v e t h e n G(M) = G ( h ) 5. The f u n c t o r G commutes w i t h d i r e c t sums, p r o d u c t s and i n d u c t i v e
limits. Proof.
A s s e r t i o n s 1, 2, 3, a r e c l e a r . F o r a s s e r t i o n 4 we use P r o p o s i t i o n
11. 3 . . I n o r d e r t o p r o v e s t a t e m e n t 5, l e t
f a m i l y o f o b j e c t s of R - f i l t , We have :
and p u t V
=
8 11 a6J a
be an a r b i t r a r y
.
288
D. Filtered Rings and Modules
Hence G(M) = 8 G(Ma). I n a s i m i l a r way we may e s t a b l i s h t h a t E commutes
acJ
w i t h d i r e c t p r o d u c t s and i n d u c t i v e l i m i t s . 111.2. Remark. an
I f R i s a graded r i n g and C
c
R - o r t h e n we can d e f i n e
e x h a u s t i v e and s e p a r a t e d f i l t r a t i o n on R ( r e s p . P) by means o f t h e (resp. F P = Pi). p 15p
subgroups F R = 8 R . P i2p 1' obtained
L e t us denote t h e
f i l t e r e d r i n g ( r e s p . f i l t e r e d module) by R ' ( r e s p . C'). I t i s
s t r a i g h t f o r w a r d t o p r o v e t h a t G ( R ' ) 2 R, G ( M ' ) 2' M. S i m i l a r l y , t h e subgroups F ' R = 8 Ri P i3-p
(resp. F'M = 0 M i ) d e f i n e a P iz-p
f i l t r a t i o n on R ( r e s p . If). D e n o t i n g by R" ( r e s p . M " ) t h e o b t a i n e d f i l t e r e d r i n g ( r e s p . module) t h e n
a g a i n G(R") 2' R and G(M")
M.
I n s t u d y i n g t h e e f f e c t o f I: o n morphisms we need : 111.3.
D e f i n i t i o n . L e t M,
N c
R-filt.
A f i l t e r e d morphism f : M + N
i s s a i d t o be s t r i c t i f f ( F M) = I m f n F N f o r each p c Z. P P f A sequence L -M -.!+ N i n R - f i l t i s s t r i c t e x a c t i f i t i s an e x a c t sequence i n R-mod such t h a t b o t h f and g a r e s t r i c t morphisms i n R-filt. I t i s c l e a r t h a t a f i l t e r e d morphism f : C
and o n l y i f f i s b i j e c t i v e and f 111.3. Theorem.
+
i s strict.
L e t R be a f i l t e r e d r i n g and c o n s i d e r t h e f o l l o w i n g
0-sequence i n R- f i l t :
(*)
L
4M
N,
as w e l l as t h e sequence G(*) i n F ( R ) - g r : G(L)
G(f)
N i s an isomorphism i f
)
G(M)
G(9L
F(N).
Then : 1. I f (*) i s s t r i c t e x a c t t h e n G(*) i s e x a c t .
111. Filtration and Associated Gradation
289
I f G ( * ) i s e x a c t and FM i s e x h a u s t i v e t h e n g i s s t r i c t .
2.
3. I f G(*) i s e x a c t w h i l e FL i s complete and FC i s s e p a r a t e d t h e n f i s strict.
4. 1.f G (*) i s e x a c t
and FP i s d i s c r e t e t h e n f i s s t r i c t . FM i s e x h a u s t i v e and separated, o r i f FF? i s e x h a u s t i v e
5. I f L i s complete,
and d i s c r e t e t h e n (*) i s e x a c t . Proof. 1. C l e a r l y G(g) o G ( f ) = 0. Look a t an x c F F1 w h i c h i s such t h a t G(g)x =O. P P T h i s says ( g ( x ) ) = 0 and t h e r e f o r e g ( x ) t Fp-l N . Now s i n c e g i s s t r i c t P
V such t h a t g ( x ) = ~ ( x ’ )i . e . x - x ’ = f ( y ) must e x i s t an x ’ F F P- 1 w i t h y f F L, b u t t h e n G ( f ) ( y p ) = xp i . e . Im G(f) = K e r G(g). P there
2 . L e t y c F N n Img and y F N . There e x i s t s an x F P such t h a t P P- 1 P for g ( x ) = y and s i n c e FM i s e x h a u s t i v e we may assume t h a t x 6 F P+S
some s 2 0. I f s = 0 t h e n we a r e done. I f s > 0 t h e n G ( g ) ( x the f a c t that Z C F
P+S
L. Thus
i s exact implies t h a t x
x
-
f ( z ) C Fp+s-l
P and
P+S
=
G(f)(zp+s)
y = g ( x ) = !(x-f(
2))
=
g(x’) with
R e p e a t i n g t h i s p r o c e d u r e we f i n a l l y Get t h a t t h e r e i s an
x ‘ c Fp+s-lM. li.
G(*)
) = 0 and P+S f o r some
c F Y such t h a t y P
=
G (u).
3. Take y F F M n I m f . Exactness o f G(*) y i e l d s : G ( g ) ( y p ) = g ( y p ) = 0, P M t h e r e f o r e y = G ( f ) ( x p ( p ) ) f o r some x ( ~ )c F L. Hence y - f ( x ( ’ ) ) F Imf n F P P P- 1 By i n d u c t i o n we f i n d a
c F
x(’-’) y
P-s
-
X(~-’),...,X(~-’)
sequence
L such t h a t :
f(x(P)) -
. .. -
f(x(P-’))
c Imf n F ~ - ~ - ~ Y .
Completeness o f FL a l l o w s t o d e f i n e x = Hence y
-
with
f(x) = y - l i m S-t-
f(x(p)
+
b
s=o
x(~-’)
+
x ( ~ - ’ ) 6 F 1. P
... +
s i n c e FM i s separated, and i t f o l l o w s f r o m t h i s
x ( ~ - ’ ) ) = 0,the l a t t e r t h a t y c f ( F L). P
D. Filtered Rings and Modules
290
F
M fl I m f i s o b v i o u s . P
That f(FpL) c
4. Along t h e l i n e s o f 3. 5. S t r i c t e x a c t n e s s o f (*) i m p l i e s e x a c t n e s s o f G(*)
< M,
suppose t h a t G(*) i s e x a c t and l e t y
because o f 1. Conversely
y # 0 be such t h a t g ( y ) = 0.
and y , I! F f o r some p c Z . P- 1 We o b t a i n t h u s : G ( g ) ( y ) = 0 and so y = C . ( f ) ( x p ( p ) ) = f ( x ( p ) ) f o r P P P some x ( p ) 6 F L. S i n c e FM i s e x h a u s t i v e we have y : F
P
-
Consequently : y
< F
with x(~-’)
P-S
P
M
f(x(p)) 6 F P. By i n d u c t i o n we o b t a i n X ( ~ ) , . . . , X ( ~ - ’ ) P- 1 L and such t h a t :
I f FM i s d i s c r e t e t h e n f o r some i n d e x s, Fp-s-l y = f(x(’)
+
... +
we may t a k e x =
?
s=o
V = 0; hence
and t h u s (*) i s e x a c t . I f FP i s complete t h e n x ( ~ - ’ ) and g e t y = f ( x ) . The f a c t t h a t f and g a r e
s t r i c t morphisms w i l l f o l l o w f r o m 2 . and 3 111.5.
Corollary.
Let f : M
--f
N be a morphism i n R - f i l t and suppose t h a t
FF: and FN a r e s e p a r a t e d and e x h a u s t i v e , w h i l e FC i s a l s o complete. Then : G ( f ) i s b i j e c t i v e i f and o n l y i f f i s an isomorphism i n R - f i l t .
111.6.
Corollary.
L e t f c HornFR ( # , N ) .
I f t h e f i l t r a t i o n FV i s s e p a r a t e d
and e x h a u s t i v e t h e n : G ( f ) i s i n j e c t i v e i f and o n l y i f f i s i n j e c t i v e and s t r i c t . 111.7. C o r o l l a r y .
L e t f f HornFR (M,N).
hypotheses i s s a t i s f i e d 1)
M
Assume t h a t one o f
the following
:
i s complete, FN i s s e p a r a t e d and e x h a u s t i v e .
2 ) FF: i s d i s c r e t e and e x h a u s t i v e . Then : G ( f ) i s s u r j e c t i v e i f and o n l y i f
f i s s u r j e c t i v e and f i s s t r i c t .
29 1
111. Filtration and Associated Gradation
Now c o n s i d e r a morphism f :
M
+
N o f t h e category R - f i l t .
f i s said t o
i f f o r every n F Z there e x i s t s a natural
have t h e A r t i n - R e e s p r o p e r t y number h ( n ) F Z such t h a t : f(FnM)
3 Im
n
f
Fh(,,]N.
One can see t h a t i f f i s s t r i c t t h e n f has t h e A r t i n - R e e s p r o p e r t y .
111.8. Theorem.
Consider i n R - f i l t t h e exact
:
sequence
b
L f M
where t h e f i l t r a t i o n s FL, FM and FN a r e e x h a u s t i v e . I f t h e morphisms f and g have t h e A r t i n - R e e s p r o p e r t y t h e n t h e sequence
-
f
L-
M- 9 - i
i s exact. Proof.
F i r s t we c o n s i d e r t h e case where L i s a submodule
i n d u c e d by M and N = V/L w i t h q u o t i e n t f i l t r a t i o n , FiN
=
FiM
+
( i F 2 ) . Thus
L/L
that i s
we have t h e e x a c t sequence
A FI/L--$O.
M
O - L L
with f i l t r a t i o n
Since t h e p r o j e c t i v e l i m i t functor i s l e f t exact, i t i s s u f f i c i e n t t o prove t h a t
i s s u r j e c t i v e . We have
I?
:M
+
xs = y,. where u1 F
t 1
+
F M i s t h e c a n o n i c a l morphism). L e t s F 2 and denote P we may w r i t e y s - l - y, = ul+vl S i n c e ys-l - ys F L + FsM , M /L
L
and v1
c
FsM. L e t xs-l
where u2 c L, v2 F Fss1P. xs-2 - xs-l
lim(M/L + Fi).
,p F Z) be an e l e m e n t o f MIL.
- x1 F FsM. S i n c e yS-*
xs-l
= -
L e t q = (n;lN(yp) (fN
M/L
= v2 € FS-IM.
-
= ys-l
Y,-~
c
L e t xs-2 =
-
ul.
L + Fs-lM,
It i s then
Y ~ - ~ - U ~ - UI t ~ .i
clear that
Y,-~
-
Y,-~
= u 2+v 2
s clear that
F o l l o w i n g t h i s p r o c e d u r e we can
c o n s t r u c t f o r e v e r y t 2 s t h e element xt w i t h t h e p r o p e r t y x ~ - ~xt - F FtM and “(x,)
5
=
= yt.
I f f o r every t
z
s we l e t x t = x,
t h e n t h e sequence
i s an element o f M. C l e a r l y , r i ( c ) = q.
292
D. Filtered Rings and Modules
Consider the following diagram :
91.V/Ker g
f2 f 1 > L/Ker f d Im f = Ker g -M f 3
L
where f
=
f 3 o f2 o f l , g
=
g 3 o g2 o gl;
f l , g1 a r e the canonical
s u r j e c t i o n s ; f 2 , g 2 are b i j e c t i v e and f 3 , g,
a r e the canonical i n j e c t i o n s .
The f i l t r a t i o n s of L/Ker f and M/Ker g are Fi(L/Ker f ) Fi(M/Ker g )
=
FiM
Fi (Im f ) = Im f
9
=
9, 0-
o
9,
-
o
9,
Ker g
=
FiL + Ker f/Ker f ,
+
Ker g/Ker g . The f i l t r a t i o n s o f I m f a n d Im g a r e
n
FiM,
Si ( I m g )
and have t h a t fg
- --+
M
=
-
-
Im g r l FiN. Me have : f = f 3 o f 2 o f I
the sequence :
M/Ker g j 0
i s exact, f l i s s u r j e c t i v e and j, i s i n j e c t i v e . Since f and g have the Artin-Rees property, f 2 and g2 a r e isomorphisms of topological modules a n d therefore f 2 and
i - L M - L N
g2
a r e isomorphisms. Hence the sequence i s exact.
-
+ 92 1%g 93
N
293
IV: Free and Finitely Gemrated Objects of R-filt
L e t R be a f i l t e r e d r i n g , t h e n L R-mod and has a b a s i s (xi)icJ
c
i f it i s free i n
c o n s i s t i n g o f elements w i t h t h e p r o p e r t y
t h a t t h e r e e x i s t s a f a m i l y (ni)iFJ R.xi F L = Z F P i c J P-ni
R-filt i s filt-free
o f i n t e g e r s such t h a t :
8 Fp+. R . x . . icJ 1 Note t h a t f o r any i c J , xi c Fn.L and xi f Fni-lL. =
1
We say t h a t (xi,
ni)ie
i s a f i l t - b a s i s f o r L.
N e x t lemma f o l l o w s from t h i s
v .-
d e f i n i t i o n and Theorem 111.4.
IV.l.
Lemma.
L e t R be a f i l t e F e d r i n g , L
1" L i s f i l t - f r e e w i t h f i l t - b a s i s (xi,
c
R-filt,
i f and o n l y i f L 2' 8 R(-ni).
ni)ieJ
2" I f L i s f i l t - f r e e w i t h f i l t - b a s i s (xi,
then :
ni)icJ
ic J is a free
t h e n G(L)
graded module i n G(R)-gr w i t h homogeneous b a s i s {(xi)?,
icJ}.
3" I f G(L) i s a f r e e o b j e c t i n G(R)-gr w i t h homogeneous b a s i s {(xi)ni,icJ} and i f FL i s d i s c r e t e , t h e n L i s f i l t - f r e e i n R - f i l t w i t h f i l t - b a s i s
nii)icJ' 4" I f M c G(R)-gr i s f r e e graded t h e n t h e r e e x i s t s a f i l t - f r e e L c R - f i l t such t h a t G(L) '2
P.
5" L e t L c R - f i l t be f i l t - f r e e w i t h f i l t - b a s i s (xi,ni)ieJ, If fi {xi.
icJ}
+
M i s a f u n c t i o n such t h a t f ( x i )
let HFR-filt.
F Fs+,,,M,
then there i s
1
a u n i q u e f i l t e r e d morphism o f degree s, g:L
+
M w h i c h e x t e n d s f.
6" L e t M F R - f i l t and suppose t h a t L i s f i l t - f r e e .
I f g:G(L)
+
G(M) i s
a graded morphism o f degree s t h e n t h e r e i s a f i l t e r e d morphism f : L o f degree s such t h a t G ( f ) = g.
+
P1
294
D. Filtered Rings and Modules
c
7" L e t L
R - f i t be f i l t - f r e e ,
and o n l y i f FR
s e x h a u s t i v e ( s e p a r a t e d ) . I f FR i s d i s c r e t e and
c Z, i c J }
ini
t h e n FL i s e x h a u s t i v e ( s e p a r a t e d ) i f
i s bounded below t h e n FL i s d i s c r e t e . I f J i s f i n i t e and
FR i s complete t h e n FL i s complete. 8" I f FR i s e x h a u s t i v e and M
c
R-filt is
then t h e r e i s a f r e e r e s o l u t i o n o f
(*)
...+
L2-
f2
fl
L1
such t h a t FM i s e x h a u s t i v e ,
P in R-filt.
,Lo
fo> K + O
where e v e r y L . i s f i l t - f r e e and e v e r y f . i s a s t r i c t J J
morphism. P o r e o v e r
i f FR and FM a r e d i s c r e t e t h e n we assume t h a t e v e r y FL., j 3 0, i s a l s o J
discrete.
9" I f FR i s e x h a u s t i v e and complete and i f G(R) i s l e f t N o e t h e r i a n and G(M) b e i n g g e n e r a t e d as a l e f t graded R-module t h e n we may s e l e c t t h e L . i n t h e r e s o l u t i o n (*) t o be f i n i t e l y g e n e r a t e d t o o . J Note t h a t an M c R - f i l t i s s a i d t o be f i n i t e l y g e n e r a t e d i f t h e r e i s a f i n i t e f a m i l y (xi,
ni)ieJ
w i t h xi
c
M and ni
or filt-f.g.
c
Z, such
that F P = F R.xi P i = l P-ni
IV.2.
Remarks.
1" If L c R - f i t - f r e e as w e l l as f i n i t e l y g e n e r a t e d i n R-mod t h e n R i s fiIt-f.g. 2" I f M
c
R - f i t i s such t h a t FP i s e x h a u s t i v e t h e n M i s f i l t - f . g .
if
and o n l y i f t h e r e e x i s t s a f i l t - f r e e L w h i c h i s f i n i t e l y generated, and a s t r i c t epimorphism n : IV.3.
L
+
V.
P r o p o s i t i o n . L e t R be a f i l t e r e d complete r i n g and l e t 11 C R - f i l t
be such t h a t FM i s s e p a r a t e d and e x h a u s t i v e . Then V i s f i l t - f . g . i f o n l y i f G(M) i s f i n i t e l y g e n e r a t e d as a graded G(R)-module.
Furthermore,
i f G(M) may be g e n e r a t e d by n homogeneous elements, t h e n 11 may be
g e n e r a t e d as an R-module by l e s s t h a n n g e n e r a t o r s .
and
295
IV. Free and Finitely Generated Objects of R-filt
Proof.
i t follows that, i f i4 i s f i l t - f . g ,
From t h e remark I V . 2 . 2 "
then
G(M) i s f i n i t e l y g e n e r a t e d . Conversely, l e t G(M) be c e n e r a t e d by homogeneous g e n e r a t o r s x ( ~ ) , 1 5 i 5 n. C o n s i d e r L c R - f i l t , L f i l t f r e e Pi . D e f i n e f : L -t Fq by f(yi) = x ( j ) . S i n c e w i t h f i l t - b a s i s (yi, pi)i G ( f ) : G(L)
+
G(F1) i s an epimorphisrn,
Theorem 111.4.5. y i e l d s t h a t
f i s a s t r i c t epimorphism and t h e n Remark IV.2.2" f i l t - f . g and g e n e r a t e d by IV.4.
Corollary.
L e t I1
c
as an R-module. R-filt,
entails that
is
u
such t h a t FFI i s s e p a r a t e d and e x h a u s t i v e
and suppose t h a t FR i s complete. I f 6 ( P ) i s l e f t N o e t h e r i a n as an o b j e c t o f G(R)-gr,
I 5 t h e n 14 i s l e f t N o e t h e r i a n t o o . Moreover, we have : K.dim P R
K.diyR)G(V).
I f K.dimRC = K.dim
G(R)
G(V) = a and E ( V ) i s a - c r i t i c a l ,
t h e n M i s an a - c r i t i c a l R-module. Proof.
L e t N be a submodule o f P I equipped w i t h t h e i n d u c e d f i l t r a t i o n ,
F N = F M n N. S i n c e G(N) c G(ES), F(N) i s f i n i t e l y g e n e r a t e d . P r o p o s i t i o n " P I V . 3 . e n t a i l s t h a t N i s f i n i t e l y generated, so M i s l e f t N o e t h e r i a n . Consider submodules N and P o f P, N c P, b o t h equipped w i t h t h e i n d u c e d { x ( ~ ) ,1 5 i 5 n } i s a Pi s e t o f homogeneous g e n e r a t o r s f o r G(N) where x i i ) c N f o r a l l i, 1 5 ; y n , f i l t r a t i o n , and assume t h a t G(N) = E ( P ) . I f
then, a q a i n f r o m p r o p o s i t i o n IV.3.,
we i n f e r t h a t
g e n e r a t e s N as w e l l as P. Thus N = P.
{x(~),1
5 i 5 n }
T h i s y i e l d s K.dimRI.l 5 K.dimF(R)F(!?)
I n o r d e r t o e s t a b l i s h o u r l a s t c l a i m , l e t N # 0, N c
I.4
and f i l t e r P/N
by p u t t i n g : Fp(M/N) = (N + F F)/N. C l e a r l y , we o b t a i n an e x a c t sequence P i n G(R)-gr : O j G ( N ) -G(M)
C(M/N)--+O.
S i n c e G(N) # 0, t h e K r u l l d i m e n s i o n o f G(M/N) i s l e s s t h a n a K.dimR M/N 2 K.dimG(R)C(M/N) c
a.
T h i s shows e x a c t l y t h a t M i s an a - c r i t i c a l module.
,
hence
296
D. Filtered Rings and Modules
IV.5. Corollary. Let R be a complete f i l t e r e d ring such t h a t F(R) i s a l e f t Noetherian ring, then : 1" R i s l e f t Noetherian and K.dim R
5
K.dim G ( R )
i s l e f t Noetherian and K.dim FOR c K.dirn P ( R )
2 ) Fo R
+ 1.
Proof
1. Follows from the foregoing corollary. 2 . Fo R i s a subring of R and i t i s c l e a r l y complete; Corollary IV.4. together with the r e s u l t s of Section A . I I . 5 . IV.6.
submodules of P are closed, i . e . i f N c F then N Consider submodules N
1. I f G ( N )
=
c
o
< R - f i l t with FP being exhaustive. Assume t h a t
Let M
Proposition.
f i n i s h the proof.
=
fl ( N PCZ
+
F H). P
P c Fc, then :
G(P) then N
P . I n p a r t i c u l a r i f the Krull dimension o f
=
G(M) i s well defined then the Krull dimension of C i s defined a n d K.dimRM
5
K.dim
G( R )
2 . I f K.dimRM = K.Jim i s an
G(F).
G(M)
F( R )
= a
and i f G ( M ) i s a - c r i t i c a l then
a - c r i t i c a l module.
3. I f G ( M ) may be generated by n homogeneous generators then F? may be generated by n generators. Proof.
1. Take x F P , then x c FiM f o r some i , x d F i - l P . Xi c G(P)i = G(N)i = ( P there i s a y1 c N
n
FiM) + Fi-lb'/Fi-l
Pl = ( H
Because :
n FiFI) + Fi-lM/Fi-lF Hence x c y1
n F i F such t h a t x-y1 c Fi-lM.
+
PnFi-l!l.
Repeating t h i s process we and up with elements y l , y 2 , . . . , y o < N such
t h a t x-(yl + y p +...+ys) i s i n Fi-s M n P , s Thus x c N
N
=
+
Fi-,F f o r s
P follows.
>
0. I t follows t h a t x
0.
c n (N+F M ) pcz
= N , whence
I V. Free and Finitely Generated Objects of R-filt
2 . Proceed as i n the proof of Corollary.
x ( ~ ) 1, c i 5 n } be a family of homogeneous aenerators f o r Pi G(M). Write rl' f o r the submodule of M generated by the elements x ( ~ ) ,
3. Let
{
1 5 i 5 n . Obviously x ( i ) 6 E ( M ~ )
Pi
Pi
=
(171
n F M)/(MI n F ~ , - ~ M ) . Pi 1
Therefore G ( M ' ) = G ( M ) and by 1. t h i s y i e l d s PI'
IV.7.
=
!I.
Remark. The above proposition i s generally applied i n case F!
i s exhaustive
and d i s c r e t e .
297
298
V: The I-adic Filtration
The I-adic f i l t r a t i o n i s very important f o r the study of
commutative
rings ( s e e [ 0 . 1 I ) . I-adic non commutative f i l t r a t i o n i s useful, f o r example in the
The study
of Lie Algebras
or f o r the study of the integral group ring
Z(G) where G i s a nilpotent group ( s e e exercises).
Let R be a r i n g , I a n ideal of R, Fc c R-mod. Using the I-adic f i l t r a t i o n as introduced before, we obtain the I-adic completion b o f P a s
fi
=
1jm M/I~M.
n Let i : M
+
M be the canonical f i l t e r e d morphism, iInC)n20is the
f i l t r a t i o n of
i.
I n case M
=
R- l i n e a r
I\ '\
c
t h a t the diagram
R>M-
: R b~~
B!
+
i s a ring homomorphism,
as an R-bimodule.
by means of which we regard The
R, then R
--f
El, a(5
C% m) =
5 i(m) has the property
: j
RR
F1 a
where j ( m )
=
160 m
i s commutative. We s h a l l now present some of the important properties of the I-adic filtration.
1" I f M i s of f i n i t e type then a i s s u r j e c t i v e . I n p a r t i c u l a r and M i s a n ;module of f i n i t e type. Indeed, there e x i s t s a f r e e module L of f i n i t e type such t h a t LLM-+O
(TT
surjective)
i(F1) =
299
V. The I-adic Filtration
We have t h e comnutati ve diagram
i s surjective, since
The morphism n
TI
i s strict.
As t h e completion f u n c t o r comnutes w i t h f i n i t e d i r e c t sums,
then
p
i s an isomorphism. Hence u i s s u r j e c t i v e . 2.
?
i s an i d e a l and
if E
t o proof t h a t
Let
5.1 c
i
0. Indeed, i t i s s u f f i c i e n t
E ~ E~, ,...,5,
c
n I
.
R + R / I p i s t h e canonical morphism). Since P .P P a (nP: i i then a; c I f o r 1 i i c n. On t h e o t h e r hand, s i n c e a2 - al C I
i
F I.B u t a3
a;
i
therefore
c1
-
i
2
.
i
a2 F I y i e l d s a3 F I, 1 s i 5 n. F i n a l l y , 1
F I, f o r a l l 1 5 i 5 n. Hence an
we o b t a i n an
7
2
= (n ( a ' ) )
we have
3.
in c Inf o r a l l n ~ E~ , ,...,5, F I , then
c2...En
6
n I
series 1 + E
^I.
+ 52
As gn F t
a:
C In and
.
i s contained i n t h e Jacobson r a d i c a l o f
Indeed l e t 5 F
2 a,...
in c In , lim
... converges.
n+m
i.
En = 0 and t h e r e f o r e t h e
We have ( l - < ) - '
= 1t
+ 52
t
and t h e r e f o r e 5 belongs t o t h e Jacobson r a d i c a l .
4. I f I i s o f f i n i t e type as a l e f t and as a r i g h t i d e a l , then : i )
in = i
=
i
f o r a l l n 2 0. I n p a r t i c u l a r
i i ) I f M i s an R-module o f f i n i t e t y p e t h e n : I"M Proof. -__
= I ~ M=
in ti, f o r
a11 n z 0.
For t h e a s s e r t i o n i ) we apply 1.
i i ) Since I n M i s o f f i n i t e type we have :
rn
=
in.
..
300
D. Filtered Rings and Modules
1% =
i
i (I'M)
=
i
I n i(M) = I n
h(M)
=
In
i. i f f o r any
The ideal I i s sa i d t o s a t i s f y the l e f t Artin-Rees property
f i n i t e l y generated l e f t R-module M , any submodule N of M and any natural number n , there
e x i s t s an i n t eg er h ( n )
3
0 such t h a t I h ( n ) M n N c I n N
I n other words I s a t i s f i e s the l e f t Artin-Rees property i f the I-adic topology of N coincides with the topology induced i n N by the I-adic topology of M 5 ) If I has the Artin-Rees property then f o r every module M of f i n i t e
type : Ker i = n
n> 1
InM = {x 6 M, (1-r)x = 0 f o r an a r b i t r a r y r
6
I}
In p a r t i c u l a r , i f I i s contained i n the Jacobson radical of R, then
n I ~ =M 0.
n? 1
Obviously we have the inclusion :
{x
€
M,(l-r)x
=
0 f o r an a r b i t r a r y r
Conversely, i f x c
n
n? 1
I'M,
6
I} c rl
n? 1
I'M.
then because I has the Artin-Rees property,
there e x i s t s a n 6 N such t h a t I n M fl Rx c I(Rx) = I x . Hence Rx c I x and therefore there ex i s t s an r c I such t h a t x = rx; therefore (1-r)x = 0 . 6 ) I f I has the Artin-Rees property and R i s a domain then
n In n? 1
= 0
7) I f R i s a l e f t Noetherian ring, I a n ideal of R sa tisfying the l e f t Artin-Rees property, then the following statements a r e equivalent : i ) I i s contained i n the Jacobson radical o f R . submodules of M a r e
i i ) I f M 6 R-mod i s f i n i t e l y generated then the closed i n the I-adic topology. i ) = i i ) We apply 5 ) i i ) = i ) Let A Hence R
=
A
be a maximal l e f t i d e a l . Assume t h a t I
+ 1.R
= A
+
I(I+A) = A
+
I* = A
+
2
I (I+A)
Q A. Then R =
A
+
I3 =
=
...
I + A. =
A+In =
...
V. The Ii+dic Filtration
Since A i s closed, A = Hence I c A
I7
n? 1
(A
+
301
I n ) and t h e r e f o r e A = R; c o n t r a d i c t i o n .
finishes the proof.
8) L e t R be a l e f t Noetherian r i n g and I an i d e a l s a t i s f y i n g t h e l e f t A r t i n-Rees p r o p e r t y . f u n c t o r t a k i n g M t o M i s exact i n t h e category o f l e f t R-modules
i ) The
o f f i n i t e type. i i ) I f M C R-mod i s f i n i t e l y generated t h e n
iz
@R M.
i i i ) R i s a r i g h t f l a t R-module. Proof. __ For i ) we apply Theorem 111.8. i i ) There e x i s t s an e x a c t sequence i n 0-K-LM-0 where L i s a f r e e module o f f i n i t e type and K i s o f f i n i t e type.
- -
From t h i s we deduce t h e f o l l o w i n g comnutative and e x a c t diagram : R@K->
R@L
11
0-K-
.
!p
i
L-M-0
R@M
ri
0
la
Since p i s an isomorphism and y i s s u r j e c t i v e , i t f o l l o w s t h a t u i s i n j e c t i v e and u i s an isomorphism.
,
Obviously
ii)
iii)
9 ) L e t R be a l e f t Noetherian r i n g . Assume t h a t I i s c o n t a i n e d i n t h e Jacobson r a d i c a l o f R and has t h e l e f t Artin-Rees p r o p e r t y . Then R i s faithfully flat.
Proof. Since
-
i
It
i s s u f f i c i e n t t o prove t h a t
i
BRM = 0 i m p l i e s M = 0.
i s a r i g h t f l a t R-module, i t i s s u f f i c i e n t t o
-
case M = R scRM = 0. Since M/I^M M = I M , t h e r e f o r e M = 0.
E
c o n s i d e r the
M/IM we o b t a i n M/IM = 0, hence
302
0. Filtered Rings and Modules
10) Let R be a r i g h t a n d l e f t Noetherian ring. Assume t h a t I has the
r i g h t and l e f t Artin-Rees property. I f
then the image of a i n Proof. $,(A)
k
i s also a nonzero-divisor.
The homomrphism R =
a c R i s a nonzero d i v i s o r i n R ,
>R,
cpa
(X) =
Xa and
JIa
R
>
R,
ah a r e i n j e c t i v e . Then we apply 8 ) , i i i ) .
Definition.
Let R be a ring and a E R. We say t h a t a i s a normalizing
element of R i f a R = R a
. The
s e t of normalizing elements of R i s
denoted by N ( R ) . Definition.
Let R be a ring. The elements al , a2, ...,a n a re a normalizing
s e t of elements of R i f i )a l E N ( R )
i i ) ai
+
( a l , a 2 ,..., a 1. - 1 ) CN(R)Xal, a 2 ,..., a i - l )
f o r i = 2 ,...n .
The elements a l , a*, ...,a n form a cen t r al i zi n g s e t of elements of R i f i ) a l E Z ( R ) ( t h e cen t er of the ring R ) i i ) ai
+
(al,
...,
a i - l ) c Z(R)/(al. a 2 ,..., a i - l )
for i
=
2 ,...,n .
A generating s e t of a n ideal I of R which i s al so a normalizing s e t of elements of
R i s cal l ed a normalizing
s e t o f generators of I . Similarly
one defines a c e n t r al i zi n g s e t of generators f o r an ide a l. V . 1.
Proposition.
Let R be a l e f t Noetherian ring, I an ideal generated
by a central system, then I s a t i s f i e s the l e f t Artin-Rees property. Proof.
s
C
Let N be a submodule o f a f i n i t e l y generated
N and consider a submodule M ’ of
t h a t M i fl N = I’N.
M
c R-mod. Let
M maximal w i t h the property
This choice makes M/M’ i n t o an essential extension
of N/ISN, where Is (N/IsN)
=
0 . I f we a r e able t o e sta blish t h a t under
t
the circumstances M/M‘ may be annihilated by some I t then I (M/M’) = 0 y ie l d s 1% n N c ISN. Note a l s o t h a t i t i s s u f f i c i e n t t o do t h i s f o r s = 1,
V. The I-adic Filtration
303
because f o r a r b i t r a r y s i t wi l l follow from t h i s case applied to
IN,
...,
instead o f N together with the conjunction of the inclusions
IS-'N
t h u s obtained a t each s t e p . I f I = Rcl a central system, l e t
p :
M
+
+...+ Rc,,
M be the map given by m
Since M i s l e f t Noetherian, there ex i s t s r e N for all k c N
i.e.
Ker pr and N i s an
where (cl
Im pr fl Ker kr = 0 .
an i n j e c t i v e R-morphism M/Ker 1
+
Ker
clm f o r a l l m c M. r+k such t h a t Ker pr = Ker p
However N i s contained i n
pL
pt-'.
Let
pr = 0 .
= 0 , then p gives r i s e to
To prove our original claim
i t will therefore be s u f f i c i e n t t o prove t h a t both Ker p and Ker
kt-l
can be annihilated by l ar g e powers of I . Now c1 being c e n t r a l , Ker i s an R/Rcl-mdule and we apply f o r some p Ker $/Ker
p +
p
induction on n t o deduce t h a t I p Ker p = O
N. Since there i s an i n j e c t i v e morphism
C
is
--t
es s en t i al submodule of M, t he re fore Im
t be the smallest natural number such t h a t
,..., c n )
Ker pm-',for 1 c m
5
:
t , the f a c t t h a t Iq Ker
f o r some q t N willfollow by induction on t .
kt-'
= 0
o
I t i s easy t o see t h a t (exactly as in Proposition V . l . )
V.2. Remark.
one may deduce the following as s er t i o n : Let R be a r i n g and I be an ideal of R which i s generated by elements of N ( R ) ( I i s not necessarily generated
by a normalizing s e t of generators). Then I has the Artin-
Rees property. Proposition.
V.3.
and denote
R
Let R be a r i g h t Noetherian ring and l e t J be a n ideal
= R/J.
i ) I f J C I i s a n ideal and
T -
=
I / J , then the completion of
R
f o r the
adic topology i s i d e n t i f i e d w i t h R/RJ.
i i ) I f I i s generated by a cen t r al i zi n g system ( a l , a*, ..., then
an)
I i s generated by the cen t r al i zi n g system ( i ( a l ) , i ( a 2 ),...,i ( a n ) )
where i : R i s the canonical morphism.
304
D. Filtered Rings and Modules
-P -roof.
in = J + In/J, t h e n t h e completion o f t h e l e f t R-module
i ) Since
t h e I - a d i c topology i s i d e n t i f i e d w i t h t h e c o m p l e t i o n o f
R
R
for
-
f o r the
a d i c- t o p o l ogy . From t h e e x a c t sequence o f l e f t R-modules :
R-0
O j J - R A we o b t a i n
t h e exact sequence o f l e f t R- modules :
0-j-i
-
L
P
i -0
- -
B u t ir i s a r i n g homomorphism. Since J = RJ
ii)I f I = (al)
then s i n c e al
t
i t follows t h a t
Z(K) and i ( R ) i s dense i n
R
-
^
^
R/RJ.
R, i(al)
€ Z(R).
Then we procede by recurrence, t a k i n g i n t o account i ) .
If I i s an i d e a l o f a r i n g R, such t h a t R / I i s l e f t
V.4. Lemma_,
Noetherian and I = ( a ) i s generated by a s i n g l e element o f t h e center o f R, then R i s l e f t Noetherian.
--Proof.
We see t h a t GI(R)
. Since
ring R/I [ X I
r i n g . Hence GI(R) C o r o l l a r y IV.5.)
i s isomorphic t o t h e q u o t i e n t r i n g o f t h e
R / I i s a b e t h e t i a n ring, R / I [ X I
i s a Noetherian r i n g . Since Gi(R)
%
i s a Noetherian GI(R)
(from
i t f o l l o w s t h a t R i s a l e f t Noetherian r i n g .
V.5. Theorem.
L e t R be a r i n g and I an i d e a l w i t h a c e n t r a l i z i n g s e t
of generators
I f R i s l e f t Noetherian, t h e n R i s l e f t Noetherian.
Proof.
Suppose t h a t I = (al,
a*,
...,
an) where al,a2
,...,an
is a
c e n t r a l i z i n g s e t o f generators. The p r o o f i s given by i n d u c t i o n on t h e number o f generators o f I . I f I = (al)
then R
i s l e f t Noetherian by Lemma V.4..
From statement
8) and P r o p o s i t i o n V.3. i t f o l l o w s t h a t t h e r e e x i s t s an exact sequence :
OjRi(al)+
R-R-0
(* 1
305
V. The I-adic Filtration
where
R
=
R/Ral
generators o f I
and t h e i(al) ,...,:(an)
are a centralizing s e t o f
From ( * ) i t f o l l o w s t h a t
R i s l e f t N o e t h e r i a n by t h e
induction hypothesis. ^
^
Hence R/Ri(al) J = Ri(al)
i s a l e f t - N o e t h e r i a n r i n g . I f we denote by J t h e i d e a l
-
-
t h e n f r o m Lemma V.4. t h e J - a d i c c o m o l e t i o n R o f R i s a l e f t
N o e t h e r i a n r i n g . Because J c I we have a commutative diagram
where
U,
i and j a r e t h e c a n o n i c a l morohisms. S i n c e j i s an isomorohism,
i t f o l l o w s t h a t a i s a s u r j e c t i v e rnorohism. S i n c e R i s l e f t N o e t h e r i a n ,
R i s l e f t Noetherian.
306
VI: The Functor HOM&,-).
Filt-projective (-injective) Modules
The p r o o e r t i e s o f HOMR mentioned e a r l i e r make i t i n t o a f u n c t o r
(R-filtf+Z-filt;
and i t i s c l e a r t h a t , by d e f i n i t i o n o f t h e f i l t r a t i o n i s e x h a u s t i v e f o r any M, N C R - f i l t .
i n HOMR(-,-),
F HOMR (M,N)
VI.l.
L e t R be a f i l t e r e d r i n g and l e t M, N E R - f i l t be such
Lemma.
that M
i s a f i n i t e l y generated o b j e c t w h i l e F N i s exhaustive, then
HOMR(M,N) Proof. f
c
=
HomR(M,N).
L e t { x ( ~ ) ,ni}
HornR (F1,N).
be a system o f g e n e r a t o r s f o r F1 and t a k e
There e x i s t s s 6 Z such
that f(x(i))
E Fni+s
N for
any 1 5 i 5 n. I t f o l l o w s f r o m t h i s t h a t f C Fs HOPR(M,N) and hence f E HOMR (M,N).
Note t h a t , i f FN i s n o t e x h a u s t i v e and i f N ' i s t h e
exhaustion o f N then HOMR(M,N) = HOMR (M,N') VI.2.
Remark.
=
HornR ( M , N ' )
I n g e n e r a l HOMR(C,N)
# HomR(M,N). F o r e x a m p l e l e t R be a
d i s c r e t e l y and e x h a u s t i v e l y f i l t e r e d r i n g such t h a t R = FiR
f o r a l l i when
Fi(R) # O . T a k e M t o b e f i l t - f r e e w i t h f i l t - b a s i s {X(~),O} w'ith lcicm, a n d t a k e N = R . Define f : M
+
N by
f(x(i))
= a(i)
f i s p. l e t io 6 Z be such t h a t FiN
i < io -p. VI.3.
hence f ( x ( i ) )
Proposition.
Let
= a(i)
M, N <
where a(i) =
1 FiR.
0 f o r any i
<
I f t h e degree o f
Consider io.
= 0, c o n t r a d i c t i o n .
R-filt,
then :
1. I f FM i s e x h a u s t i v e and FN i s separated, t h e n F HOMq(C,N) i s s e o a r a t e d
2.If M i s a f i n i t e l y g e n e r a t e d R-module and i f FY i s e x h a u s t i v e and FN i s d i s c r e t e , t h e n F HOMR(M,N) i s d i s c r e t e . 3. IfFK i s e x h a u s t i v e and FN i s complete t h e n F HOPR(P,N)
i s complete.
307
Vl. The Functor HOMR(-,-I
Proof. 1. L e t f
;
c n
F HOMR (M,N). Then f(FpP) c +F ,s, N = 0. T h e r e f o r e P P f ( M ) = f p ( U F M) = 0. P P 2. l e t x(’), ..., x ( ~ )be g e n e r a t o r s f o r 11. S i n c e FM i s e x h a u s t i v e , t h e r e i s an ro F Z such t h a t x ( ~ )C Fr M f o r a l l i 0
:‘
{ l ,...,n}. S i n c e FiN
c
0
=
f o r a l l i < no f o r some no F Z, i t f o l l o w s t h a t f o r any f 6 F i HOMR(M,N) where i < no
-
r
0
we have f ( x ( i ) )
=
= 0 i f i c no-ro
0, hence F . HOMR (M,N) 1
3. L e t q c p F Z and c o n s i d e r t h e a r o j e c t i v e system HOMR (M,N)
HOMR(M,N)/FDHOMR(Fn,N) ‘Ip/
Tl
pq
HOMR(P,N)/Fq
We c l a i m t h a t HOMR(M,N) = l j m HOMR (Fn,N)/F 0
HOMR(M,N).
P
HOMR(M,N) = X.
Indeed, l e t ( f ( p ) , p F Z ) be an e l e m e n t o f X such t h a t rioq ( f ( D ) ) = f f o r p < q. L e t f ( p ) = , o o ( g ( D ) ) 3 w i t h g ( p ) 6 HOMR (C,N). g(p)-g(q)
(4)
If D c q then
c
F HOMR(M,N) and t h e r e f o r e ( g ( D ) - g ( q ) ) ( F S M ) c FS+qN f o r 9 s F Z. S i n c e FM i s e x h a u s t i v e , f o r any x C M t h e r e e x i s t s some t C Z such t h a t x F FtM and t h u s g ( D ) ( x ) - g ( q ) ( x ) sequence { q ( O ) ( x ) ,
D
c Z}
I f x F FtM,
C F HOMR(M,N).
selected q
VI.4.
.
complete t h e n :
Let M
+
N d e f i n e d by f ( x ) = l i m
n--
:
<
g(D)(x)
Ft+qN.
Hence f
4
Corollary.
+
then
implies that f(x) F g(q)(x) o r f-g(q)
Consequently t h e
i s a Cauchy sequence i n N. Comoleteness o f N
e n t a i l s t h a t t h e mapping f : M well defined.
C FttqN.
c
c
(J(~)(X)
+
g(p)x, i s Ft+qN f o r
Therefore (f-g(q))(FtM) HOMR(M,N) and
0
q
D 5
c Ft+q N,
( f ) = f(q) f o r the
R - f i l t be such t h a t FM i s e x h a u s t i v e and
q
308
D. Filtered Rings and Modules
i )The r i n g HOMR(M,M) i s comnlete i i ) The r i n g HomFR(M,V) i s complete. Proof. i ) D i r e c t l y f r o m p r o p o s i t i o n VI.3.3. i i ) S i n c e HornFR (M,M) = FoHOMR(M,V) we see t h a t HornFR (F1,N) i n HOMR(M,M). HornFR (M,N)
i s closed
i s complete.
We now c o n s i d e r f i l t - p r o j e c t i v e modules. We say t h a t a P t R f i l t i s f i l t - p r o j e c t i v e i f t h e f o l l o w i n g diaqram i n R - f i l t ,
where f i s a s t r i c t
morphi sm, D
may be completed by a f i l t e r e d morphism such t h a t t h e diagram i s commutative. I f P i s f i l t - p r o j e c t i v e ,
,f)
HOM ( I p
: HOMR(P,M)
+
t h e c a n o n i c a l maoping
HOMR(P,N") i s an eoimorohism.
VI.5. P r o p o s i t i o n . L e t R be a f i l t e r e d r i n g , such t h a t FR i s e x h a u s t i v e . Suppose P C R - f i l t i s such t h a t FP i s e x h a u s t i v e . Then t h e f o l l o w i n g statements h o l d : 1. P i s f i l t - p r o j e c t i v e i f and o n l y i f P i s a d i r e c t summand i n R - f i l t o f a f i l t - f r e e object. 2. I f P i s f i l t p r o j e c t i v e , t h e n G(P) i s p r o j e c t i v e i n G(R)-gr.
3. I f FR i s complete and if P i s f i n i t e l y g e n e r a t e d f i l t - p r o j e c t i v e t h e n FP i s comolete. Proof. 1. L e t
L
be f i l t - f r e e w i t h f i l t - b a s i s
diagram i n R - f i l t :
( x ( ~ ) ,ni)icJ.
Consider t h e f o l l o w i n g
309
VI. The Functor HOMRI-,-I
lg L
> M"-0
M
where f i s a s t r i c t eoimorphism. Since x ( ~ )F F
c Fni
g(x(i))
M" = f ( F 0
L, i t f o l l o w s t h a t
c Fn.M such t h a t
M ) . Hence t h e r e e x i t s "i
Lemma I V . l .
= f(y(i)).
g(x('))
'i
morphism o f degree 0, h : L hence L i s f i l t - D r o j e c t i v e .
+
1
5" y i e l d s t h e e x i s t e n c e o f a f i l t e r e d M such t h a t h ( x ( i ) )
= Y ( ~ ) .C l e a r l y
If P i s any f i l t - D r o j e c t i v e o b j e c t then f
t h e r e e x i s t s a f i l t - f r e e o b j e c t L and a s t r i c t eoimorphism L (see Lemma IV.1.8"). morphism g ' : P
--+
9 = f o h,
Projectivity o f
P-0
P imDlies t h e e x i s t e n c e o f a
L o f degree 0, w i t h f o g ' = l p . We have g ' ( F i P ) c F i L n
w h i l e conversely x c FiL
n
Im
CJ'
j m p l i e s x = g ' ( y ) F FiL
i.e.
Imp' ,
f(x) =
= fo g ' ( y ) = y F FiP. Therefore x c g'(FiP)
o r g ' (FiP)
= FiL
means t h a t g ' i s s t r i c t . Note t h a t
n Im
g ' f o l l o w s . The l a t t e r
i n R-mod we have :
L = I m g ' 8 Ker f. Now we c l a i m t h a t : FiL
=
I m g ' 8 Fi Ker f = (Im g ' f! FiL) 8 (Ker f
Fi
n
FiL).
Indeed, i f x 6 F.L then x = y+z w i t h y F Im g ' and z 6 Ker f . Hence f ( y ) = f ( x ) c FiP and, w h i l e y c Im 9 ' . we g e t t h a t g ' o f ( y ) = y or y F g'(FiP)
thus
L
=
= FiL
Im g
8
n Im
g ' . On t h e o t h e r hand, z = x-y F FiL
Ker f ' i n R - f i l t .
n
Ker f and
F i n a l l y since q' i s a s t r i c t m
monomorphism and s i n c e we have I m g ' =
P i n R - f i l t i t follows t h a t P
i s a d i r e c t summand o f a f i l t - f r e e o b j e c t .
The converse i m p l i c a t i o n i s easy. 2. D i r e c t l y from 1 and t h e p r o p e r t i e s o f t h e f u n c t o r G 3. The c o n s t r u c t i o n o f L i n 1 shows t h a t , i n case P i s f i n i t e l y generated
310
0.Filtered Rings and Modules
i n R-filt, L = P
Q
we may choose L t o be f i n i t e l y g e n e r a t e d t o o and we have t h a t
Q f o r some Q F R - f i l t .
S i n c e R i s comolete, L i s comolete and so
a Cauchy-sequence ( x , , ) ~ ~ ~i n P converges t o some x F L. W r i t e x = o + q w i t h p F P, q F Q. S i n c e
FP i s e x h a u s t i v e we have p F FkP
f o r some
k C Z and i t i s c l e a r t h a t xn-P w i l l be i n FtL f o r t l a r q e enouqh i . e . and x F P. T h a t P i s comnlete f o l l o w s .
q = li+m (xn-p) = 0 n I f M,N
C R-filt,
we i n t r o d u c e a n a t u r a l man.
G(HOMR(M,N))
'P
+
H0MG.R)
D(M,N),
(G(M),G(N),
as f o l l o w s : f o r f 6 F HOWR(M,N), P
VI.6. Lemma.
'0 =
x 6 F M out q
,p(fD)(x ) = f(x)o+q. q
i s a monomorphism Moreover, .p(M,N) i s an isomorphism
'$(M,N)
i f M i s filt-projective. P r o o f . F i x i n g M, N
c
R - f i l t we w r i t e
'P
n s t e a d o f *D(M.N). I f
m ( f ) = 0,
P
0 f o r e v e r y x F F M, q F Z. I t f o l l o w s t h a t f ( F M) c Fo+q-lN Pf9 9 q HOMR(M,N). Consequently fD = 0. I n o r d e r t o f o r any q c Z i . e . f F F P-S p r o v e t h e second s t a t e m e n t , f i r s t assume t h a t M i s f i l t - f r e e and l e t then f ( x )
( x ( ~ ) ,P,)icJ
=
be a f i l t - b a s i s f o r
M.
I n t h i s case { x ( i ) , i
F
Oi
J} i s a
homogeneous b a s i s f o r G(M). L e t t i n g g F HOMG(R) (G(M), G(N))D, we o b t a i n g ( x ( i ) ) = yp+pi ( i ) D e f i n e f : Ivl + N by p u t t i n g f ( x ( i ) ) = Y ' ~ ) , w i t h t h i s Pi d e f i n i t i o n i t i s c l e a r t h a t f F FoHOMR(El,N) and a l s o t h a t f p ( f o ) = 0 . T h i s e s t a b l i s h e s s u r j e c t i v i t y o f 0. I f M i s f i l t - o r o j e c t i v e t h e n t h e r e i s a f i l t - f r e e L such t h a t L = M Q V ' i n R - f i l t .
We know t h a t HOMR
commutes w i t h f i n i t e d i r e c t sums t h e r e f o r e we may use t h e o r e v i o u s p a r t o f the proof.
VI.7.
o
Definition.
L e t FR be an e x h a u s t i v e f i l t r a t i o n o f
R.
A filt-injective
o b j e c t i s a n o b j e c t F! F R - f i l t such t h a t , i f I i s a l e f t i d e a l o f R e q u i p e d w i t h t h e i n d u c e d f i l t r a t i o n t h e n any f i l t e r e d morohism f : I extends t o a f i l t e r e d morohism R
+
M.
+
N
31 1
Vl. The Functor HOM&,-I
Remarks. 1. L e t FR be e x h a u s t i v e and complete and suDpose t h a t G(R) i s l e f t Noetherian. I f M i s
R - f i l t i n j e c t i v e t h e n i t i s i n j e c t i v e i n R-mod,
because o f Lemma V I . l . 2 . L e t FR be e x h a u s t i v e . I f Q
<
R - f i l t i s i n j e c t i v e i n R-mod t h e n i t i s
injective i n R - f i l t . VI.9.
Theorem.
L e t FR be e x h a u s t i v e and l e t 0 E R - f i l t be comolete
If G(Q) i s i n j e c t i v e i n G(R)-gr t h e n Q i s f i l t - i n j e c t i v e . Proof.
Equip t h e l e f t i d e a l I o f R, w i t h t h e f i l t r a t i o n FoI = F R P
Since i : I
+
n
I.
R i s a s t r i c t morphism we o b t a i n t h e f o l l o w i n g commutative
diagram :
I
G HOMR (i,lQ)
G(HOMR(R,Q))
G(HOMR(I,Q))
m(I .Q)
.p(R,Q)
H O M ~ ((G(R) ~ ) ,G(Q))
,I >
HOMG(R)(G(i) , G ( ~ Q ) )
HOpG(p)(G(I) ,G(Q))
S i n c e G ( i ) i s a monomorphism, i n j e c t i v i t y o f G(Q) e n t a i l s t h a t HOMG(R)(G(i),G(lQ)) i s an eoimorohism
.
Hence
(1,Q) i s an enimorohism
'p
t o o , t h e r e f o r e an isomorphism. Thus G HOMR(i,l ) i s an isomorahism,
Q
w h i l e F HOM (R,Q) R
and F HOMR(I,Q)
a r e e x h a u s t i v e and comillete, t h e r e f o r e
b y Theorem I I I . 4 . , HOMR(i,l ) i s e o i m o r o h i c .
Q
o
312
VII: Projective Modules and Homological Dimension of Rings
We w i l l say t h a t x
t xn, n c N
c
R i s t o o o l o g i c a l l y n i l n o t e n t i f the sequence
} i s a Cauchy sequence converginq
t o 0. R e c a l l t h e f o l l o w i n g
lemma about t o p o l o g i c a l r i n g s :
VII.l. Lemma.
L e t R be a complete t o o o l o g i c a l r i n g and c o n s i d e r a
fundamental s e t o f neighbourhoods o f zero, c o n s i s t i n g o f a d d i t i v e subgroups.
If
'J
: R
+
S i s a r i n g homomorohism such t h a t every x C Ker
t o p o l o g i c a l l y n i l p o t e n t , then every idemnotent element o f I m
'D i s '0
may be
l i f t e d t o R. Proof.
Let e
c
I m 'D be idemootent, e =
2
' ~ ( x )f o r some x c R.
2
Since ,p(x - x ) = 0, a = x - x t Ker q . Put b = x
+
X ( l - 2 x ) w i t h X t R, and
2 determine X such t h a t b = b w h i l e X commutes w i t h a. We g e t Put :
x2 -X) (1+4a)+a=O.
7
k
where C;k (1-1)2k =
i s the c o e f f i c i e n t o f 1 ( - l ) k i n t h e binomial exoansion o f 1 k 0 i.e. i s an i n t e g e r . Therefore X may be viewed as a Dower C2k
s e r i e s w i t h i n t e g e r c o e f f i c i e n t s which i s e a s i l y seen t o be a convergent one. Hence (*) determines A and i t commutes w i t h a. Consequently b i s idempotent. Furthermore ,p(b)
=
.(x)
= e.
V I I . 2 . Lemma.
,
a
c
Ker
'0
y i e l d s X C Ker
L e t R be a f i l t e r e d r i n g ,
M
= G(f).
C R - f i l t such t h a t FM i s --f
'p
Consider t h e morphism : : HOMFR(M,M)
-f
M i s a f i l t e r e d morphism
Then t h e r e e x i s t s a s t r i c t morphism g : M
i n R - f i l t s a t i s f y i n g : g2 = g and G(g) = G ( f ) . Proof.
and thus
0
exhaustive and complete. Suppose t h a t f : M such t h a t G ( f ) [
'p
Hom G(R)-gr
( G ( W 9
G(W)
--f
M
313
VII. Projective Modules and Homological Dimension of Rings
g i v e n by , p ( f ) = G ( f ) ; =
F-H l OMR
tp
(M,M) = { f : M
t h e n fn c F-n HOM,(M,M) VII.l.,
i s a r i n g homomorphism. We see t h a t Ker +
M , f (FpM) c F
M,V
D C
'D =
Z }. Hence i f f C Ker
'0
t3-1
t h a t l i m fn = 0. Hence, by Lemma n+m can be l i f t e d . I f o n l y remains t o be
and we f i n d
e v e r y i d e m p o t e n t i n Im
$0
p r o v e d t h a t any i d e m p o t e n t f i n HornFR (P,Y) i s s t r i c t . Suppose t h a t f ( x ) c F M D hence f i s s t r i c t . VII.3.
f C F-l inverse
f . Then f ( f ( x ) ) = f ( x ) so t h a t f ( x ) C f ( F
HOMR (M,M) c J(HOMR(EI,M)). HOMR (M,M)
c
o5n< m
(J
=
Jacobson r a d i c a l ) . Indeed i f
t h e n f i s t o u o l o g i c a l l y n i l o o t e n t and so 1-f has an
fn.
VII.4. C o r o l l a r y .
I f M i s as i n t h e f o r e g o i n g lemma t h e n any c o u n t a b l e
s e t o f o r t h o g o n a l i d e m p o t e n t elements i n Im VII.5.
M),
P
I f M c R - f i l t i s such t h a t FM i s e x h a u s t i v e and comolete
Remark.
t h e n F-l
n Im
Theorem.
'0
can b e l i f t e d t o HomR-filt(M,P)
L e t R be a f i l t e r e d r i n g such t h a t FR i s e x h a u s t i v e . L e t
P be o r o j e c t i v e i n G(R)-gr and suuuose t h a t , e i t h e r FR i s d i s c r e t e and
9 t h e g r a d a t i o n o f P i s l e f t l i m i t e d o r P i s f i n i t e l y q e n e r a t e d w h i l e FR g g i s complete. Then t h e r e i s a f i l t - p r o j e c t i v e module P 6 R - f i l t such
t h a t G(P) = P
I f M c R - f i l t , t h e n f o r any mornhism g : P + G(P) 4' 9 o f degree p t h e r e i s a f i l t e r e d morohism f : P -L M o f degree p such t h a t g = G(f). Proof. The hypotheses i m o l y t h a t t h e r e i s a f r e e o b j e c t L
q
i n G(R)-gr
+ L o f degree z e r o , such t h a t h2 = h and I m h = P 9 g g' If P i s f i n i t e l y g e n e r a t e d t h e n L may be chosen t o be f i n i t e l y g e n e r a t e d 9 9 t o o . Assume t h a t L = G ( L ) , where L c R - f i l t i s f i l t - f r e e . I f Po has l e f t 9 l i m i ' t e d g r a d i n g , t h e n L may be chosen so t h a t i t has l e f t - l i m i t e d g r a d i n g g t o o and we suppose FL t o be d i s c r e t e s i n c e FR i s d i s c r e t e . I n t h e case
and a morphism h : L
314
0. Filtered Rings and Modules
where P
i s f i n i t e l y generated a n d FR i s complete, we obtain t h a t FL i s g complete. By Lemma D.7., there e x i s t s a s t r i c t f i l t e r e d morphism f : L + L n
with f L = f and G(f)
=
h . P u t P = Im f . Since
f i s s t r i c t i t follows t h a t
G(P) = P
a n d t h a t P i s f i l t - p r o j e c t i v e as required. Moreover i f P i s 9 f i n i t e l y generated then FP i s complete. To prove the remaining statement
we may w r i t e g : G ( P )
+
G ( M ) a n d up t o taking a s u i t a b l e
we may assume deg g = 0 . We have L morphism h : G ( L )
+
P
Q
Q and hence there i s a graded
G ( M ) such t h a t h l G ( P ) = g . However h = G ( k )
some f i l t e r e d morphism k : L VII.6. Corollary.
=
suspension
+
for
M. Putting f = h ] P we a r r i v e a t g = G ( f ) .
Let R be a f i l t e r e d ring such t h a t FR i s exhaustive.
Let P E R - f i l t be such t h a t FP i s separated and exhaustive. Supoose t h a t i s d i s c r e t e and G ( P ) i s l e f t limited o r G ( P ) i s f i n i t e l y generated
FR
while FR i s complete. I f G ( P ) i s projective i n G ( R ) - g r , then P i s f i l t projective.
_-Proof.
By the Theorem VII.5. we may s e l e c t a
Q e R - f i l t such t h a t G ( Q )
rr
projective o b j e c t
G ( P ) ( Q may be chosen i n b o t h cases such t h a t
FQ i s complete and exhaustive); l e t g : G ( Q ) There e x i s t s afilteredmorphism f : Q
+
+
G ( P ) denote t h i s isomorphism
P such t h a t G(f) = g . We can
apply Corollary 11.5. and deduce t h a t f i s a n isomorphism.
VII.7. Corollary.
Let R be a f i l t e r e d ring such t h a t FR i s exhaustive
and d i s c r e t e a n d l e t M E R - f i l t be such t h a t FM too i s exhaustive and d i s c r e t e . Then we have : p.dim R M
5
p.dimG(R)G(M)
By Lemma IV.1.8", there e x i s t s a f r e e resolution f o r M i n R - f i l t : f f M -0 f n - l , Ln-2+ ...- L 11 , L0 ... Ln-l where F L . a r e d i s c r e t e f o r a l l j . Write Q f o r Ker f n - l , where
Proof. -
+ J
n
=
p.dim
G(R)
G(M), then G ( Q ) i s a projective G(R)-module with l e f t
315
VII. Proiective Modules and Homological Dimension of Rings
l i m i t e d g r a d i n g . By t h e theorem we may s e l e c t a p r o j e c t i v e o b j e c t Q 6 R - f i l t such t h a t G(Q)
G(Q'), l e t g : G(Q')
There e x i s t s a f i l t e r e d morphism f : Q '
--f
G ( Q ) denote t h i s isomorphism.
+
Q such t h a t G ( f ) = g. Now we
end up w i t h t h e s i t u a t i o n where Q' i s f i l t - p r o j e c t i v e ,
FR i s d i s c r e t e
t o b e d i s c r e t e j u s t as w e l l ) w h i l e G ( f ) i s an
( n o t e t h a t FQ' t u r n s o u t
isomorphism, t h e r e f o r e we may a p p l y C o r o l l a r y 111.5. and deduce t h a t f i s an isomorphism, whence i t f o l l o w s t h a t Q i s f i l t p r o j e c t i v e and t h e r e f o r e p r o j e c t i v e i n R-mod.
VII.8. C o r o l l a r y .
L e t R be a f i l t e r e d r i n g w i t h e x h a u s t i v e and d i s c r e t e
f i l t r a t i o n , then gl.dim Proof.
R c
gr.gl.dimG(R).
E q u i p a l e f t R-module M w i t h t h e f i l t r a t i o n FiM = FiR.M,
then
FM i s e x h a u s t i v e and d i s c r e t e . By t h e f o r e g o i n g c o r o l l a r y we have : p.dim M c p.dim G(M) 5 g r . g l . d i m G(R).
VII.9. C o r o l l a r y .
L e t R be a graded r i n g , t h e n :
g r . g l .dim R 5 g l .dim
I f t h e gradation o f R i s l e f t l i m i t e d then : gr.gl.dim R
=
gl-dim
z.
1
P r o o f . The second s t a t e m e n t f o l l o w s f r o m t h e f o r e g o i n g c o r o l l a r y a p p l i e d t o t h e r i n g R f i l t e r e d w i t h t h e a s s o c i a t e d f i l t r a t i o n . The f i r s t s t a t e m e n t from t h e graded t h e o r y i n C h a p t e r A .
VII.10. C o r o l l a r y .
____ _ _ .
L e t R b e a f i l t e r e d r i n g such t h a t FR i s e x h a u s t i v e
and complete, l e t M € R - f i l t be such t h a t FM i s e x h a u s t i v e . Suppose t h a t G ( R ) and G(M) a r e l e f t N o e t h e r i a n , t h e n : p.dimRM 5 p.dimG(R)G(M).
Proof.
Apply
Lemma I V.l . and Theorem VII.5. and use t h e
o f t h e p r o o f o f C o r o l l a r y VII.7.
VII.11. C o r o l l a r y .
argumentation
o
L e t R be a f i l t e r e d r i n g such t h a t FR i s e x h a u s t i v e
and complete. Suppose t h a t G(R) i s l e f t N o e t h e r i a n . We have gl.dim R 5 gr.gl.dim G(R).
316
Proof.
D. Filtered Rings and Modules
Let M be a f i n i t e l y generated l e f t R-module and t a k e xl,
t o be a s e t o f g e n e r a t o r s . F i l t e r M w i t h t h e f i l t r a t i o n : FiM =
...,xm n
2 FiR.xk. k=l Doing t h i s we o b t a i n a f i n i t e l y generated f i l t e r e d module with an exhaustive
f i l t r a t i o n . Corollary V I I . 9 . f i n i s h e s t h e proof.
o
317
VIII: Weak (Flat) Dimension of Filtered Modules
L e t us s t a r t t h i s s e c t i o n w i t h a b a s i c lemma on t e n s o r products o f f i l t e r e d modules.
If M
c
R-filt,
N F f i l t - R then t h e Z-module N mR M
M) i s t h e Z-submodule o f N mR M generated has a f i l t r a t i o n : F (N P by a l l elements o f type n 60 m where n F FtN, m € FsM and p = s+t. This c o n s t r u c t i o n y i e l d s a f u n c t o r : 8 : filt-R R x R-filt
Z-filt.
+
R
VIII.l. Lemna.
With n o t a t i o n s as above, t h e f o l l o w i n g statements a r e
true :
1" I f FM and FN a r e exhaustive then so i s F(N 2" I f FM and FN are d i s c r e t e then so i s F(N
Qo
M). R M).
bD
R 3" For a l l m, n C Z : N(n) 8 M(m) = (N 8 M)(mtn). R R 4" I n t h e category R - f i l t we have t h e f o l l o w i n g isomorphism R 8 M 2 M; R s i m i l a r l y N QpR R 2 N i n f i l t - R . 5" This f u n c t o r 60 commutes w i t h d i r e c t sums and i n d u c t i v e l i m i t s . R 6" For M F f i l t - R , N € R - f i l t - S , P c f i l t - S , we have t h e f o l l o w i n g
isomorphism i n Z - f i l t : HOMS(M 8 N,P) 2' HOMR (M, HOMS(N,P)). R
-Proof. Statements lo, 2",
3" a r e c l e a r .
: N 8 R + N, cp(x.r) = x r . R N @ R, $ ( x ) = x 60 1. I t s i n v e r s e i s g i v e n by $:N R Now, i f x b~ r F F (N 8 R) w i t h x F FiM, r F F.R, i + j = p, then P R J ~ ( 8x r ) = x r F FiN.F.R c F N. I n a s i m i l a r way one e s t a b l i s h e s t h a t J P $ i s a morphism i n f i l t - R .
4" Consider t h e R-module morphism,
cp
--f
5" L e t N = 8
(a,B)CAxB
8 a FA
N
Na, M = 8 pFB
@MzN
a R
8 M.
R
M
then
Indeed, l e t
318
0.Filtered Rings and Modules
be t h e R-module morphism which i s g i v e n by :
v ( ( c xa) o CA
60 (
2 Y,))
xa @ Y p
= 2 (a,3)CAxB
pCB
Then i t i s w e l l known t h a t rp i s an isomorphism o f a b e l i a n aroups. L e t x @ yc F
z
x =
( N m PI) where x c FiN, y c F.N and i P R J xa, y = 2 y P t h e n xa C FiN, and y
P
0CB
iL cA
immediately t h a t
xa 60 y
c F
0
P
(Y
M ), P
a R
+
j = p. If F j Mp.
It
follows
hence
0 N3 &3 M ) . I n a s i m i l a r way one e s t a b l i s h e s t h a t P ( a,P ) CA xB -1 q i s a c t u a l l y a morphism i n 2 - f i l t . The a s s e r t i o n f o r i n d u c t i v e l i m i t s
IQ(X m y )
c
Fp (
may be p r o v e d i n an e q u a l l y s t r a i g h t f o r w a r d w a y .
6" D e f i n e : HCMS(F.l CQ N,P)
+ HCM (P,H0FIS(N,P)) R by : ( v ( f ) ( x ) ) ( y ) = f ( x @ y ) , and d e f i n e
Q
JI : HOMR(M,HOMS(N,P))
HOMS (!I
+
by : $ ( g ) ( x m y ) = ( q ( x ) ) ( y ) .
c
Hence w ( f ) ( x )
Fp+t HOMS(N,P)
T h i s expresses t h a t
O,
R
N,P)
If f C F
x 6Oy 6 Fttr
then i t f o l l o w s from
c%
P
60
P
HOMS ( V 60 N,P), x C FtF.", Y C FrN R N t h a t f ( x 6 n y ) C Fo+t+rP=
R and t h u s Q ( f ) 6 FD HOWR (V,HOPIS(N,P)).
i s a morphism i n Z - f i l t
. Along
t h e same l i n e s one
v e r i f i e s t h a t $ i s a morphism o f degree 0 and ji i s t h e i n v e r s e o f rp. o
F.1 6 f i l t - R .
Let N C R - f i l t , q~ =
q(N,M)
: G(P1)
60
D e f i n e a graded morphism G(N)
+
G ( M 60 N ) ,
by p u t t i n g
G(R) Q
(ms
60 nt) = (m
clear that
~p
QO
n)s+t. O b v i o u s l y t h i s i s
i s surjective.
w e l l d e f i n e d and i t i s
VIII. Weak (Flat) Dimension of Filtered Modules
VIII.2.
Lemma.
M or N i s a free object i n f i l t - R or
If e i t h e r
r e s p e c t i v e l y t h e n q(M,N)
319 R-filt
i s an isomorphism.
Knowing t h a t 60 and G commute w i t h d i r e c t sums, t h e p r o o f may R be reduced t o t h e case where M = R ( n j R . B u t t h e n Lemma VIII.l. a l l o w s Proof.
us t o reduce t h e p r o o f f u r t h e r t o t h e case M = R R . I n t h i s case we have t h e f o l l o w i n g commutative d i a g r a m o f graded morphisms, v e r t i c a l arrows r e p r e s e n t i n g isomorphisms :
So, q(M,R)
VIII.3.
i s a l s o an isomorphism.
Lemma.
L e t R be a f i l t e r e d r i n g and l e t M C R - f i l t be such
t h a t FR and FM a r e d i s c r e t e and e x h a u s t i v e . I f G(M) i s a f l a t o b j e c t i n G(R)-gr t h e n Proof.
P is
a f l a t R-module.
IfJ i s a r i g h t i d e a l o f R, e q u i p i t w i t h t h e f i l t r a t i o n
F J = F R n J . L e t i : J + R be t h e c a n o n i c a l i n c l u s i o n morphisms ; P P n o t e t h a t i i s a s t r i c t morphism. A p p l i c a t i o n o f Theorem 111.4. y i e l d s t h a t G ( i ) i s i n j e c t i v e , and s o we o b t a i n t h e f o l l o w i n g commutative diagram :
G(J @ M ) R
G(i
@
lM)
>
G(R
@
R
M)
320
D. Filtered Rings and Modules
The f a c t t h a t (p(M,R) i s an isomorphism, whereas G ( i ) monomorphism, e n t a i l s t h a t Then G ( i
@
ip(M,J)
66
lG(M) i s a
i s a monomorphism, hence an isomorphism.
has t o be monomorphic and r e p e a t e d a p p l i c a t i o n o f
lM)
Theorem 111.4. a l l o w s t o d e r i v e f r o m t h i s t h a t F ( J f V ) i s d i s c r e t e . Hence, i @ lM i s a s t r i c t monomorphism and i t f o l l o w s t h a t PI i s f l a t .
VIII.4. C o r o l l a r y .
L e t R be a f i l t e r e d r i n g , M C R - f i l t ,
such t h a t FR
a r e d i s c r e t e and e x h a u s t i v e . Then we have : w.dim Proof.
(*)
R
M
gr.w.dim
5
G(R) G(M).
P u t n = gr.w.dim
0-L-
f
where Lo,
...,
G(R)
G(f1).
Ln-l-...
Ln-l
C o n s i d e r an e x a c t sequence :
-L1-
fl
Lo-
a r e f r e e o b j e c t s , w h i l e FLi,
f0
i
=
M-0
,
0,...,n-1, and FL
a r e e x h a u s t i v e and d i s c r e t e . By e x a c t n e s s o f t h e f u n c t o r G we d e r i v e f r o m
(*) an e x a c t sequence : O-G(L)-
...
G(Ln-l)+
-G(Lo)
-
G(M)-
0
Our assumptions i m p l y t h a t G(L) i s a f l a t o b j e c t o f G(R)-gr and t h e f o r e g o i n g lemma e n t a i l s t h a t L i s f l a t . T h e r e f o r e w.dimR M c n.
o
VIII.5. C o r o l l a r y . F o r a f i l t e r e d r i n g R w i t h FR b e i n g e x h a u s t i v e and d i s c r e t e we have : g1.w.dim R 5 g r . g l . w . d i m Proof.
G(R).
P u t n = g r . g l . w.dim G(R). Any M C R-mod may be f i l t e r e d by t h e
filtration F
M
P The i n e q u a l i t y
=
F
P
R.P w h i c h i s e a s i l y seen t o be e x h a u s t i v e and d i s c r e t e .
o f C o r o l l a r y VII.4. t h u s h o l d s f o r any M w i t h t h e i n d u c e d
f i l t r a t i o n , and t h i s Droves o u r c l a i m .
321
VIII. Weak (Ffarl Dimension of Filtered Modules
VIII.6. Corollary.
Let R be a graded r i n g , then
g r . g l . w. dim.R
gl.w.dim
5
E
I f moreover R i s l e f t l i m i t e d then : g r . g l . w dim R Proof.
=
gl.w.dim
E.
Easy combination of foregoing r e s u l t s
VIII.7. C o r o l l a r y . Let R be a p o s i t i v e l y graded r i n g and assume t h a t Ro i s a Von Neumann r e g u l a r r i n g : gl.w.dim R = left-w.dimR Ro
=
r i g h t - w.dim R R 0'
Take bl c R-mod and equip i t w i t h t h e t r i v i a l f i l t r a t i o n
Proof.
F M = 0 f o r p c 0, F M = M f o r p
3 0. Endow R w i t h t h e f i l t r a t i o n P 8 R i . In t h i s s e t t i n g M i s a f i l t e r e d R-module and c l e a r l y isp = 0 f o r a l l D # 0. Therefore G(M) i s a n n i h i l a t e d by t h e i d e a l
P
F R P
=
G(M)
R,
=
P
8 Rn.
n 31
w.dim M
R
5
From Corollary V I I I . 4 .
w.dim G(M)
R
5
, we
obtain :
l.w.dimRRo + Ro + 1.w.dim
and the l a t t e r i s j u s t l.w.dimR Ro.
RO
G(M)
I t follows t h a t gl.w.dim R = l.w.dimR Ro.
The o t h e r e q u a l i t y may be e s t a b l i s h e d i n e x a c t l y the same way.
VIII.8. C o r o l l a r y .
u
Let R be a f i l t e r e d r i n g such t h a t FR i s d i s c r e t e
and e x h a u s t i v e . Let N c R - f i l t , M c f i l t - R be such t h a t FM and FN a r e d i s c r e t e and e x h a u s t i v e , then : Tor:(R)(G(M),G(N)) = 0 i m p l i e s Proof.
Tor,R ( b l , N ) = 0 .
Consider a f r e e r e s o l u t i o n of N :
where FiL i s d i s c r e t e and e x h a u s t i v e , f o r a l l i, and where fo, f l , f 2 , ... a r e s t r i c t rnorphisms. Consider the following commutative diagrams, the v e r t i c a l arrows of which r e p r e s e n t isomorphisms :
322
D. Filtered Rings and Modules
... --,G(M) . .. We
3
60
G(R)
G(L2)+G(M)
I L2)
G(M 60 R
G(L1)
69
G(R) G(M
3 -
C-(Lo) G(R)
.1
@
R
L1)
0
j G(W) 60
__j
G(M
1
60
R
Lo)
0
have : G*,n ( M 60 L*) = HnG(M 60 L*) = Hn (Gflur)
R
If T o r i ( R ) ( G ( M ) , G ( N ) ) = 0 t h e n G*”(M i s exhaustive
R T o r (M,N) n
,
and d i s c r e t e = Hn(M 69 L),
R
69
G(L,))
= Tor:(R)(G(M).G(N))
G(R)
R
=
0.
o
60
R
L*) = 0 and, s i n c e F(M m L), R
'323
IX: Exercises. Comments. References
1. L e t R be a f i l t e r e d r i n g w i t h t h e f i l t r a t i o n FiR such t h a t FOR = R, M
c
R - f i l t w i t h t h e f i l t r a t i o n FiM.
submodule e q u i p e d w i t h
such t h a t FoY = M and N c Y a
t h e induced f i l t r a t i o n . L e t
'pF,
:
M
+
M
be
t h e c a n o n i c a l morphism. We s h a l l denote by N ' t h e adherence o f qM(N)
1) N
in s
M,
that i s N' =
-1
N ' and wM ( N ' )
=
11 (cpM ( N )
iCZ
n
iCZ
+
M).
Fi
Prove t h a t :
(N + FiM).
2 ) I f N,P a r e submodules o f M t h e n (N fl P ) ' = N ' 3) L e t f:M
+
N be a morphism i n R - f i l t .
n
P ' and (N+P)' = N ' + P ' .
I f G i s a submodule o f N and
1
we denote by H = f- (G) t h e n f - ' ( G ' )
=
H'.
4 ) I f N,P a r e submodules such t h a t P i s f i n i t e l y g e n e r a t e d t h e n (N: P )
' =( N' : P I )
5 ) I f I i s a f i n i t e l y generated l e f t i d e a l , then ( N : I ) ' =(N':I'). 2 . L e t I be a t w o - s i d e d i d e a l o f a r i n g R and E be a l e f t R-module; we s h a l l denote by GI(E)
t h e graded module a s s o c i a t e d t o t h e f i l t r a t i o n .
c o - i n d u c e d by I - a d i c f i l t r a t i o n ( i . e . G I ( E ) = En = AnnEIn). GI(E)
i s a GI(R)-module
(GI(R)
5
n=l = 5
En/En-l
where
I n - 1/ I n ) ; t h e n
n =O 1) The f u n c t i o n s e n d i n g an R-submodule F o f GI(F) 2 ) AnnEI
of
GI(E)
is strict
if
E =
U
n?l
i s an e s s e n t i a l GI(R)-submodule
3. L e t R be a r i n g w i t h f i l t r a t i o n FiR
E t o t h e GI(R)-submodule
AnnE I". of CI(E).
such t h a t FOR = R. L e t P be a
t w o - s i d e d i d e a l o f R. I f G ( P ) i s a p r i m e i d e a l i n G(R) t h e n t h e adherence p =
II (PtFiR) o f P i s a o r i m e i d e a l . 1'0
I n particular i f
t h e f i l t r a t i o n o f R i s seDarated and G(R) i s a p r i m e r i n g t h e n R i s a prime r i n g .
0. Filtered Rings and Modules
324
4. L e t R be a r i n g w i t h f i l t r a t i o n FiR such t h a t FOR = R and FiR i s s e p a r a t e d . I f G(R) i s a domain t h e n R i s a domain.
Z, p #
5. L e t M = Z w i t h t h e p - a d i c f i l t r a t i o n (p C
f i l t r a t i o n FiN = Z,
v
0)
and
i 5 0. Then t h e morphism lZ : M
b u t i t i s n o t s t r i c t . Moreover G ( l Z ) : G(P)
G(N)
+
N
--f
= Z with
N i s bijective
i s injective nor
s u r j e c t i ve . 6. L e t R be a f i l t e r e d r i n g w i t h f i l t r a t i o n FiR such t h a t FOR = R and l e t M be a f i n i t e l y g e n e r a t e d R-module. f i l t r a t i o n FiM = FiR.M.
Consider on M t h e i n d u c e d
Then :
1) G(M) i s a f i n i t e l y g e n e r a t e d G(R)-module. 2 ) M i s a f i n i t e l y g e n e r a t e d R- module.
- -
3) FiM = FiR.M (90,
: M
+
= FiR.g
M (M)
- -
M- R vM(M)
and
M i s t h e c a n o n i c a l morohism).
( H i n t : There e x i s t s a f r e e module L f i n i t e l y g e n e r a t e d and a s u r j e c t i v e homomorDhisrn L
M
+
+
0; f u r t h e r on we use t h e p r o p e r t i e s
o f the completion functor).
7. L e t M,
N,
P F R-filt
and c o n s i d e r t h e s t r i c t morphisms M
f
N
g
P.
I f f i s s u r j e c t i v e (resp. g i s i n j e c t i v e ) then g o f i s s t r i c t . 8. L e t M,
N
E R - f i l t and l e t f : M
- -
1) I f f i s s t r i c t t h e n f : M
-
+
+
N be a morphism i n R - f i l t .
N is strict ^
^
2) I f f has t h e A r t i n - R e e s p r o D e r t y t h e n f : M
^
+
N has t h e A r t i n - R e e s
property.
9. 1) I f R i s a r i n g , show t h a t i f I i s t h e i d e a l o f R [ X 1 , t h a t i s g e n e r a t e d by Xi,
,..., Xn]
t h e n t h e I - a d i c comDletion o f R [ X1,X 2,....Xnl
i s t h e power s e r i e s r i n g R [ [ X1,X2
,..., X n l l .
2 ) I f R i s a N o e t h e r i a n r i n q , show t h a t R[[ X1,...,Xn]]is ring.
X2
a Noetherian
IX. Exercises. Comments. References
325
10. Let R be a l e f t Noetherian ring a n d I an ideal generated by n
elements belonging t o the center of the ring R. Consider the I-adic f i l t r a t i o n on R . Then : 1) The associated graded ring G I ( R ) i s l e f t Noetherian
2 ) The completed R i s a l e f t Noetherian ring. 11. 1) Let R be a ring ( n o t necessarily Noetherian) and I a n ideal
generated by central elements. Then I s a t i s f i e s the Artin-Rees property f o r l e f t Noetherian modules. 2 ) I f 0-
M'+
M-
R-modules , then 0
-
0 i s an exact sequence of Noetherian
MI'---,
M'
+ M-
Mi'-
0
is
exact.
12. 1) Assume t h a t R i s a l e f t and r i g h t Noetherian ring and I an ideal which s a t i s f i e s the l e f t and r i g h t Artin-Rees properties. For any f i n i t e l y graded l e f t Rwodule M we have : Torn(R/I,M) R
h
Tor R n ( R / 1 9 P ) , n 2 0
( M i s the completion with respect t o the I-adic f i l t r a t i o n ) 2 ) The following statements a r e equivalent f o r a map f : M
+
N of
f i n i t e l y generated R-modules :
R a ) The map f i = Tori(R/I,M)
--f
R Tori(R/I,N) induced by f i s an
isomorphism f o r a l l i 2 0. b) f o
i s an isomorphism a n d f l i s an eoimorphism.
-
-
c ) f induces a n isomorphism f : M
-
--f
N
13. Let R be a f i l t e r e d comolete ring and 0-
MI-
f
M & P"--,
0
an exact sequence i n R - f i l t where f and g a r e s t r i c t and such t h a t FM'
, FM, FM" a r e separated and exhaustive.
1) I f M i s f i 1 t . f . g then M" i s f i 1 t . f . g . 2 ) I f M ' and M" a r e f i 1 t . f . g then M i s f i 1 t . f . g .
D. Filtered Rings and Modules
326 14. See [ 66 1 .
1) L e t U(g) be t h e u n i v e r s a l e n v e l o o i n g a l g e b r a o f a f i n i t e d i m e n s i o n a l s o l v a b l e L i e A l g e b r a g o v e r an a l g e b r a i c a l l y c l o s e d f i e l d K o f c h a r a c t e r i s t i c z e r o and l e t I be an i d e a l o f U ( g ) . Then
I has a n o r m a l i z i n g s e t o f g e n e r a t o r s . 2 ) Moreover, i f g i s a n i l p o t e n t L i e Algebra, t h e n I has a c e n t r a l i z i n g s e t o f generators. ( H i n t 1 ) ) C o n s i d e r U(g) as a l e f t g-module by s e t t i n g x
r = xr-rx = [ x,r]
,
c
x 6 g, r
o f x and r i n U ( g ) . L e t U(g),
U ( g ) , where x r i s t h e o r o d u c t
be t h e R-subspace o f U(g) soanned
by t h e i n - f o l d p r o d u c t s o f elements o f g where 0 5 m 5 n. Then U(g)n i s a finite-dimensional
Let 0 # x c I
U U(g),.
A l s o U(g) =
space and i s a submodule o f U ( g ) .
n2o
and l e t g ( x ) denote t h e
g-submodule o f U(g) g e n e r a t e d by x. From above g ( x ) i s f i n i t e d i m e n s i o n a l , so by L i e ’ s theorem
([D.Z.], pag 60)
such t h a n K y1 i s a submodule o f U ( g ) . Thus y1 Suppose now t h a t yl,
yp,
U(g) which b e l o n g t o I .
c
3 0 # y1 C g ( x )
N(U(q))
n I.
...,ui
i s a n o r m a l i z i n g s e t o f elements o f L e t U(g) U(g)/J where J (yl,y *,...,yi). =
=
I f J # I then, by a o p l y i n g t h e above argument a g a i n , 3 6
# yi+l
c
-
N ( b ( g ) ) fl I .
The
set
Iyl,
y *,..., yi+lj
is a
n o r m a l i z i n g s e t o f elements o f I . S i n c e U(g) i s l e f t and r i c l h t N o e t h e r i a n I has a n o r m a l i z i n g s e t o f q e n e r a t o r s .
I f G i s a f i n i t e l y generated and n i l p o t e n t group t h e n
15.
t h e i d e a l I o f t h e r i n g Z [ G I , where I = < 1-4, g C C- > has t h e A r t i n - R e e s p r o p e r t y . ( H i n t : G has a c e n t r a l s e r i e s G = G 0 ) G 1 J . . . ’ C =
{ e ) such t h a t each q u o t i e n t G./Gi+l
i s cyclic.
The
proof
c o n t i n u e s by i n d u c t i o n on t h e m i n i m a l l e n q t h n o f such a s e r i e s .
=
IX. Exercises. Comments. References
327
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336
Index
Abelian regular ring Almost strongly graded Artin-Rees property Artin-Rees property on the l e f t B i l a t e r a l pre-kernel functor Birational algebra
260 62 29 1 300 248 249
Centralizing s e t of elements Centralizing s e t o f generators f o r an ideal Co cy c 1es Cocycl e s of f i rs t degree Cogenerated torsion theory Cohen-Macaulay module Cohen-Macaulay ring Co-induced f i l t r a t i o n Complete f i l t r a t i o n Completely projective ring Completion functor Conjugate of a bimodul e C r i t i c a l module
30 2
Dehomogenized element Discrete f i l t r a t i o n Discrete graded valuation Divisor c l a ss group Divisorial ideal
120
Exhaustive f i l t r a t i o n Exhaustion functor
279
Factor s e t Fi 1 t-basi s Fi 1 tered b i m du l e Fi l t e r e d f i n i t e l y generated Fi 1tered f r e e module Fi 1 tered module F i l t e r e d morphism of degree P
30 2 26 28 134 20 7 207 282 283 212 286 34 96 279 170 173 173 282 23 293 280 294 29 3 2 78
280
337
Index
Fi I tered ring
Filtration Fi 1 t r a t i on degree Fil t - p r o j e c t i v e module Fi ni t e type (kernel functor) Fractional ideal Generalized Rees ring Globally birational algebra Goldie’s graded kernel functor Gorens t e i n module Gorenstein ring G-pri me Graded abel ian regular ring Graded algebraic closure Graded algebraic element Graded Artinian Graded bi rational a1 gebra Graded bi regul a r r i ng Graded bounded ring Graded Dedeki nd ring Graded division ring Graded f r e e Graded f u l l y bounded ring Graded f u l l y ( l e f t ) idempotent Graded Goldie ring Graded ideal Graded i n j e c t i v e Graded integral element Graded kernel functor Graded Graded Graded Graded Graded
Krull domain Lambek torsion theory l e f t ideal l e f t module l e f t principal ideal ring
Graded local ring Graded maximal ideal Graded maximal submodule
278 278 280 308 14 1 172 30, 92 249
138 20 7 207 68 260 158 155 54 253 271 2 38 179 38 5 238 271 222 8
a
160 132 175 137
a
3
34 198 198 52
338
fndex
Graded morphism Graded Noetherian Graded prime kernel f u n c t o r Graded p r o j e c t i v e Graded r e a l i z a b l e f i e l d Graded r e a l i z e d f i e l d Graded r e g u l a r r i n g Graded r i n g Graded semisimple Graded simple Graded s u p p o r t i n g module Graded t e n s o r product Graded uniformly simple Graded v a l u a t i o n r i n g Grade o f an i d e a l GZ-ri ng
3 54
15 1 7 158 158 258
1 47, 52 47, 52 15 1 12 47 154 20 7 254
Hereditary t o r s i o n theory Homogeneous e l ement (components)
130
Homogenized element (module)
120
I-adic f i l t r a t i o n I.B.N. property
278 33
Idempotent kernel f u n c t o r
130
Invariant I n v a r i a n t graded module Invariant ideal I n v e r t i b l e bimodule Invertible ideal Jacobson graded r a d i c a l
Kernel f u n c t o r Kernel f u n c t o r of f i n i t e type Kernel o f a prime Krull dimension Limited ( l e f t or r i g h t ) graded module Negatively graded Nilgraded r i n g Normalizing s e t of g e n e r a t o r s
1
35 44 58 17 173 52 130 14 1 153
95 79 79 270 30 2
339
Index
Object of quotients 0-group Opposite graded ring Ordered group Perfect torsion theory P . I . algebra
Polynomial i d e n t i t y Positively graded Pre-kernel functor Primary decomposition f o r submodules Prime of a ring Property C . P . Property T Property T for r i g i d kernel functors
14 1 78 1 78 147 231 231 79 248 118 163 2 12 147 148
Ri gi d kernel functor Rigid prime kernel functor
248 118 270 92 92 208 135 131 152
Rigid torsion theory
131
Semi r e s t r i cted prime
163 279
Radical pre-kernel functor Reduced decomposition Reduced gradation Rees module Rees ring Regular local ring Rigid cogenerated kernel functor
Separated f i l t r a t i o n Stable kernel functor Standard polynomial S t r i c t exact sequence S t r i c t f i l t e r e d morphism Strongly graded ring Suspension functor Symmetric r i g i d torsion theory Symmetric torsion theory System of homogeneous parameters
140 231 288 288 15 282 139 139 215
340
Index
T- f unc t o r Topologically n i l p o t e n t element Torsi on theory
147
Trivial f i l t r a t i o n T r i v i a l l y graded
2 7%
Weakly isomorphic i n R-gr
312 130
2
43
Zariski c e n t r a l r i n g
252
ZG-ri ng
254.
