Etale Homotopy

Lecture Notes in Mathematics A collection of informal reports and seminars Edited by A. Dold, Heidelberg and B. Eckmann, Z0rich

100 M. Artin Massachusetts Institute of Technology, Cambridge, Mass.

B. Mazur Harvard University, Cambridge, Mass.

Etale Homotopy 1969

Springer-Verlag Berlin. Heidelberg. New York

This research has been supported by NSF.

All rights reserved. No part of this book may be translated or reproduced in any form without written permission from Springer Verlag. 9 by Springer-Verlag Berlin" Heidelberg 1969 Library of Congress Catalog Card Number 75-88710 Printed in Germany. Title No. 3706

Contents

.

A g l o s s a r y of the categories in which we shall work, and fibre resolutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2.

Pro-objects

3.

Completions.

4.

Cohomologlcal

5.

Completions

6.

Homotopy

7.

Stable

in the h o m o t o p y

category. . . . . . . . . . . . . . . . . . . . . . .

.............................................. criteria

for

6 20 25

~ -isomorphism. ................

35

...............................

60

of completions . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

70

results.

...........................................

75

8.

Hypercoverlngs.

...........................................

93

9.

The V e r d l e r

functor. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

111

and fibratlons.

groups

lO.

The f u n d a m e n t a l

group. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

117

ll.

A proflniteness

theorem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

12@

12.

Comparison

theorems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

129

1.

Limits. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

147

2.

Pro-objects

3.

Morphisms

of pro-objects. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4.

Exactness

properties

Appendix:

and p r o - r e p r e s e n t a b l e

functors.

of the pro-category.

................ 15@ 159

. . . . . . . . . . . . . . . . . . 163

References .....................................................

167

-iThese notes are an expansion of the results announced in [2].

The material was presented in a seminar at Harvard

University during the academic year '65-'66. ~ Our aim is to study the analogues of homotopy invariants which can be obtained from varieties by using the etale topology of Grothendieck.

Using the constructions of Lubkin [21]

or Verdier [3], we associate to any locally noetherian prescheme

X

a pro-object in the homotopy category of simplicial

sets (cf. 2), which we call the etale homotopy type of the prescheme

X .

For a normal variety over the field of complex

numbers, we show that

Xet

is a certain profinite completion

of the classical homotopy type.

This comparison result, to-

gether with a number of others, is in section ]2. Much of our work consists in setting up a reasonable theory of homotopy for pro-simplicial sets.

This homotopy category

turns out to be quite amenable to the techniques of classical algebraic topology.

One may establish the analogues of Hurewicz

and Whitehead theorems (4.3), (4.4), (4.5), and also one has available the techniques of Postnikoff decomposition.

There is

*We thank G. Borkowski for her flawless typing of the manuscript.

-2 -

O.2

a direction in which this pro-category the classical homotopy category:

is more flexible

It is sufficiently

rich so

as to admit pro-flnlte and p-adic completion functors (3.4)).

than

(cf.

Many of our results about schemes are phrased in

terms of these functors. For instance, an algebraically let

X1 , X2

from

X

let

X

be a connected,

closed field

K

of

A

of characteristic

K

in

where the subscript A

cl

denotes

denotes pro-flnlte

there is an example due to Serre singular variety into

C

X

damental groups. homotopy type.

1 Xcl

and

i Xcl

Thus However,

type of a nonslngular

~ .

Then

'

the classical homotopy type, completion.

We remark that

[28] of a projective non-

over a number field

such that

independent

zero, and

A

XI 2 cl ~ Xcl

K

scheme over

be the schemes over the complex numbers obtained

via two embeddings

and where

pointed

2 Xcl

K

and Imbeddings

have non-isomorphlc

of

fun-

do not necessarily have the same

as Serre indicated

to us, the homotopy

simply connected projective

of complex imbedding.

surface is

This gives an affirmative

response in that case to a question raised in [ 2 ]. know whether the same holds for higher dimensional

We do not simply con-

nected nonsingular varieties. Atlyah has suggested analogous Let

L/K

questions for bundles:

be an extension of algebraic number fields, with

a subfield of the complex numbers.

Let

V

denote a variety

K

-3defined over

K , and

over

An imbedding

V~L

.

E

an algebraic vector bundle defined c

of

induce a complex vector bundle Vcl = in

(V~C)cl

K(Vcl )

.

depends on

schemes over a field

of characteristics

let

K

p ~ q

xl,~

X,Y

rings

will

over the topological

space

Rp,Rq

over

of

zero.

K

Ec

Suppose given

with residue fields

and that

reductions modulo

X,Y

have

each of the valua-

be the schemes over the complex numbers obof

K

in

are simply connected. A

Then

1

"

Note that in this case we may conclude X1

and of

C , and assume finally

A

Xcl

motopy groups of

~

be smooth proper connected

respectively,

tained from some imbeddings X 1,~

K

of characteristic

isomorphic non-degenerate

that

Ec

into

c .

two discrete valuation

Let

L/K

The problem is to study how tbe class of

As another example,

tions.

0.3

y1

that the classical ho-

are isomorphic

(abstractly)

since they are finitely generated abellan groups with isomorphic profinite

completions

This result is in the spirit of the question originally posed by Nashnitzer: zero with isomorphic

Given two varieties

reductions modulo p , what additional

ditions will imply that the varieties or, to go even further, stract question:

in characteristic

diffeomorphic?

How much information

from an honest simpliclal

set

K

con-

are themselves homotoplc, We are led to the abis lost when one passes

to its proflnite

completion

4 -

K ?

Suppose

CW-complex.

that

K

is of the homotopy type of a finite

Then we conjecture

number of homotopically

that there is at most a finite

distinct

completion is isomorphic

to

version of the conjecture, gebraic description

0.4

simplicial

K .

In section 7, we prove a stable

and moreover,

we give a complete al-

of the set of distinct

with isomorphic profinite there exist distinct

sets whose profinite

completions.

stable homotopy types

We show by examples

that

stable homotopy types whose completions

are

i s omo rphi c. If where

V E

is a variety over a field is the separable algebraic

on the ~-adic homotopy type of variety,

this representation

K , and closure,

~ = V~

.

G = Gal~/K) then

If

G

V

operates

is an abelian

yields no further information

the operation on ~-adic cohomology.

However,

than

for a more general

V , it may be expected that more precise information is contained in this representation. the Grassmanlan

Perhaps the special case

of m-dimensional

V = G m'n ,

planes in n-space,

K = Q , is

of interest. Indeed,

if

the comparison

^

theorem

the pro-simplicial space

m,n GC

denotes

set

~-adic completion,

(12.9) an operation of ~,n

.

one obtains using G = Gal~/Q)

In other words,

is endowed with a natural action of

~-adic completion.

on

the topological G

on its

The recent ideas of Quillen [13] (b) with

regard to the conjecture of Adams suggest that an analysis of the above action may provide fruitful results in topological K- the o ry. We have not attempted

to develop an analogue of topological

-

O.5

5-

K-theory associated to the etale topology.

Quillen's ideas

cry out for the development of such a functor, however.

Look-

ing even further ahead, one may wonder whether analogues of the Novikov-Browder invariants

(or the more recent theory of

Sullivan) describing the dlffeomorphy types representing a given tangential homotopy type are amenable to definition for an arbitrary scheme, proper and smooth over a field.

This

would bring one closer towards an understanding of the above problem of Washnitzer. We should like to signal two errors in our announcement [ 2 1.

Namely,

(5.4) of [ 2 ] is obvious nonsense.

statement is (8~8) below.

Theorem

A correct

(6.3) of [ 2 ] lacks a co-

homologlcal dimension hypothesis, as in (12.5).

6

w

i.i

-

A glossary of the catesories in which we shall work,

and fibre resolutions.

We will make free use of the classical theory of simpllcial objects with values in an arbitrary category. references are [8], [12], [17], [23], [25].

The principal If

ST

gory of simplex types (objects being ordered sets and morphisms are monotone maps), and gory generated by gory

for

[O,...,n]

is the full subcate-

i < k , then we have, for any cate-

ST

ST(C) . to

(ST/k)(C) . to

C .

C ,

It is the category of contravarlant functors

C .

Similarly we have the category of simplicial

objects truncated at level

ST/k

An=

C , the category of slmpllcial objects with values in

denoted from

Ai

ST/k

is the cate-

k ~ with values in

C , denoted

It is the category of contravarlant functors from The functor which "truncates at level

k ,"

*/k- ST(C) --> (ST/k)(C) will have a left (resp. right) adJolnt, denoted cOSkk)

provided

C

sk k

is closed under finite inductive

nlte projective) limits.

(resp. (resp. fi-

We also consider the compositions,

coskk: ST(C)--> (ST/k) (C) --> ST(C) sh: ST(C)--> (ST/k) (C) --> ST(C) and note that the adjointness properties give us functorial morphlsms, SkkX--> X ,

and

X--> coskkX .

-

Both of these functors understandable

(and for general

different).

Briefly,

that of

1.2

C

the simpllclal

set

SkkX

whose truncation at level

k .

The slmpllcial

able from

X

than

In fact, in all dimensions

k .

is the simplicial

set

is that subsimk

agrees with

simplices in dimencOSkkX

plainer language,

Hom(skkAn, x)

An

pliclal map

of

is the n-simplex,

cOSkkX

greater of

where in this formula

set, which as a functor from

The reader should interpret

n-simplices

is obtain-

n , the n-slmpllces

is represented by the object of the same name in

set).

is the cate-

by adding certain simpllces in dimensions

are given by the set

cOSkkX

C

admit

the situation is hardly any

X , and which has no nondegenerate

sions greater than

An

and coskeleton)

in the case where

gory of sets

X

-

(the skeleton,

visualizations

pllclal set of

7

ST ST

to Sets (or in

regarded as a simpllcial

the above formula for the

in this way:

Every time we have a sim-

SkkAn--> X , we are given a canonical way of extend-

ing it to a simplicial map

An --> X .

The reader should become at home with the coskeleton functor. It is the exceptionally

good functorial properties

allow one to achieve a functorial for slmplicial

sets

obtain, ultimately, pro-simplicial Since

(and functoriality

Let morphism,

cosk k

that

decomposition

is crucial if one is to

any sort of Postnikoff-type

decomposition

for

sets).

cosk k

is defined as a (finite) projective

commutes with projective inductive

Postnikoff-type

of

limits.

Similarly,

sk k

limit,

it

commutes with

limits. C

be the category of sets.

or synonymously,

fibre map

We have the notion of Kan (cf. [23], or for a more

-8general treatment, (B,b)

If

is given as a pointed

be referred

if

one-polnt

f: E--> B simpllcial

to as the fibre of

The functor = Am

[25]).

cosk k

k ~ m .

Thus

1.3 is a fibre map, where f-l(b)

set, then

f .

preserves Kan morphlsms.

Also

cOSkk(e ) = e , k ~ 0 , where

slmpllcial set, and

will

cOSkk(Am ) e

is the

cOSkk(I).. = I , k > 0 , where

I

is the one-slmplex. If

F--> E--> B P

is a sequence of pointed simpllclal morphism,

and

F

is its fibre,

cOSkkF-->

has the property that

sets, where

p

is a Kan

then

cOSkkE p--~ cOSkkB

cos~F

is the fibre of

p' , since

cosk k

commutes with fibre products. It is of course also true that for homotopies,

which suggests

simple interpretation

that

cosk k

k > 0 , cosk k (k > 0)

preserves

should have a

as a functor on the homotopy category,

and

indeed we shall see this below. Let agreeable

S

denote the category of simplicial

simpliclal

sets.

The most

sets with which to work are those which sat-

isfy the Kan condition.

(A simpllcial

set is said to satisfy the

9 -

Kan,

1.4

[17], [12] 3.1 or Extension condition if

morphism. plexes. )

K--> e

is a Kan

These simplicial sets will be referred to as Kan comLet

So

denote the full subcategory of Kan complexes.

The notion of homotopy has good properties only when restricted to the subcategory

So

(see Kan, On C.S.S. Complexes [18] for

more discussion of this), and the technique one has for extending homotopy notions to all of tain adJoint functor:

S

is the construction of a cer-

Let

So/(,,,)A> sl (~) denote the inclusion functor of the homotopy categories. (On C.S.S. Complexes) Kan constructs a left adJoint to he calls

Ex ~ .

In [18], i , which

See also [12] 2.3.1 for a more complete treat-

ment from the point of view of "Calculus of Fractions."

Explic-

itly, Homo(EX~K,L) ~ Hom(K, iL)

where the homs are taken in the homotopy categories, plicial set, and

L

a Kan complex.

Remarks:

(i)

K

a sim-

Kan had not

formulated the notion of adJointness yet when [18] was written, and no such explicit statement as above is made in [18].

However

one easily checks this adjointness, using the morphisms of functors (loc. cit. 5.2):

Ex ~.i ident.

> ident. > i .Ex ~ e @@

10-

1.5

together with homotopy commutativity

for any map

f

of the dia-

gram

K

> Ex~L

e

e

Ex~K

>

L

f

The main category in which we will do homotopy theory is the extended homotop~ K

are simpllcial

morphisms

from

category of simplicial

sets, and for simplicial

X

to

Y

in

K , denoted

sets.

sets [X,Y]

be the set of homotopy classes of morphisms

from

~ ,

The objects of

X, Y

the set of

is defined to X

to

Ex'Y.

Note that

[x,x] = [x, Ex|

[Ex X, Ex'Y]

and the natural inclusion of the homotopy category of Kan complexes into

~

is an equivalence

of categories.

For short we refer to

as the simplicial category. When we wish to speak of a simplicial map from one simplicial set to another

(rather than a morphism in the category

there is the possibility of confusion, former by referring

~) and

we signal that we mean the

to it as an actual simplicial map.

As usual, we shall have occasion to use certain closely related homotopy

categories

orate one that we need,

of diagrams.

first, and obtain

We define

the most elab-

the others as full

-

11

1.6

-

subcategories. ~sq:

The extended homotopy category of simpliclal squares:

Objects are actual commutative squares of s~mpliclal sets, and actual slmpllclal maps:

f

A

i

C

> B

> D

A morphlsm is an equivalence class of actual commutative diagrams,

Ex~A,

Ex~f I

> Ex~B ,

f S

Ex~g'

A m >

B

k

D

C m ~

J

Ex~C '

.> Ex~D , Ex~i,

The equivalence relation is homotopy for such morphlsms of diagrams. Ko,pairs:

The extended homoto~y category of p o l n t e d c o n n e c t e d

simpllclal pairs: slmpliclal sets

Objects are pointed maps of pointed connected X --> Y

regarded as a square as follows:

-

e

12

I.Z

-

> e

1: [

X

> Y

The extended homotopy category of simplicial pairs:

~pairs:

X --> Y

Objects are maps of simplicial sets

regarded as a square

as follows:

[

l

.> Y

X

K

0

:

The pointed connected simplicial category (for short):

Objects are maps

e --> X

simplicial set, and category of

X

Kpair s .

where

e

is the one point constant

is connected; regard (Thus the subscript

~o

as a full sub-

indicates more gen-

O

erally "pointed connected objects.") ~an sq:

The full subcategory of

generated by those

~o, sq

squares whose horizontal maps are fibre maps. ~anpairs:

The full subcategory of

~o,pairs

generated by

those pairs whose horizontal maps are fibre maps. The category

~

CW-complexes, denoted

is equivalent to the homotopy category of ~ .

One passes from

~

to

by

the geometric realization functor, and from

~

to

by

the singular functor (these functors being adjoint: of [19] and also

[22],

related categories to

[12]).

As with

K

II,

S ,

cf. Prop. 9.1

there are closely related

, which we will use.

13 ~o:

1.8

The pointed homotopy category of connected pointed

CW-complexes. ~pairs:

The homotopy category of CW-pairs.

continuous maps of CW-complexes, commutative

squares,

f: X --> Y

Objects are

where morphisms are

again,

f

X

> Y

,l

I

X'

..> yt

considered up to homotopy equivalence. The singular functor to

~o

and

~pairs

to

S

may be interpreted as bringing

~pairs

We may form the categories

" pro-~ , p r o - ~ o , prO-~pair s , etc.

These will be referred to as the pro-simplicial pointed pro-simplicial etc.

category,

By a pro-simplicial

We will also pass,

~o

category,

the pro-simplicial

the

pairs category,

set we will mean an object of pro-~ .

on occasion,

to the equivalent

categories pro-~,

pro-~ o . (The reader may wonder why we do not stick either to the category of CW-complexes The reason is this.

or to the category of Kan complexes.

On the one hand, we have occasion to make

very strong use of fibre resolution and coskeleton functors, ing rise to Postnikoff

decompositions

have been systematically Kan complexes.

niques of attaching

etc., and these techniques

and quite elegantly developed

On the other hand, cells,

giv-

only for

we also use the "adjoint"

to kill specific obstructions,

tech-

etc.

- 14 -

1.9

which have been systematically exposed only for CW-complexes. ) Fibre Resolutions:

The problem is to "replace a map, up to

homotopy, by a fibre map. Proposition 1.1:

,!

The inclusion functors

Kan--> Ko,pair s ~ansq --> ~o, sq

are equivalences of categories. Proof:

One may give various procedures for functorially

associating to any actual simpliclal map of pointed connected pairs

p: E --> B

actually-commutative

E

(1.2)

c

>

triangles,

Et

p~a B

such that ~o,pairs

p' "

[12], where

is Kan, and

a: p --> p'

is an isomorphism in

For example, we may take the construction a

is, in their terminology,

and therefore an isomorphism in

(5.5.1) of

an anodyne extension,

~o,pairs "

The construction is

functorial and hence yields isomorphy-inverses

to the inclusion

functors appearing in the statement of proposition i,i. that we do not need that pl : E' --> B

E

and

to be a fibre map.

B

Note

be connected in order for

All that is necessary is that

-

E --> B

15

-

l.lO

be surjective on connected components.

pointed connected pairs to diagrams

The functor from

(1.2) will be referred to as

fibre resolution. The 3 by 3 Lemma:

Given a diagram of pointed connected

simplicial sets and actual simplicial maps,

f Xll

> X12

(6) X21

such that

g,h

h

'> X22

induce surJections on fundamental groups, we will

form, in a certain sense, a double fibre resolution of (8).

What

is meant is that we will end up with a three by three square array of simplicial sets and maps, any vertical or horizontal line of which will form, up to homotopy, a fibre triple. To begin, let us fibre-resolve

(5) vertically

(i.e. make the

vertical maps fibre maps) and consider the fibre triple obtained.

Xo1

(6v)

l ,L v

> X02

L l v

Xll

-> X12

~l

> X22

- 16 -

The base point of

XI2

i.ii

actually lles in

X02 .

a surjectlon on fundamental groups, we have that

Since X02

g

induces

is connected.

Therefore we may form the horizontal fibre resolution of the above diagram, yielding:

00 ~ >

(6vh)

Lemma 1.2: mental groups.

Ol

X02

i

ih

X20 ~ >

X~I ~ >

Suppose that

g, h

__>

I

X22

induce surJections on funda-

Then the left-hand row of (6vh) is, up to homo-

topy, a fibre triple. Proof:

Denote (6vh) by

~i " and form the three-by-three

diagrams c~I c

as follows:

a2

c~2 c

is made from

m3

c

(~#

ml ' and

~#

is made from

fibre-resolving the bottom three vertical arrows, from

~2

~i "

by

is made

by fibre-resolving the right-hand three horizontal arrows.

Denote by of

aS

~3

Then

Remark:

r i, c i , the upper row and left-hand column, resp. rl, rS, c2, c4 A morphlsm

~

are flbre-triples. of fibre-triples

- 17

-

1.12

A

> A'

B

>

C

> C'

,[ [ ,l l

with connected

fibres

A,A'

B'

w h i c h induces

on two out of three of the s i m p l i c i a l induces

a homotopy

exact h o m o t o p y classical

equivalence

sequence

Whitehead

vi

and

ci

and

sets involved,

on the third.

of a fibre

triple,

This

equivalence actually

is by the long

the five lemma and the

theorem.

If, w h e n d e a l i n g with triples as the "fibre"

a homotopy

g

A --> B --> C , we refer to g

as the "proSection,"

have a common

"fibre,"

A

then we m a y say that

which we denote

Xi .

The

n a t u r a l maps

c I c c 2 c c 3 c c$

induce h o m o t o p y - i s o m o r p h i s m s our remark

,

v I c v2 c v 3 c v4

of the "projections."

We m a y a p p l y

to the i n c l u s i o n maps:

x 1 = x2 = x 3 ~ x4

where

the composite

isomorphism

since

of the first vI

and

v3

two maps must be a homotopyare fibre-triples.

of the last two maps must be an i s o m o r p h i s m

since

The composite c2

and

c#

18 -

are fibre-triples.

1.13

Thus all maps are homotopy-isomorphisms,

con-

cluding the proof of the lemma. Examples:

Let

X

be in

Kan condition.

Let

n ~ 0 .

Ko

and suppose it satisfies the

Form the object

X --> c o s ~ X

in

Ko,pairs ' and fibre resolve it, to obtain the fibre triple,

X (n) --> X ' - >

giving us the canonical .

If

X

cos~X

(n-l)-connected flbre d e c o m p o s l t l o n o f

is a general object in

the Kan condition, by applying

Ko 9 we first make it satisfy

Ex ~

to it, and then follow the

above procedure to obtain its canonical composition.

(n-1)-connected fibre de-

To indicate that we have applied

Ex ~ , we denote

the sequence obtained:

x(n) _> X' --> c o s ~ X ' p'

.

We also obtain sequences,

X (n) --> c o s ~ + i X '

where

Pn+l = c~

--> Pn+l

' using that

cos~X'

coskn+iCOSk n = cosk n .

This latter sequence, is the n-th fibre triple in the canonical Postnikoff decomposition of Maclane space o f type

X .

(~n(X),n)

Clearly

X(n)

is an Eilenberg-

.

This being functorlal, we have the identical situation in pro-K o .

Thus, for

X

in

pro-~ o

we have objects of

pro-Van ,

-

19-

1.14

x(n) _> x' -> cos~x' X(n)->

for all

cos~+IX'-->

n ~ 0 , where, again

trivial except in dimension

X(n)

cos~X'

has homotopy pro-groups

n , where

~n(X(n)) --> ~n(X) .

-

w

20

-

2.1

Pro-objects in the homotopy category.

We work in this section with

~o ' the category whose objects

are connected pointed CW-complexes and whose morphisms are pointed homotopy classes of maps between them. phisms from

X

to

Y

by

IX,Y] .

functorial constructions in

~o

Denote the set of mor-

We are going to recall some

which yield

(cf. App.) analogous

constructions in pro-~ o : To begin with, the homotopy and homology groups are functors on

~o ' hence if

X = [Xi]

is a pro-object,

its homotopy and

its homology with values in some abelian group

A E (Ab)

are nat-

urally defined as pro-groups and/or pro-abelian groups:

v n(X) = {Vn(Xi ) } (2.1) Hn(X,A ) = [Hn(Xi,A)]

,

A E (Ab) .

For our purposes, it is preferable to take the actual limit group when defining cohomology. and since

lim

Since cohomology is contravarlant,

is an exact functor,

this works well.

Thus the

cohomology is (2.2)

Hn(X,A) = lim Hn(Xi,A)

A E (Ab)

i

which is an abelian group. Homology and cohomology with values in twisted abellan groups can be defined in the obvious way: Vl(X) = [Vl(Xi ) ]

on

Suppose given an operation of

A , i.e., a map

2.2

- 21 -

Vl(X

in pro- (grps) .

) . T_> Aut(A)

Then

Hn(X,AT) =

lim -@

(i,r (2.3) ~ ( X , AT) = [Hn(Xi,A~) } (i,$)

where the index category consists of pairs is a homomorphism

~l(Xi) --> Aut(A)

(i,$)

representing

such that ~ .

It is

immediately seen that this category is cofinal with the original index category

[i] , hence that twisted coefficients may for

practical purposes be thought of as obtained from a compatible family of maps

~l(Xi) --> Aut(A)

.

Although fibre sequences and related constructions are not functorial in ial. For

~o

in general,

certain important cases are functor-

For instance, as remarked in w

there is the coskeleton.

X 6 ~o ' its coskeleton is characterized by the following

universal property:

(2.4)

The homotopy groups of

cosknX

vanish in dimensions

> (n-l) , and the canonical map

x - > cOSknX is universal with respect to maps into objects whose homotopy is zero in dimension > (n-l) . Thus we can define cosknX

for

X = [Xi] 6 pro-~ o

as the

22 pro-object Let

{cos~Xl} X --> Y

2.3

-

.

be a map in

~o

which

(i) induces an isomor-

phism on homotopy in dimensions < n , and (ii) such that trivial homotopy in dimensions > n .

F

is (n-1)-connected.

The homotopy classes of maps of any (n-1)-connected 0Y

sion > (n-l) .

are trivial since

0Y

has

Then if we represent the

map up to homotopy by a fibratlon, its fibre

loop space

Y

W

to the

has zero homotopy in dimen-

Therefore the exact sequence of homotopy classes

of maps ([29]w [w, oY]-> [W,F]-> [W,X]

shows that (2.5)

F

F

is characterized in

is (n-l)-connected,

trivial, and Thus if

Xi --~ Yi

F--> X

~o

as follows:

the composed map

is

is universal for these properties.

is a filtering inverse system of maps in

each of which satisfies the above assumptions fixed

F--> X--> Y

~o

(i), (li) for some

n , then there is a canonically determined inverse system

of maps Fi --> Xi --~ Yi

where

Fi

is the fibre of the map

Xi --> Yi " etc...

This situation arises for instance when

Yi = c~

' so

that Postnikov decompositions and related constructions extend naturally to the pro-category,

with the obvious definition that

(2.6)

G-

K(G,n) = [K(Gi,n)}

[Gi] E pro-(grps)

.

23

2.#

-

A similar notion is that of covering space.

H

l(X)

is a subgroup, where for the moment

Suppose X

is a CW-

complex.

Then the covering space

group

is characterized in the homotopy category

H

~

corresponding to the sub~o

by the

following property, as is easily verified:

(2.7)

For any

W

the map

the subset of the subgroup Of course if

H

X H --> X

[W,X] H

[W,~]

identifies

of maps which carry

~l (W)

with into

(x) .

of

is a normal subgroup, then

~

is Just the

fibre (2.5) of the map X--~ K(~I(X)/H,I ) .

Now let

X = [Xj)

a sub-pro-group.

be in pro-~ , and let

Replacing

assume (App. 3.3) that

H

X, H and

X

H~-> ~I(X)

be

by isomorphic objects, we may have the same filtering cate-

gory and that we are given compatible in~ections

Hjrepresenting the map.

l(Xj)

Via (2.7), we obtain an inverse system

of maps (Xj)Hj--> Xj ,

hence a map of pro-objects X H --> X .

-

For any

W = [Wi ] E

(Xj) H

24

pro-~ o , a map

iff some representing map

-

2.5

W--> Xj W i --> Xj

factors through does, and this will

J occur (2.7) iff the map Since

~l(W) --~ ~l(Xj)

(W, XH) = lim~ (W,(Xj)Hj)

factors through

Hj .

, we obtain

J Corollary

(2.8):

Let

X E pro-~ o , and let

a monomorphlsm of pro-groups. with a map for each

X H --~ X

There is an

W E pro-~Eo , [W,X HI

Note that in particular,

together

is carried to the subset of ~l(W) --~ ~l(X)

this characterizes

of the choices we made in the description. XH

X H E pro-~ o

be

which is characterized by the property that

of maps such that the induced map

refer to

H--~ ~l(X)

[W,X]

factors through H. XH

independently

We will, naturally,

as the coverlng space determined by

H .

25 w

Completions.

(3.1):

Definition gory of

(groups)

A class

@

of groups

is a full subcate-

satisfying

(o)

0

(i)

A subgroup

Ee_.,.

0--> A--> B--> B 6 @

3.1

iff

of a C-group

C--> 0

is in

is an exact

@ .

sequence

Moreover, of groups,

if then

A, C E ~ .

A class

@

is called a complete

class if in addition

the

following axiom holds (ii)

If

A , B 6 @ , then the product

self indexed by Of course,

B

is in

the index

B

AB

of

A

with it-

@ . is just to give a bound

on the cardinal-

ity which is required. We will be primarily class of finite groups, ucts of primes

@

consists

Corollary

in the case that

@

is the

or of finite groups whose orders are prod-

coming from a given set of primes.

clearly complete that

interested

classes.

However,

These are

as we shall see,

the case

of all groups has some interst for us.

(3.9) :

A complete

class

~

has the following

properties :

(i) G

and

then

If K

K

GDNmK

are groups

is normal in

N , and if

contains

a subgroup

H

such that G/N , N / K

normal in

G

N

is normal in are ~-groups,

such that

G/H E ~ .

-

(ii) that

If

G 6 ~ , and if

26

3.2

-

A

is an abelian G-module

A E @ , then the cohomology groups

Hq(G,A)

To see (i), note that the normalizer of N .

Therefore

the set

dexed by a ~-group hence is a ~ g r o u p ~-group

too.

the standard apply

C .

of conjugates

Thus

N/nK a

by (2.1)(ii).

of

are in in

K

G in

contains

G

G/aK a

is in(N/K) c ,

is a subgroup of

By (3.1)(i),

@ .

is

a

(ii), it suffices to remark that the terms in

cochain complex are of the form

A G•215

, and to

(3.1) (ii). Let

of maps map

For

[K a]

K

such

G = G~>

~ --~ ~'

[Gi]

be a pro-group,

C , where

C

and consider the category

varies in a class

is a commutative

triangle

9

G

~ , and where a

> C

C'

It follows easily from axiom

(i) above that the opposite

gory of this category is filtering; cofinal sub-category.

Therefore

group which we denote by Clearly

~

cate-

and it obviously has a small

the range

C

determines

and call the C-completion

of

a proG .

is a pro-object in the category of ~-groups, and the A canonical map G --> G is universal with respect to maps into pro~-groups.

~

We restate this fact as

Cor011ary

(3.3):

Let

~

be a class of groups.

The inclusion

27

3.3

of p r o - @ into pro~grps) has an adjoint

A9

9 pro-(grps) --> p r o - ~ .

Equivalently,

for any

G E pro-(grps),

restricted to the category If

C

~

is pro-representable

is a class of groups, we will denote by

subcategory of

~o

~ .

in

G .

C~ o

the full

Our object in this section is to

prove the analogue of (3.2) for

pro-~ o

Hom(G,.)

consisting of pointed CW-complexes whose homo-

topy groups are all in

Theorem

the functor

(3.4) :

Let

~o

:

X E pro-~.

There is an object

A X--> X , called the th~ C-completion of

and a map

is universal with respect to maps from

X

A X E X , which

to elements of p r o - G ~ o .

Equivalently, (3.4'):

The inclusion of p r o - G ~ o

in pro-~ o

has an adjoint

A 9 p r o - ~ o --> p r o - ~

(3.4"): (X,.)

Let

o

.

X 6 pro-~/o , and consider the covariant functor

restricted to the category

representable in

~o

~o

"

This functor is pro-

"

The equivalence of the three assertions follows immediately from the definitions of pro-objects.

Of course,

pro-~ o

the object in

which represents the functor (X,.) of (3.4") is just A the Crcompletion X of (3.4). Thus we may think intuitively of A X as obtained from X by ignoring all information except that pertaining to maps to objects of

~o

"

28 -

3.#

We will prove the theorem in the form of (3.4"). to be shown category

J

(App. 1.2) is that given

X 6 pro-~ o , the opposite

to the category of maps

gory, where a morphism

~ --> ~'

What has

X ~> W

is a filtering

is a commutative

cate-

triangle

r X

> W

"

WI

and that

J

has a small cofinal subcategory.

lim~ Hom(Xi,W )

when

X = IX i} E pro-~ o

and

Since

Hom(X,W)

=

W E ~o ' one sees

immediately that it suffices to prove the theorem for each of the X i , i.e., in the case that Now it follows from

X 6 ~o "

(3.1,

@

is closed under

products

of groups.

product

W 1 x W2 .

(1.2b).

This axiom reads as follows in our situation:

(3.5)

is a diagram in

Hence if

(i)) that

Thus

J

are in

satisfies

(1.2a).

@~o " so is their It remains to show If

> Wl_7_ > w2

x

~o

W 1 , W2

with

i@ = j~

and with

there is a factorization

X

> W

WI

Wl, W 2 E @~o ' then

29

with

ik = jk

such that

Let us represent complexes

-

W E ~o

"

(3.5) by a diagram of actual maps of CW-

such that

i~

is homotopic

letters for the representing maps). h: i~ N j$ .

3.5

to

j~ .

(We use the same

Choose such a homotopy

We are going to construct inductively a sequence

of factorizations

Lqq

X

X

W1

for each jSq .

q

such that

~n(Xq)

More precisely,

we take

E ~

Xq_ 1

and the homotopy

hq: i~q ~ j~q

of

iSq-1 ~ J~q-1 "

Xq_ 1

Xq

is to

(q+l)-cells,

is to extend the homotopy

Then the direct limit

~n(X)

E ~

W

of the system

Denote by

N

, and let

on a set

S

Since

Gs: Dq+l -~ W1

Replacing

X

by

~: F - ~

(q+l)-cells

N

to

X . ~: ~q(X) --~

be a surjec~iive map of a free group

of generators

~(s)

of

be a pointed

N

onto

N .

For each

~s

"

s E S ,

continuous map representing

= 0 , we may find a continuous

extending

Xq_ 1 , we

n ( q , and we are to construct

the kernel of the homomorphism

ms: 3D q+l --~ X

~(s) .

Xq , we may by induction assume

for

by the addition of some

~q(W1)

of

is already constructed.

may assume that

let

q ~ 0 ,

i~q N

Xq 's, has the required properties.

that

F

For

by the addition of certain

To prove the existence

Xq

n ~ q , and that

X = Xo .

be obtained from

hq-l:

for

Fix such a

Gs

pointed map

for each

s E S o

-

Construction the pointed

i:

Z

Z

Z

For

which is the

the base point,

(q+l)-skeleton

= (Dq+lxo)U(SDq+ixI)O(Dq+lxl)

is homeomorphic

mo rphi sm.

denotes

of

form Dq+lxl.

as follows:

Z = 8 (Dq+ixI)

Thus

3.6

-

d E D q+l

If

CW-complex

We may describe

3 0

with

S q+l .

.

Fix such a (pointed homeo-

s 6 S , define a pointed map

fs: Z --> W 2

as

follows :

(•

On

Dq+ix0

(ii)

On

8Dq+ixI

between

i@

and

(iii) On

, let , let

Dq+lxl

~q+l(W2)

the h o m o t o p y

h

' where

fs = has

h

is the h o m o t o p y

JW . , let

After the i d e n t i f i c a t i o n class in

fs = iBs "

of

which

to

fs = J6s " Z

with

represents

XU m D q+l .

sq+l ' fs

determines

the obstruction

Since

S

a

to extending

freely generates

S

we obtain in this way a h o m o m o r p h i s m Construction Let

(A,B)

Y

2:

Let

V

f: F ->

be an integer

denote the pointed

CW

greater than

pair

Ay = D q+l V D q+l v ... v D q+Iv By = S q

v Sq

v ... v S qY

We must fix an explicit pointed map

cv:

(Dq+l'sq) ->

(A,B)y

q+l(W2) 0 .

F ,

31

-

This map should have the property represents

el + r

+ "'" + cy

than

1 , and it represents

Here

ei

is the class in

Let

d

denote

-

3.7

that the class

~c

in the case where

the product ~q(B)

r162

represented

the base point of

q

cy by

E is greater when

q = 1 .

S~ c B

. Define, in analogy Y with construction 1 , Z(y) to be the (q+l)-skeleton of A xI . Y Then we may also describe Z(y) as follows:

z(y)

where

Zi

is the

the class

is represented

v z2

(q+l)-skeleton

m a y then be interpreted represents

-- z I

by

Construction

3:

as a minlmal word in

D~+lxI

6y in

Given an element

are

.

The map .

CyXid

This

c

Y

~q+l(Z(y)) , where

6i

z 6 F , we may write

z

S , in a unique manner:

eI

r

Y

.

z = S1

where the

v z

c : Z --> Z(y) Y

61+62 + . . . +

Z i c Z(y)

v ...

of

as a map

A

~ 1 .

az: By--> X , 8z:

... sy

r

Define maps

A

Y

--> Wl ' fz:

Z(y) --> W 2

as follows: (i) 6s i

on

If

r

= 1 , let

D q+l c Ay

(ii) If

r

and let

= -i , let

~z = ms i fz = fs i ~z = ~si J

on

S q c By

on on

let

Z i c Z(y) siq , let

Bz =

. 8 z = BsiJ

-

q+l Di , and let

on

fz = fsi J

32

-

on

3.8

Z i , where

evident pointed linear involution of

denotes the

J

D~+ll , S~

or

Zi

about

a hyperplane. By construction,

the map

fzCy: Z--> W 2

represents the element

q+l(W2) 9 Choose a set of generators for the image group in

Vq(X) , where

G = ker(f: F--> Vq+l(W2))

, and let

be a set of representatives of this generating set. z

in

Z

define a map

c z . S q --> X

by

c z = ~zCy .

denote the pointed CW-complex built from cells

D q+l z

--> W 1

as the pointed map extending

on

D q§

for all

X

z E Z , via the maps ~

of

X

G

Z c G

For each Let

Xq

by attaching cz .

on

~G

(q+l)-

Define

given by

~q: Xq 6zC u

.

Z

Since Let

hz:

hq: Xq•

z E G , the map

(Dq+l•

--> W 2

--> W 2

by

X q , ~ q , hq

Then

E ~ , and since F/G

is in

K c ~q(Xq)

~q(X) c N c K .

hence is in in

.

i~q

and

D q+l z •

j~q c

satisfy the necessary properties,

F/G ~ .

~q(Xq) E ~ .

Thus

~ , and

Now since

is isomorphic to a subgroup of The group

~q(Xq)

by annihilating the elements of the image Let

Define

hz .

this group,

q .

fzCy .

h , and which is given on

and it remains only to check that

~q(X)

is homotoplc to zero.

to be the pointed homotopy between

We claim that

~q+l(W2)

Z --> W 2

be an extension of

which is an extension of Xq•

fzCy

is obtained from ~Z

of

be the normal subgroup generated by The quotient ~q(X)/N

N/K

is a quotient of

is a subgroup of

~q(Xq) = ~q(X)/K E ~ .

Z

under ~Z . F/G ,

~q(Wq) , hence

This completes the proof

-

that the category construction of

J W

33

3.9

-

is filtering,

and it is clear from the

above by spanning in some cells,

has a small cofinal subcategory,

which proves

that

J

(3.4).

We want to state explicitly the construction of the cofinal index category: Corollary

(3.6):

Let

~

be a class of groups and

The following construction yields maps

X --> W

which are coflnal among maps to elements of Suppose first that fine

X

X E ~o

Choose a normal subgroup tient group is in Let

X

q

X

ically to If

W

be the limit of the

o

: De-

= X .

N c ~q(Xq_l) S c N

such that the quo-

which generates

N . q+l Ds

by spanning a (q+l)-cell

q-i

into some sphere representing Now let

X

~ , and a set

be obtained from

~o

W E ~o

is an actual CW-complex.

recursively as follows:

q

with

X E pro-~o 9

s

in X

~q(Xq_ l)

for each

's, so that

q

X

s E S .

maps canon-

W . X E pro-~ o , say

X = {Xi} , first choose an

then apply the above construction,

replacing

From this explicit description, Corollary

(3.7):

by

Xi .

we get immediately

In the notation of (3.3, 3.4),

A

Notice that moreover for in corollary

X

and

i

(x)

X E ~o

.

the maps

X --> W

described

(3.6) induce surJections on fundamental groups and

34 -

yield a pro-object of maps,

IX --> W]

(~o,pairs)

3.10

in the category of homotopy classes

such that the range

is isomorphic

[W]

to

A

X .

Let us denote this pro-object by

to extend naturally to a functor from

X --> ~ . ~o

It is easily seen

to pro-(~o,palrs)

.

Thus we obtain Corollary

(3.8):

There is a functor

pro-~ o --> pro- (~o,pairs)

carrying an

X 6 pro-~ o

d-completion

of

Remark

where

A

to an object

x->

whose range is the

X .

(3.9):

For

X E pro-~(o , X = {Xi] I E l

denotes ~-completlon,

a sense as in App.

(4.4).

and the rig~t-hand

' we have

side is given

-

w

35

-

4.1

Cohomologlcal criteria for

It is not true that a map

~-iLsqmorphlsm.

X --> Y

of objects in pro-~ o

which induces isomorphlsms on homotopy an isomorphism.

To see this, let

various coskeletons

cosknX i

~q(X) ~ ~q(Y)

X = IXi]

is itself

be in pro-~ o .

The

form a pro-object indexed by pairs

(i,n) , with the obvious canonically induced maps between them. Let us denote this functorlally determined pro-object by

(4.1)

X ~ = [ c o s ~ X i]

when

It is clear that the canonical map on all homotopy and homology.

XW :

X = [Xi] E pro-~ o .

X --> X @

But if

X

induces isomorphisms

is an actual CW-complex,

then [X ~ ,X] = llm [cosknX, X] -@ n h X--> X"

and it follows immediately that the canonical map be invertlble unless all the homotopy groups of a certain point, i.e., unless Definition

(4.2):

A map

X = cosknX f: X --> Y

a ~ - i s o m o r p h i s m if the induced map

X

cannot

vanish above

for large n . in pro-~ o

f@ : X~

--> Y @

is called is an iso-

morphism. It is clear that this is equivalent with the assertion that for every

n , the induced map

cosknX--> cosknY

is an isomorphism.

The theorem below gives cohomological criteria for a map f: X --> Y

which determine for a complete class

~

whether or

-

36

#.2

-

A

not the induced map

f

of @-completions

We will use the following terminology: twisted coefficient M E @

and a map

group

~l(X) --> Aut M

through some subgroup of ficient groups

: Vl (~) (3.7) on

M

Aut M

X

X 6 pro-~ o , a Ct-

is an abelian group

in pro-(grps) contained in

therefore correspond M , i.e.,

on

on

If

which factors @ .

Such coef-

to operations of

~l(X)

to twisted coefficient groups

M 6

~ . Let

c Vl(~)

H : ~XAlV 1 X .

M E

is a ~ - i s o m o r p h i s m .

in

Vl(X)

be a sub-pro-group.

Its inverse image

determines a covering space (2.7)

XH

of

Such a covering space will be called a ~-covering space. A

Thus

@-covering

spaces are classified by

Th=e o r e m ( 4 . 3 ) : f: X--> Y

Let

@

~v~(X) = ~l(~)

.

be a complete class of groups, and

a morphism in pro-~ o .

The following assertions are

equivalent: (i)

9: ~--> ~

is a ~-isomorphism.

(ii) Vl(X) --~> Vl(Y) , and for every d-twisted abelian coefficient group

M 6 @ ,

Hq(Y,M) ~> Hq(X,M)

(iii)

~l(X) ---> ~l (Y)

sponding ~-covering spaces abelian

for all

q .

and for every induced map of correX' --> Y'

and every (untwisted)

A 6 @ ,

Hq(Y',A) --~> Hq(X',A)

for all

q .

37

If we let

~

-

4.3

be the class of all groups, we obtain the fol-

lowing corollaries: Corollary

(4.4):

Let

f: X--> Y

be a map in pro-~ o .

The

following are equivalent: (i)

f

is a ~-isomorphism.

(ll) cos~f oos~x -> cos~Y ( ~ ) ~n(f): ~n(X)--> ~n(Y)

is an isomorphism for each n.

is an isomorphism for each

n .

The equivalence of (4.4)(i) and (ii) is trivial, and so is the implication

(1) ==> (iii).

and we will verify condition in passing to the limit: patible maps resenting

f .

It remains to show (iil) ==>

(ii) of (4.3).

We may suppose

fi: Xi -> Ti

This is an exercise

(App. 3.3) given com-

with a single index category

Consider the morphlsms of fibre sequences

K(~n(Xl),n)

> cos~+IX i

K(~n(Yi),n )

,

> cosknX i

> cos~+lY i ~ >

Let us denote it by

_>

Ki

El

> Ei ~ >

> B1

Bi

(ii),

cos~Y i

I

rep-

-

38

4.4

-

We may assume by induction that (4.3)(ii) is proved for To prove it for cient group.

coskn+ 1 , let

M

cosk n .

be a twisted abelian coeffi-

We may assume given compatible operations of

Vl(Xi) , Vl(Yi)

on

M (2.3).

Then we obtain a filtering system

of morphisms of spectral sequences indexed by

I

H p (Bi,Hq(Ki, M ) ) ---=> HP+q(Ei,M)

H p ( B I , H q ( ~ , M )) ===> HP+q(EI,M)

which converges to a morphism of limit spectral sequences whose abutment is the map

HP+qCcoskn+iY, M )

,, ,> HP+q(coskn+iX,M ) 9

Hence it suffices to show that the limit is biJectlve on Consider the index category of maps

lim H p

J

(Bj,Hq(Kj,M))

~

(ilimJ)

lim ( lim HP(Bi,Hq ) J ili'J (Kj,M) = lim J H p (B, H q (Kj,M))

lim HP (B' ,Hq(Kj,M) ) J

i --> J

in

I .

E~ q

We have

H p (Bi, Hq (Kj, M ) ) (App. l. ll(b) )

where

B = cosknY .

where

B t = cosknX

the last isomorphism being by the induction hypothesis.

Hence

6

-

39

4.5

-

we need only show

l~m H p(B',Hq(Kj,M)) ~ i ~

H p(B',Hq(K~,M))

.

But H p(B',Hq(K(r

is a functor of

r , and the two pro-objects

Vn(Xl)

and

~n(Yi)

are isomorphic, which completes the proof. Corollary Let n

(4.5):

X E pro-AGo

(Hurewlcz Theorem in the pro-category).

and suppose that

is an integer > 1 .

Vq(X) = 0

for

q < n , where

Then the canonical map

Hn(X) is biJective. For, say

X = {Xi} , and consider the fibre sequence

F i --> X i --> c o s ~ X i 9

Put

F = IFi] .

it does for F --> X

F

Since the Hurewlcz theorem holds for each as well.

Fi ,

Thus it suffices to show that the map

is a ~ -isomorphism.

Now the pro-object

cosknX

has

trivial homotopy, hence is trivial by (4.4), and thus has trivial cohomology.

Therefore for

A E (Ab) , the cohomology spectral

sequence of the above fibre sequence for the coefficient group A

collapses to yield

4.6

- 40

Since

X

for all

~> Hq(F,A)

Hq(X,A)

is simply connected and

A E (Ab)

q .

is arbitrary, we

can apply (4.3) (li). Proof of (4.3):

(li) => (i).

Suppose (ll) holds. Notice first of all that to show A A A f: X --> Y is a ~ -isomorphism is equivalent with showing that for every

W E ~o

which is of finite homotopy dimension, i.e.,

whose homotopy vanishes in high dimensions, the map

f

the composition with

induces a blJection

Ix, w] <-~ Ix, w]

For, if we let

W

have homotopy dimension < n , then

Ix,w]

[~,w] _ [cOSknX,W] ^ A

and similarly for

Y .

Therefore,

ized by its maps to objects of

since

@~o

cos~X

of homotopy dimension < n ,

this will show ^

cOSknX ~

for each

cos

n .

The problem is the following:

X

~>W,

is character-

Given a map

41 -

W

as above, to factor it through

4.[

Y , i.e., to find a commuta-

tive diagram

f

X

.> Y

(#.6)

and to show that it is unique. given a filtering category

By (App. 3.3), we may suppose

I

of diagrams in fi

Xi

(#.7) W

Since by (4.3)(ii), ~l(W)

~l(X)--~> ~I(Y) , the map from

factors uniquely through

Vl(Y ) .

~l(X)

to

Therefore we may (App.

3.3) further suppose given compatible commutative triangles for each

i ~i (Xi)

> ~i (Yi)

i(w) where of course the map

Let

J

~l(Xi) --> ~l(W)

be the category of pairs

map making the triangle

is the one induced by

(i,$)

such that

r

is a

-

42

-

~.8 fl

Xi

i

> Yi /

(4.8) W

commute a map

(up to homotopy),

(i,$) --> (i',~')

is

The following lemma will prove both the existence

and

i--> i'

in

I

where a morphism

such that

Yi t

"> Yi

(4.8') W

commutes.

the uniqueness

of (4.6), hence will complete the proof that

(li) => (i): Lemma

(4.9):

J

is filtering and is cofinal with

I .

We leave it to the reader to verify that this is sufficient. For our particular index categories,

the assertions

of (4.9)

are implied by the following two facts, as is easily seen: (4.10): such that (b) triangle

~i'

For any

i E I , there is an

factors through

Given two maps

$~'$~: Yi' -> W

i--> i'

in

I

fi': Xi' -> Yi' "

$1" ~2: Yi -> W

(4.8)', there is an

composed maps gory).

(a)

i--> i t

in

are equal

making a commutative I

such that the

(in the homotopy cate-

-

43

)4.9

-

We shall show (a) by an inductive argument on the homotopydimension of

W , and an application of obstruction theory: vanishes for

q > n .

Suppose that (#.10)

has already been demonstrated for

cos~_iW

.

Suppose

~l(W)

q

(W)

factors through

(xi) ->

Since

~l(Yi) , we may also suppose

n>2.

Thus we have a homotopy-commutative diagram,

Xi

"'

>

W

Yi

> cos~_lW

Our problem is to find a morphism a homotopy-lifting,

such that one has

as indicated:

Xi"

'

W

I.

i --> i"

>

Yi"

> coskn_iW

An elementary construction:

A continuous map

f: X --> Y

cellular ([29] page 403) if

of CW-complexes will be called

f(S~X)

c SknY

for all

One has a notion of cellularity for homotopies,

n . as well (loc.

cit.) and the well known fact (loc. cit., chap. 7, sec. 6, thm. 17) that if

f

then

is homotopic rel A to a cellular map.

f

is cellular when restricted to a subcomplex Let

A c X ,

44

#.i0

-

X I

.> yw

X

-> Y f

be homotopy-commutatlve, F: tf' N fs .

with a given homotopy-commutatlvlty,

Suppose that

f,f'

are cellular.

form the CW-complex mapping cylinders of

f,fl

Then we may to get the dia-

gram

XI

> CI fl

L

Lt

X

> C

f

where the horizontal arrows are now the natural inclusions as CW-complexes.

Using

obtain a new map

~

F , we shall describe how to modify and an actually-commutative

XI

diagram,

> C'

s[

X

~ , to

f'

~

It p

-f

Recall that the mapping cylinder

> C

C

is the fibre sum indicated

in the diagram below, in the category of CW-complexes and continuous maps, (where

Xxl

in the diagram below is given its

-

natural CW-structure

45 -

4.11

(loc. cit. p. 401, ex. i0)):

x

x "'~''tUJ > xxl

Y

>

C

J We now consider the map (i)

7: X'xl --> C

~(x',r) = (s(x'),2r-1)

(ii) T ( x ' , r ) =

jF(x',2r)

given as follows:

for

for

1/2 < r < 1 .

0

<

r

<

1/2

.

Since

X'

x{o? i~ > X tx I

ft

Y'

jt

>

is commutative, we get a unique map struction makes tion,

s

(yF)

C

tF: C' --> C

above actually-commutative.

is cellular, we might modify

retaining the actual-commutativity of lar.

If

F

(for short:

F

If, in addi-

to a cellular homotopy,

(yF) , making

tF

is a homotopy relative to the q-skeleton of

celluX'

a homotopy rel q ), the above may itself be taken

to be a homotopy relative to Remark:

which, by con-

SkqX' x l .

The homotopy-commutative diagram (~) induces no

homomorphism on relative cohomology,

-46

H*(f,G)--> but

H*(f',G)

,

certainly does:

(YF)

H* (f, g)

>

H* (f' ,G)

II

II

H*(C,X;G)

> H*(C',X';G)

Thus, a homotopy-commutatlve diagram of homotopy-commutativity

F

(Y)

together with a choice

determine a morphism on relative

cohomology.

2.

K%llln~ obstructlon classes:

Having replaced

Xi -->

W

Y•

> coskn_lW

by some actually-commutative diagram, we may apply classical obstruction theory ([29], Ch. VIII) to the fibre bundle

W-->

coskn_lW , whose fibre is isomorphic to

The obstruc-

K(~n(W),n) .

tion to the existence of the homotopy-lifting,

47 -

Xi

W is a class

>

~.>

Yi

cos~_iW

~(i) 6 En+l(Yi,Xi;]--~)

local system of coefficients, is a ~-twisted

%. 13

where

{Vn(W)} .

~-~

is the induced

By our assumptions,

system of ~-groups.

Consider the long exact sequence,

~O:i,T- D _. ~(xi,]-71 ~ ~+l(q,xi~T- D ~ ~§ By hypothesis, i -, i'

En+l(Y,~)

-, E n + l ( x , ~ )

such that the image of

But the "elementary

construction"

So there is a morphism

.

r~(i)

_. ~+l(xl,T-D,

vanishes

in

Bn+l(Yi,

allows us to represent

the choice of a homotopy-commutativity

m , after

F , by an actually commuta-

tive diagram Xi ,

> Yi i (u

Xi --

> Yi

inducing a morphism on relative cohomology. ~(i' ) , and therefore,

~(i')

Hn(Xi , , ~ )

5x' = ~(i' ) .

such that

such that the image of y" 6 Hn(Yi,,,]- ~)

.

= 0 .

x'

be

construction"

another actually-commutative

diagram,

a

since

yF~(i) =

class in

Find a morphism

x' , Mr' E Hn(Xi.,, ~ )

This is possible,

Use the "elementary

Let

We have that

i' -~ i"

comes from some Hncy,~-I)

to represent (7p,) 9

~ Hn(x,V-~)

i' -r i"

by

.

48

By construction, the obstruction class

~(1)

goes to zero

under the map, ->

9

Thus we obtain the required extension,

Xi,, ~ m >

Yi"

w

oo.,

_lw

The proof of (b) is similar, and we leave it to the reader.

(ill) --> (ll): Assume that (ill) holds, and let abelian coefficient group on

X

M 6 ~

and

be a ~-twlsted

Y , given by a map

~l(X) ~ ~l (Y) -> Aut M . We can suppose (App. 3.3) given a filtering system of maps fl

xl in

~o

representing

> Yi

f , and a compatible sequence of maps

7rl(Xi) --> ~l (Yi) -> Aut M representing the operation of arrow is the map induced by

~I(Y) fl "

denote the kernel and image of and let

X~--> X i , Y~--> Yi

and ~ i

respectively.

on

Let

~l(Xl)

Aut M , where the first Hi , Gi

(resp. ~&i '

(resp. ~l(Yi))

in

~i )

Aut M ,

be the covering spaces induced by The

X~

(resp. Y~)

form a pro-object

49 X'

4.15

(resp. Y') which is the covering space of

termined by the kernel of the map

(resp. Y) de-

X

~l(X) --> Aut M

(resp. ~l (Y)

--> Aut M). Using the covering resolution of

M

X!l

of

X i , one constructs a canonical

in the well known way:

If

X!l -> Xi

sented by an actual covering space of CW-complexes, tion is the one obtained from the simplicial

xi <

x,

(xl3• Xi

on the various fibre powers of

X i' 9

the resolu-

object

<

I)

by taking the direct images on

is repre-

<

@

@

9

of the coefficient

groups

M

Let us denote this resolu-

tion by 0--> M--> L~--> Li1 --> ...

V

Each on

Li

is the direct image of an untwisted

coefficient group

X[ , namely V

L~ v ~ M Gi

Gv

where

M

is untwisted

product of

M

on

X! l

with itself,

by construction,

and

indexed by elements of

M i

is the

V

G i = Gix...xG i.

Denote by

o

l_>

0--> M--> K i --> K i

the resolution of

M

from the covering

Y~ , so that

coefficient

group

on

Yi

...

obtained in the corresponding Ki

way

is the direct image of the

-

50-

4.16 ~V

K[ v = M i

By functorality,

there is a filtering inductive system of

morphisms of spectral sequences relating the spectral sequences of these two resolutions:

H p(Hq(Yi,K i) ) ====> HP+q(yi,M )

H p (Hq(XI,L [) ) ---~=--> HP+q(xi,M)

hence a morphism of the limit spectral sequences, whose abutment is just the map

HP+q(y,M)--> HP+q(X,M)

We are to show that this map is biJective. to show that for each

(*)

is biJective.

q

and

.

Therefore it suffices

v , the map

V

lim Hq(Yi,K.Vm) --> lira Hq(Xi, L i) -@

.-~

i

i

Now we have

and

qcxi,

qcxb Iv)

-

51

-

4.17

Consider the index category of maps

i --> J

in

I .

Clearly

we have isomorphisms

lim_, H q (Y~, K~ v )

J V

lim_, H q

(q, K~v)

(where

K I. = M ~j) J

i-,j V

= lira (lira j

(App. 1.11)

i~j V

lim (Hq (Y', M J))

J V

Now (4.3)(iii) asserts that

V

Hq(y',M ~j) ~ Hq(x',M ~j) .

Hence we

are reduced for (*) to showing that v

(**)

Gv

lim H q(X',M ~ j) --~> lira H q(X',M J) . -@ -@

j But

{~j} , {G j}

J are isomorphic as pro-groups,

since they repre-

A

sent the images of the corresponding maps Vl (~) -> Ant M a functor of

and since

v l(~)

v l(~)

~I(Y) -> Aut M .

Since

and

Hq(x ',M Fv

)

is

F , (*) follows.

(i) => (iii): This implication follows from the following result: Theorem (4.11): A

X 6 pro-~ o 9

Let

Let

~

be a complete class and let

A

H c ~I(X)

be a sub-pro-group and let

H c ~l(X)

A

be its inverse image. space

XH

of

X

Then t h e O ~ c o m p l e t i o n

determined by

H

Xtt

is canonically

of the covering ~-isomorphic to

52 -

the covering

A X~

4.18

A X .

of

To obtain the implication

(i)=>

(iii) from (#.Ii), we may

apply the theorem to the corresponding C-covering spaces X',Y' A A of X,Y respectively 9 Since X ~ Y , it implies that also A A X w = Y' Hence we are reduced to the case X = X' i e. to showing that if

f

is a

~-isomorphism,

Hq(Y,A) --~ Hq(X,A)

then

for

A E ~

.

But Hq(Y,A)

[Y,K(A,q)] =

[~,K(A,q) ]

=

[X,K(A,q)]

=

Hq (X,A)

since

K(A,q)

E C,~o

.

This completes the proof of (4.3). Proof of (4.11):

Of course, the map

A A X H -->

is induced

^

by the canonical map want to show that

~A fl

--> ~

since

E pro-~ 0 9

is the ~-completion of

~

.

Thus we

We may

(App. 3.3) represent the given data by a compatible filtering family of sub-pro-groups

where

X = {Xi] , X i E ~o " A is the inverse image of H i

Since

(2.8)

XH=

[(Xi)Hi]

Let in

Hi

be the sub-pro-group which

~l(Xi)

and (3.9)

so that

H = [Hi] .

[(Xi)Hi] = X H , it follows

- 53

-

4.19

that it is enough to show that for each

N

(xi)H i Hence we may replace Write

X

by

Vl(~) = [~j]

)A

> (xi H i

X i , i.e., suppose

X

is in

(~j E ~) as a pro-OFgroup,

are quotient groups of H

^

i

~o "

where the

Vl(X) , in such a way that (App. 5.2)

is represented by a filtering family of subgroups

^ Then if #~j

denotes the inverse image of ~ j

A Vl(X),- , it is

in

clear that

^

Thus, applying

(2.8),

(3.9) again, we may replace

H

by

j

,

i.e., assume that (4.12).

H

is a subgroup of

image of a subgroup

~

Recall (3.4) that I

of objects

W

~l(X) , and is the inverse

of some quotient group A X

#

of

Vl(X)

may be regarded as an inverse system

each of which is obtained from

X

by spanning

in cells in successive dimensions so that the resulting homotopy groups are all in

~ .

Thus (2.8)

~

will be obtained as an

inverse system of the corresponding covering spaces

W'

indexed by

so that the

map

I) of these

Vl(X) --> ~

W , if

factors through

W

is "large enough" Vl(W ) .

(again

This covering space

W'

-

is obtained from in

@~o "

X' = X H

54

-

%. 20

by spanning in some cells, and it is

Therefore the theorem in our case is equivalent with

the assertion that the category of such cofinal with the category of all maps

W'

(indexed by

X' --> Y

such that

Thus we need to verify App.(1.5) in this context.

I) is Y 6@~/0 9

The conditions

read:

(a) a map

Given a map

X--> W , W

X t --> W t

with

Y E @~o ' there is

factors through

as above, such that

.

(b) maps

X' --'> Y

Given a map

V' _i> _> Y

with

as above, and a pair of

X ~> V , u 6 ~ o Y 6 ~o

such that

ir

~ jr

(where

J Ct:

_> V'), there exists a diagram

X t

X

> W W

as above

V

such that

ik' ~ jk I .

To prove

(a), we construct

W

as the limit of a compatible

sequence of CW-complexes

Xq , where

~n(Xq) E ~

for

Given

struct

by spanning in some (q+l)-cells in such a way that the

map Xq_ I'

Xq

' I --> Y Xq_

n < q .

extends to

Xq' .

by spanning in (q+l)-cells,

the map

t I) --> Y Vq(Xq_

~

factors through

X'q

and

Xq_ 1 , the problem is to con-

Since

X'q

is also obtained from

the condition is clearly that

factor through

~q(X~)

.

Let

N c ~ q ( Xq _I l )

55 -

4.21

be the kernel of this map. We treat first the case K c H c ~l(X.) Then

K c N

be the kernel of the map

is a normal subgroup of

an injection

K/K n N c H/NO-> ~l(y)

Therefore by (B.2)(i), ~l(X)

and with

Now suppose operates on

q > 1 .

is isomorphic

in

.

, hence

We can let

Then Let

~q(Xq_l)

above.

~l(X) --> K .

X1

is a @-group,

be the intersection

(~q/N)

normal in

be obtained by

under this action.

to a subgroup of

F

F .

~l(Xq_l)

F

We have

K/K G N E @ 9

contains a subgroup

E @ 9

Let

Xo = X .

H , hence of

to kill the subgroup

~q(Xq_l) N

K 0 N

~l(X)/F

spanning in 2-cells

jugates of

q = 1 , setting

and it

of the con-

Then

~q(Xq_l)/F

, hence is a @-group.

Suppose we span a (q+l)-cell into

Xq_ 1

sents the element

The homotopy classes of

s E ~q(Xq_ l) .

~q(Xq_ l) = ~q( X'q_l ) lying over

S

in

are all translates

repr@sented

Xq_ 1 of

whose boundary

by the various

Consider

F

in

s

X

under the action of

X' q-i

~l "

Therefore if

Thus we may span in

q-1 "

(4.13)(b).

closely that of (a).

spheres of

repre-

(by choosing a path to the base point)

s E F , then so are all these other classes. cells to kill

S

The proof of this assertion follows

We again construct

W

as a limit of a se-

quence of factorizations

X

,> X

W

such that

~n(Xq)

E @

for

n < q , where

Xo = X

and

Xq

is

56 -

obtained from

X

4.22

by spanning in (q+l)-cells, and where the

q-1

homotopy

hq-l: ir

|

~j

!

r

extends to a homotopy

hq: i ~

• J~

.

Following the notation of (3.4), replace N

be the kernel of the map

r

X

ms: D q+l --> X

S c N

~: F--> N onto

N .

be a pointed continuous map representing

and extend

Cms

tion of

through

~

Xq_ 1 , let

~q(X) --> 7rq(V) , and let

be a map of a free group on a set of generators Let

by

to a map

Bs: D q+l --> V .

~(s),

We obtain a factoriza-

corresponding to

X'

D q+l = X s . The covering X~ of X s s is obtained by spanning cells into each of

the spheres

of

X'

a'

SV

X Um

lying over

represents the choice of a point Choose for each For each

v

a path from

m s , where the index

Pv

above the base point of

i(pv)

to the base point of

v , we get a map

f

SV

: Z-->Y

as in construction 1 of (3.4), replacing respectively,

v

m s , 8s

by

m~v, 6~v

and, using the chosen path above, the map

determines a class tending the homotopy

fsv

in

~q+l(Y)

h

to

X s'

("" ,?sv"'" )

.

fsv

The obstruction to ex-

is represented by the vector

( q+l(Y))P

X . Y .

57 where

P = Iv]

fs: F-->

is the index set.

4.23 Thus we obtain a homomorphism

(~q+l(Y)) P .

The important thing is that since and

~q+l(Y)

6 @ , the group

(Vq+l(Y)) P

Therefore if we replace the group (3.4) by

Example:

(a)

Vq(X)

-isomorphic m>

is again a @-group.

Vq+l(W2)

of the proof of (#.13)(b)

the proof of (4.11).

Let

vanish for to

Hence

(Uniqueness of Eilenberg-Maclane

Corollary #.i#:

if

is a @-covering,

(~q+l(Y)) P , the same argument applies.

holds, which completes

groups

X' --> X

K(v,n)

X

be in pro-~ o

q ~ n , and ; cos~X

spaces)

and suppose the pro-

~n(X) = v .

Then

is isomorphic with

X

is

K(~,n) ,

n . I

Proof: and that

An application of (4.4) gives that

coskn_lX

is contractible.

X ~ cos~X

,

(4.1#) then follows from

the fibre triple: coskn_lX-->

(b)

cosknX--> K(Tr,n)

.

(Uniqueness of the n-sphere)

Corollary #.15:

Suppose

@

is the class of finite groups

whose orders have prime divisors in a fixed set Suppose if

X

P

of primes.

is in p r o - ~ o , is simply connected, and H (X,Z) = 0 A q q ~ n , and Hq(X,Z) = Z if q = n for some fixed n > 2 .

Then there is a ~ -isomorphism, A Sn

>

X

-

Proof:

Let

58

-

4.24

~

denote a topological generator of the proA jectlve limit of the pro-abelian group Z . An application of (4.4) enables us to represent each

Xi

is

n-1

X

by a system

[Xi]

such that

connected, and for which there is a compat-

ible system

Hn(Xi'Z) which represents the element

6 Lim Hn(Xi, Z ) = Lim(~) (-. (.-

We have that

by the ordinary Hurewlcz theo-

H n(xi,Z) ~ ~n(Xi )

rein, and hence the

may be interpreted as producing

T i E ~n(Xi )

a morphlsm Sn

in pro-~ o . mology groups

~>X

.

This morphism is clearly an isomorphism on all hoHq

for

We shall show that

q ~

.

Hn(Sn) ~

Hn(X)

is an isomorphism as

well, for if we evaluate, using our assumption that we find that

Sn

is a morphlsm

such that Lira #n" Lim (--

(--

--> Lira (-.

Hn(X ) ~ ~ ,

-

is an isomorphism.

59-

This latter fact implies that

an isomorphism, and (4.15) follows from (4.3, iii).

#.25

~n

itself is

60

-

w

5.1

-

Completlons~and fibrations.

Throughout this section, we let

@

denote a class (3.1)

consisting of finite groups. Let

X E pro-~ , and consider the pro-object pair

described in (3.8).

A X-->X

We can replace this object by a canonically

isomorphic fibre resolution, as in (1.1).

Let us write it the

same way. Proposition (5.1): erty that

~

Proof: X --> W

The fibre

is ~-contractible The inverse system

A X --> X

of

X

has the prop-

(4.2). A X--> X

is represented by maps

which induce isomorphisms of fundamental groups (3.8) and

thus each object in the inverse system of X is connected. DeA note by X the universal covering of , and by ~ the induced covering of tion

X

that of

of

X .

By (#.ll),

~

is @-isomorphic with the comple-

N

~

X^, and the fibre of

X--> X

X--> X .

A

is ~-isomorphic with simply connected, and

Thus we may suppose

in fact that the inverse system comprising

consists of simply

connected objects. We need to show that cohomology groups

Vl(~) = 0

and that for

A E ~

the

vanish for all q > 0 . Let us write A A X --> X = IXi --> Xi] i , and x = {xi} 9 We

Hq(X,A)

the inverse system as

obtain a directed system of spectral sequences

(5.2)

(EPq)i = H p(~i,Hq(Xi,A)) ==> HP+q(Xi,A)

where the coefficient groups

Hq(XI,A)

are untwisted since

A Xi

-

is simply connected. A Vq(X i) E ~ for all

Now q .

6~

-

5.2

~

is a class of finite groups, and A Hence X i is homotopic to a CW-complex

which is finite in each dimension. Therefore the cohomology of A X i commutes with direct limits of coefficient groups. Passing to the limit of the

E~ q

term of (5.2) we obtain

A lira HP(xi,Hq(xi,A)) i

A Hq = lim HP(xi, (xj,A)) = i-~j

A

A

lira H p (X, Hq (xj,A))

= ~P(x, lim ~q(xj,A))

=

J

J

HP(I,~q(x,A)) .

Therefore the limit of (5.2) is a spectral sequence

E~q = HP(~,Hq(x,A)) ==> HP+q(X,A) .

Since the coefficient groups are u~twisted, Hq(X,A)

.

we have

E~q=

But by (4.3), the map

~,~o= ~P(I,A)-> ~P(x,A) is an isomorphism for all and hence

E pq = 0

for

p , which implies q > 0

It remains to show that

by induction on ~l(~) = 0 .

q .

Consider

sequence

~2 (~) -> ~i (•

E~ q = 0 , q > 0 ,

-> ~i (x) -> o

the exact

62

Let

G

be a @-group,

and suppose given a (say) surJectlve map

(*)

~i (•

Sinoe

5.3

-

-> G .

=1(x) = ~l(~) = o , and since

~l(X)

maps to the group

A

0/~(~2(~)) G

, it follows that

is abelian and the map

HI(7~G)

=2(~)

maps onto

a.

Therefore

(*) represents an element of

Hom(x,G) =

which was shown above to be zero.

Lemma

(5.3):

A pro-object

of maps of CW-complexes

X --> Y

in the homotopy

category

gives rise in a functorial way to a pro-

object X

> Y

A

.>~

(5.4) x

in the category of squares,

where the vertical arrows are isomor-

phic to the ones introduced

in (3.4).

Proof: X c y

It clearly suffices

of CW-complexes.

to treat the case of a pair

We let the pro-object

the category of diagrams

(5.5)

X

.> Y

it

.>V

U

(5.4) be given b y

63 -

where

U,V E @ ~

, U

5.4

is obtained from

X

b y spanning cells in

successive dimensions as in (3.4), and

V

is obtained from

b y spanning in cells including at least those spanned into to obtain fixed,

U .

A morphism is a map of squares leaving

Y X

X --> Y

taken up to homotopy.

It is clear that the left vertical A arrow yields a pro-object cofinal with X--> X as in (3.4), and that moreover any map

Y--> W , W E @

factors through a

V

corn-

ing from a square (5.5). To show the right arrow cofinal with A Y--> Y and that the resulting pro-object (5.4) is functorlal, one reduces immediately to proving that given a diagram of ~'~complexes > X

Y

and a homotopy

h

> U

-

W

> V

> > W'

between the two squares

X

.>Y

W

.> W w

L

there is a square X

.>Y

U'

. .> V I

(5.6)

64

-

dominating

(5.5) such that

h

To do this, we first choose from

Y

to

extends to a homotopy on

V1

U' --> ~'

so that the homotopy extends

V 1 , as in (3.4).

Then we choose any square

X

.> Y

L I

(5.7)

ui

mapping to (5.5). the case that X--> U

h

so that

a diagram h

> vI

Replacing

(5.5) to (5.7), we are reduced to

extends to h

V .

extends to

add the cells spanned into

that

5.5

-

Choose

X--> U'

U' , as in (3.4).

X

to obtain

U'

to

mapping to Then we can V , obtaining

(5.6) mapping to (5.5); and it is clear by construction extends.

Let

X--> Y E prO(~pair s) be as in the above discussion.

we fibre resolve the square

(5.4), we obtain a pro-object

If

of dia-

grams F

.>X

F

> X

->Y

(5.8) A

where

F

and

Theorem

1~

are the fibres of the horizontal maps.

(5.9):

With the above notation,

is a pro-object such that each A F--> ~ is a ~ - i s o m o r p h i s m !

A

->Y

Yi

suppose

Y = {Yi ]

is simply connected.

Then

.

65 Proof:

5.6

Applying (1.2), we may imbed the square (5.8) in a

projective system in the homotopy category of diagrams

(5. lO )

F

.>X

>Y

I [ I where the vertical and horizontal arrows are flbratlons, up to homotopy.

Say we index the diagrams making up (5.10) by

i E I .

Now if we recall the construction (3.4) of a cofinal system of maps

Y --> W

(W E @ ~ ) by spanning in cells, it is clear that

for a simply connected

Y

we need only consider those

W

tained by spanning in cells of dimension ~ 3 , and then ~2(W)

is surJective.

=i(,)

We want to show that

~

= o .

is ~-acyclic, and since

sists of finite groups, it suffices to treat A ~ Z/p 6 @ , p

a prime.

@

con-

twisted coeffi-

Consider the inductive

system of spectral sequences

HP(@i,Hq(~i,A ) ---> HP+q(Xi,A ) .

We have

~2(Y) -->

Therefore in the diagram (5.10), we have

(5. ii )

cient groups

ob-

66

-

5.7

lim E pq = lim HP(r i = lim HP(r

Since

r

.

is simply connected, we can take the second limit over

those maps

i --> J

which induce the zero map

Then the induced action on

Hq(r

Vl(r

2Z/p .

Moreover,

cohomology with values in

r

Hq(r

homology commutes with products 9

o->

is trivial.

we want 9

are vector

(For, its co-

is trivial,

since co-

In the exact sequence

v->

o

(2~/p)s , hence induction on

q

shows what

Thus we conclude

lira E pq = o i This limit is the HP+q(X,A)

2Z/p

9

is again a sum of

Hq(~j,A)

is Cracyclic by (5.1), hence its

homology with values in a product of

V

.

A ~2Z/p , the coefficient groups

spaces over

E pq

if

p > o

term of a spectral sequence abutting to

which is zero for

.

is trivial, and so

lim E pq = lira H p(r J Since

--> ~l(r

p + q > 0

by (5.1).

lira E~ q = lira Hq(~j,A) = Hq(~,A) = O , i j

Therefore

q>O,

-

67

-

5.8

as well, as was to be shown. Now let twisted.

A E @ be a coefficient group on F , possibly A A Since X , Y are in pro-@, they are projective limits of

CW-complexes homotopic to ones finite in each dimension, and hence the same is true for

F .

Thus cohomology on

F

commutes with

direct limits, and the inductive system of spectral sequences

EPq = HP(Fi,Hq(~i,A)) ==> HP+q(Fi,A) has as limit

llm E2Pq = i

llm j

HP~,Hq(~j,A))

-

HP~,Hq(~,A))

= 0

if

q>0

.

Therefore the spectral sequence collapses and yields isomorphisms

H p (~,A) ---->H p (F,A)

for all

p .

To complete the proof that suffices by (4.3) to show that diagram

A F

and

~

are ~ - i s o m o r p h i c ,

~l~F) ---->~ i ~ ) .

Consider the

it

- 68-

5.9

> 0

7r2 (~)

Since

Y

> ~ra(~ )

l

,L

0

0

is simply connected,

exact functor

~l(X) = 0 , and

A

is a right

(meaning that it commutes with direct limits,

it has the appropriate

the rows and columns A of this diagram are exact, in an obvious sense. Since Y , Y are simply connected,

adJointness

since

property),

we have

H2(Y,A) = Hom(~r2(Y),A ) = Hom(Tr2('~),A ) = 1-12(~,A)

for

A E C , and so the map

d

b

is the zero map, and since

e

is an isomorphism,

a pointed,

fl .

a

we apply

We revert from

connected,

resolve the inclusion

is surJective,

as was to be shown.

To conclude this section, space functor

of (5.12) is an isomorphism.

~o

c

is zero.

Tbls completes

Thus

the proof.

(5.9) to study the loopto

~o " and suppose

simply connected simpliclal

set.

e --> K

If we fibre-

e --> K , we obtain a fibre triple: --> e' --> K

Hence

- 69

-

5.10

which is actually functorlal for pointed maps,

C~,I

and if pret

> ei~

f -- f' , we obtain ~

Of ~ Of'

(1.1) and thus we may inter-

as a functor from the connected

subcategory of

Ko

Corollary 5.13: pointed

.> K 1

to

Ko " If

e--> K

slmplicial set, then A

is a ~ - i s o m o r p h i s m .

simply connected full

is a connected,

simply connected

-

w

Homotopy

groups

In this section, h o m o t o p y groups for a complete

7 0

of completions.

we discuss

of an

the relationships

X E pro-~ o

class

6.1

-

~ .

between

the A X

and of its ~ - c o m p l e t i o n

Since

X

maps to

~ , we are given

a map

(6.1)

~n (X) --> ~n

for each n = 1 .

n , and we saw in More generally,

Corollary

(6.2) :

)

(3.7) that it is an isomorphism

when

we have

Suppose

that

~q(X) = 0

for

l(qKn.

Then

v n (X) --~> ~n (~)

This is analogous

to (4.5).

If say

X = ~X i] , let

the fibre of the map

~X i --~ coskn_iX i] .

(ill)) that

is a

X

by

X' --~ X

X' , i.e., we m a y suppose

and for each

i .

X--~ W

mension

where

at least

In addition,

that

It follows

W

(S.6) of

{X~]

from

(4.4

~q(Xi)

= 0

for

to treat the case ^ X--~ X , it suffices

i 4_ q ~ n X = Xi . to take

is obtained by spanning in cells of di-

(n+l) , from which the assertion one can describe

~2

in complete

is clear. generality

as follows : Proposition be abelian.

Let

(6-3): J

Let

X = {X i ] E

be

and so we may replace

Hence it suffices

Then in the construction maps

~-isomorphism

X' -

p r o - ~ o , and let

be the index category of pairs

J =

A E (i,H)

-

where

H c Vl(Xi)

Let

(Xi)H

71

-

6.2

is a normal subgroup such that

be the covering space of

Xi

Vl(Xi)/H 6 @ 9

determined by

H .

Then the groups

H 2 ( (XI)H,A)

form a directed system indexed by

J 9 and

A 9r2(X).. is characterized

by the property that

(6.4)

~o~(~ 2 (1),A) ~ lira H 2 ( (Xi)H,A) . J A

Proof:

Let

A

~

be the universal covering of

be the induced covering space of

X .

X , and let X A ~A Then by (4.11), X ~ X . We

have

li~ ~ ((xI)H,A) = ~ (~,A) J

and

~2(~)=

may assume

~2(~) . ~l(X)

Hence we may replace

X

by

has no non-zero maps to Crgroups.

9

i.e.,

we

Then (6.4)

is just the assertion that

He(~ 9

~> H2(X,A) 9

which follows from (4.3). Following Serre Definition pro-(groups). ~-module

[27]9

(6.5): G

A 6 ~ ,

Let

we introduce the notion of good group: ~

be a complete class9 and

G E

is said to be ~-good if for every twisted abelian

6.3

-72 Hq(~,A) --~> Hq(G,A)

where

~

is the d~-completion of

Corollary (6.6): G = [G i} completion of

for all

G . is d~-good if and only if the C~-

K(G,1) = [K(Gi,1)]

is ~ - i s o m o r p h i c

For, the completion of the map -isomorphism iff. Applying

to

K(G,1) --> K(~,l)

K(~,l)

.

is a

is C~-good, by (4.3).

(4.4), we can now prove

Thieorem (6.7): X 6 pro-~#o

G

q ,

Let

~

be a class of finite groups, and let

be simply connected.

Then

A ~q(X) ---->~q(~)

if

q (x)

is C-good for

Proof:

Suppose

actually

X = IXi]

~q(X)

~q(Xl) = 0

has for

descending induction on when

q < n

q < n-1 . is ~-good for

(4.3) to the fibre of the map case that

for

q ~ n-1 .

Applying

X --~ cosk2X , one reduces to the

~l(Xi) = 0 q < r

for all

and all

i .

i .

Suppose that

We proceed by

r > 2 , the theorem being true by (6.2)

r > n-1 : Let

(6.8)

G = ~r(X) , and consider the fibratlon

F--> X--> K(G,r)

.

By (5.9), the ~-completion of (6.8) is ~ - i s o m o r p h i c sequence,

and we may suppose the theorem true for

F

with a fibre by induction.

-

73

6.#

-

Thus we are done by the exact homotopy

sequence arising from the

completion of (6.8) if we prove the following: Proposition r > 1

(6.9):

an integer.

K(G,r)--> K(~,r) Proof: apply

(6.6),

@

E

r = i , this is

is contractible, (6.10):

0

is C~good iff the map

(6.1).

For

r > i , we may

is @-good.

E--> K(G,r)

and induction.

Let

sequence of pro-abelian A"

G

(5.9) to the fibration

Corollary

Suppose

group

~-isomorphism.

K(G,r-I)-->

where

be a class of finite groups and

A pro-abelian is a

For

Let

0--> A' --> A--> A" --> 0

groups,

and

~

be an exact

a class of finite groups.

Then the sequence

0--> ~'--> ~--> ~"-->

is exact. This follows from the above proposition,

and

(5.9), applied

to the fibration

K(A',2) --> K(A,2) --> K(A",2)

Example nected

(6.11):

X , even when

Theorem n = 2 .

of a 2-sphere and a 1-sphere, finite groups.

Then .

.

(6.7) is false for non-simply For example, and let

~l(X) = 2Z , and

~

let

~

corresponding

covering space is a wedge of

be a wedge

be the class of all

~2(X)

of copies of

X

con-

is a countable

For any finite quotient group of S1

~l(X)

sum

, the

and finitely many

74

S2 .

We apply (6.3).

-

6.5

It is immediately seen that while

H~

= ~-T A

the group Hom(Tr2 (~), A)

is the subgroup of this group consisting of vectors

(...,a o,a I,...)

ai E A

which are periodic, with some period. Example (6.12):

Take

@

to be the category of finite groups.

Then the additive group of rational numbers, K(Q,n)

is ~-contractible.

o->

to obtain that n > 2

K(Q/Z,n)

z->

Q , is e-good.

Thus

Apply (5.9) to

Q->

%/z-> o

is ~-isomorphic with

A K(Z,n+I) , if

.

An application of (5.13) gives the above for

n = i as well.

-

w

7 5

7.1

-

Stable results.

In this section we study the question of classification, up to stable homotopy equivalence,

of all pointed CW-complexes of a

fixed stable pro-finite homotopy type. to a purely algebraic one (Cot.

This problem is reduced

(7.15), below).

It is shown

that there are only a finite number of distinct stable CW-complexes having a given stable pro-finite completion. If

A,B

are pointed CW-complexes,

erated reduced suspension. integers

nj

let

SnA

denote nt---hit-

For any integer

(j = 1,2) , two maps,

k , and nonnegative nj+k fj: snj(A)--> S (B) ,

(j = 1,2) , will be referred to as "equivalent maps of degree k" n-n 1 if there is an integer n such that S (fl) is homotoplc to n-n e S (f2) . The set of equivalence classes of maps of degree k forms an abelian group which we denote

(A,B)k .

topy category we will mean the category pointed CW-complexes, the set

and morphisms from

(A,B) = (A,B)o .

By stable homo-

whose objects are A

to

B

are given by

As we have defined it, the stable ho-

motopy category is an additive category.

Although there is a

standard technique for enlarging this category slightly to a triangulated category,

(in the terminology of Verdier,

we shall not bother to do this.

cf. [16] )

(For general treatments of

stable theory, see [1], [24].) If

A

is a subcomplex of

X , and if we denote by

CW-complex obtained by constructing the cone over

A

in

obtain a triple (A) A --> X --> X/A where the arrows denote the natural inclusions.

Anything

X/A X

the 3 we

-76

isomorphic

tion.

to

such a triple

-

7.2

(A) w i l l

be

referred

to

as

a cofibra-

The cofibrations play the role of the "triangles" of our

category. We have the standard facts that any continuous map of CWcomplexes

f: X --> Y

may be continued to a cofibration,

Y->

x->

z

f and this cofibration is unique dependent only upon canonical) isomorphism. (V,)

f

Also, functors of the form

up to (non(,W)

and

applied to cofibrations yield three-term exact sequences

of groups, etc. In this section let

~n(A)

denote the functor

(sn,A) , the

stable homotopy group functor. L~mma (~3): If (A,B)k

A,B

are finite CW-complexes,

are finitely generated abelia~ groups for all

sequently so are Proof:

Well known, a~d easy.

as the ordinary homotopy groups of choice of

L .

k .

Con-

~n(B) . Here is one way of seeing it:

By the Freudenthal suspension theorem,

complex.

then the groups

But this

S~

S~

~n(B)

may be interpreted

for a sufficiently large

is a simply-connected finite ~ -

It therefore has finitely generated integral homology

groups, and the relative Hurewicz theorem applies, assuring that it has finitely generated homotopy groups. relating

Hm(A,~n(B))

lemma (7. i).

to

(A,B)k

The spectral sequence

allows one to then deduce

-

77

7.3

-

What is the relationship between suspension and completion? Let

A

denote completion with respect to a class

~ .

Let

be the pointed homotopy category of CW-complexes.

Let

E,A: ~o

--> ~o

be functors,

Z

left adJoint to

and suppose,

A

~o

further,

that both functors preserve the subcategory A

Lemma (7.2) : The natural map Proof:

EX --~ ~(~)

is an isomorphism. , form the com-

To obtain a two-sided inverse to

posite A

where the first map comes from adjointness

of

A

and

E .

Since

A

A(EX)

is in p r o - ~ o

--~ A ( ~ )

the above composite extends to a map,

which yields

(by adJolntness

of

Z,A

in prO-~o),

a

^

map

~: ~

--> ZX .

One checks that

a

and

~

are two-sided

inverses of one another. Corollary (7.3) : If completion, (Sk

A

denotes p-completion or profinite

the natural map

a: skx--> sk(~)

is an isomorphism.

denotes k-fold reduced suspension. ) From now on, unless explicitly indicated,

A

will denote

profinite completion. Corollary (7.4): Let --> ~n(~)

is pro-finite

Proof: skx

for some

Note that k

are ~-good where

X

be a finite CW-complex.

therefore use theorem

n(X)

completion. ~n(X)

is the ordinary homotopy groups of

(with a shift in dimension), ~

Then

and these groups

is the class of all finite groups. (6.7).

We may

78

-

7.~

y =

Consider two objects of p r o - ~ o , X = (Xl)iE I , and (Yj) j 6J "

Define the pro-abellan group

indexed over

(x,y) k - u m.(xi,Yj) k

J E J .

i Notice that

Lim(X,X) = Hom(X,Y) e-

J

the latter group being

Hom

Corollary (7.5): Let

in the category p r o - ~ o .

A,B

be finite CW-complexes.

Then the

natural homomorphism of abelian groups

(A,B) --> (~,B)

is proflnite completion. Proof:

Suppose

A 1 --> ~

--> A 3

is a cofibratlon.

Then the

Puppe sequence gives us a morphism of long exact sequences.

...-->

(AB,B)k-->

^

...-> for all

(A2,B)k-->

A

(A1,B) k --> (A3,B)k_ 1 --> ...

L?

(a3,B)k-> (a~,B)k--> (al,B)k--> (a3,B) k - 1 -> k 6 Z .

"'"

The top llne consists in finitely generated

abelian groups, and the bottom line in pro-abelian groups.

Note

that if we take the proflnlte completion of the top llne we obtain

7,5

-79-

a long exact sequence in pro-Ab . phisms for all

k .

Suppose

k .

This observation,

quance of cofibratlons us Cor.(7.5).

q.e.d.

(a)

and

~3

is an isomorphism for

coupled with Cor.(7.4)and

that any finite CW-complex

1A 1

were Isomor-

Then applying the five-lemma in the abelian

category pro-ab, we would obtain that all

~,~

A

may be built up from a finite se-

0A 1 --~ JA2--~ 0A 3 Note:

JA 3

the fact

for

J = 1,...,r

gives

"built up" means the following.

are spheres of various dimensions,

J = 1,...,r .

(b)

J+lA 1 =

(C)

rA2

--

for

J = l,...,r-i .

A.

A model for the profinite

Suppose Thus map

A = SB , where

A = B•215

U B•

fn: I/~I --~ I/3I

.

tlng

x E R/Z . fn(B,x)-

that this functor

fn

[A, ]

B

is a pointed finite CW-complex.

Consider,

for any integer

which is multiplication by

after the identification all

completion of a double suspension.

of

I/3I

with

for all

B E B

fn: A--~ A

and

x E I .

for by setNote

represents the n-th power map of the group-valued into itself.

Suppose now that

A = S2B , so that

[A, ]

values in the category of abelian groups. cofibratlons

That is,

R/Z , fn(X) -- nx

This map gives rise to a map (6,fn(X))

n .

n , the

of finite complexes,

we obtain

A-A~A/n fn ~

actually takes

Continuing the

fn

to

- 80 Since

we obtain compatible maps,

finn = fm~

fmn

A

A

n

> A

> A/ran

> A

> A/n

Thus we may interpret ~o "

7.6

A = (A/n)n6J

Applying the homology functor to

as a pro-object of (~)

finite homology groups in each dimension. connected, theorem, Thus

A If

so is A/n

A/n .

note that

Since

A

A/n

has

is simply

Consequently by the relative Hurewicz

has finite homotopy groups in each dimension.

is actually in pro-C~o, where @ is the class of finite groups. W

is in d~ o, then

[A,W]

the group of homotopy classes

of continuous maps is a finite abelian group.

Abelian since

A

is a double suspension; finite by induction on the number of cells of

B . Proposition (7.6): A--> ~ Proof:

map

is profinite completion.

We need only show for any

[A,W] --> [A,W]

is an isomorphism.

W

in

F

that the induced

This follows by consider-

ing the long exact sequences,

[SB, W]--> n

and using that If

p

[SB, W]-->

[A,W]

[A/n,W]->

[A,W]--> n

[A,W]

is finite.

is a prime the above argument gives us a very useful

model for the p-completion of

A .

It is given by the pro-object:

81

-

7.7

...--> A/pr+l--> A/pr-->

...--> A/p

Stable prohomotopy equivalence.

If

A,B

are CW-complexes we call stable prohomotopy equiA A valence between A and B an isomorphism ~: A--> B in the category pro-g~ o . Let

A

be a finite CW-complex.

Then

(A,A) = E(A)

given the structure of a ring by composition. as a Z-module, by lemma If

B

may be

It is of finite type

(7.1).

is a finite CW-complex,

module and a right- E(B)-module.

then

(A,B)

is a left

E(A)-

Of course, it is of finite type

over both of those rings, since it is in fact of finite type over Z

@

If the ring,

is the proflnite completion of Hom(~,~) = Lim(~,~)

.

Since

A , let

(~,~)

E(~)

denote

is a pro-object

in the category of finite abelian groups, we may regard the module Hom(~,~)

as a proflnite abellan group.

Thus

E(~)

is a topo-

logical ring and, after cot. (7.5), may be described as the profinite completion of

E(A) ; also

Cqrollary ~.7): Let

E(~) = E(A)@~ . A,B

be finite CW-complexes.

Then

Hom(i,B) = Hom(A,B)@z~ = HomCA, B)@E(B)E(~ ) .

Lemma (7-8): Let

r

A--> B

be a morphism of finite CW-

complexes having the property that Then

r

is an isomorphism.

~: ~--> ~

is a ~-isomorphism.

- 82 -

Proof:

7.8

Let us begin by seeing that the analogous statement

holds for finitely generated abelian groups.

Namely, if

~: C --> C'

is a homomorphism of finitely generated abelian groups such that A

is an isomorphism,

then

$

A

is, since

is a faithfully flat

functor on the category of finitely generated abelian groups. Now consider the two homomorphisms,

Hom(B,A) c~> E(A) Hom(B,A) _B> E(B)

induced by composition on the right (resp. left) by e. Since ~ is a ~-isomorphism,

and

Consequently

m

ing remarks.

If

m-l(1A)

~-i(~)

and

spectively.

A

A

is a finite complex, m and

and

O

are themselves isomorphisms by our open-

1ACE(A ) , 1BCE(B )

are the identity maps then

are left and right inverses to

This concludes lemma

Corqllary(7.9):

Let

A,B

r , re-

(7.8).

be finite CW-complexes which are

of the same stable prohomotopy type. a free module of rank one over B

~ are isomorphisms.

Suppose that

E(A) .

Then

A

Hom(A,B)

is

is isomorphic to

in the stable homotopy category. Proof:

Let

module, Hom(A,B)

e:

.

A-->

Then

B

A

r

denote a generator of the free E(A)is an isomorphism,

and lemma (7.8)

applies. Let

A

denote a ring (with unit, not necessarily commutative)

of finite type as a module over completion,

~ = A~

.

Denote by

Z .

Let

P(A)

~

denote the profinite

the set of isomorphism

-

classes of modules

M

A

property that

83

7.9

-

of finite type over

A

which have the

A

M = M|

is free of rank one over

~ .

A useful fact concerning completion of such rings

A

is the

following: Lemma (T.lO): Let

M,N

be A-modules,

with

M

of finite type.

Then

is an isomorphism.

Consequently a module

P

of finite type over A

A

is projective Proof:

over

A

if and only if

is projective

over

A.

Find a resolution of A-modules,

0--> R--> F--> M--> 0

with

F

free of finite type over

finite type over group.

A

since

F

It follows

that

R

is of

is a finitely generated free abellan

Form the diagram of exact sequences,

0

,

> HomA(M,N)|

A

> HornA

L" and since

F

Therefore

~A

general

A .

_

I"

is free of finite type, is injective.

HomA (R,N)|

Since

~F M

is an isomorphism.

was supposed to be a quite

A-module of finite type, we may conclude that

~R

is

84 -

also injective.

7. lO

Diagram-chasing then gives

~M

surjective.

The final statement of lemma ?.10 then follows in the nontrivial direction, for if

N 1 --> N 2 --> 0

is a surJective morphism

A

of A-modules then since

HomA(P, N1)@Z --> HomA(P,N2)|

is surJective

~p

is an isomorphism, and HomA(P, N1) --> HomA(P, N2) A surJective since Z is faithfully flat over Z . Let

A,B

motopy type.

is

be finite CW-complexes of the same stable prohoThen

is a module of finite type over A E(A) = A such that ~ is free of rank one over A by cor. (7.7). A Thus M is a projective module over A by lemma(7.10),and represents a class in this class c(A,B)

Hom(A,B) = M

P(A) .

c(A,B) .

Let us write

P(A) = P(A)

The results obtained above assure us that

plays the role of an obstruction to

the same stable homotopy type. guished element

and denote

0 E P(A)

Note that

stable pro-homotopy type.

P(A)

and

B

being of

has a distln-

represented by the isomorphism class

of the free module of rank one over Corollary (7.11): Let

A

A,B Then

A .

be finite CW-complexes of the same c(A,B) = 0

if and only if

A,B

have the same stable homotopy type. It is easily seen that a module A M

is free of rank one over

jective)

A-module such that

= A@Zp , for all primes localized at the prime

Thus

P(A)

p .

M

of finite type such that

may also be described as a (proMp = M@Zp p , where

is free of rank one over Zp

denotes the integers

(An application of NakayamaWs lemma).

is exactly the set

Pl(A)

defined in [~],

According to a result of Jordan-Zassenhaus

(lO.1).

[32] there are

85 -

only a finite number of isomorphism A

satisfying (1)

M

7. ll

classes of modules

over

the conditions: is a free Z-module.

(ii) M|

is a free

A|

of finite rank.

We are grateful to Bass for calling our attention result,

M

and pointing out that this implies

that

to this

Pl(A)

is a fi-

nite set.

Twisting finite CW-complexes.

Let us consider a category

~

which is the full subcategory

of the stable homotopy category generated by simply connected finite CW-complexes. complex,

and let

T

Let

Let

CW-complex

type as

~: 8o _> Ab

~(A) = (A,X) .

be a finite,

be a class in

finite simply connected stable prohomotopy

X

X

P(X)

: ~O

> Ab

We shall describe a

Y = XT

which is of the same

and such that

the abelian group

garded in a natural manner as a right

T

.

c(X,Y) = 9 .

be the functor represented by

Of course,

a left E(X)-module

simply connected CW-

representing

(A,B)

E(X)-module.

the class

T .

X ; thus, may be reLet

M

be

Define the functor

as follows:

cpl"(A) = (A,X)@E(x)M

(we may write symbolically, Definition: one such that

~

= ~(x)M).

A finite cohomological

functor

~: ~o _> Ab

is

86 -

(i) (ii)

a(S q) if

7.12

is finitely generated for all

A I --> A 2 --> A 3

q (> 2)

is a cofibration in

@ , then

e(A1)--> ~(A 2) --> (~(A3)

iS exact. (iii) There is an integer for all N-connected objects identified

N

such that

(where in the above, we have

W

with the functor which it represents).

W

Note that if

~

is a functor representable in

is a finite cohomological functor. representing

For, if

Y

~q(Y)

is fi-

q , (li) is implied by the Puppe se-

quence, and (ill) expresses the fact that logical dimension.

@ , then

is the object

~ , then (i) expresses the fact that

nitely generated for all

let

Homfunct(a,W) = 0

Y

has finite cohomo-

Actually we have the stronger result:

Lemma (7.12 ) : Le t

~: @ o _ >

Ab

T

P(X ) .

Then

be a class in

be represented by T

X , and

is a finite cohomological

functor. Proof:

Let

M

be a left- E(X)-module representing

Then (i) follows because (ii) follows because

M

M

is of finite type over

is projective over

E(X) .

T .

E(X) , and Since

T

satisfies both (1) and (ii), using an argument similar to that of lemma ~ . l ) , o n e

sees that

group for all objects Now let We show that

N ~T

A

~ T (A) of

is a finitely generated abelian

~ .

be the integer for which satisfies

(iii) for this

~

satisfies N .

(iii).

Suppose not.

-

87

-

7.13

Then there would be an N - c o n n e c t e d of functors T

(A) =

Consider the functor A

Applying the operation /k A T A A y: --> W . Note that y

for if

A

the zero h o m o m o r p h l s m range of

y(A)

, 'r

is

either.

defined by

9 we obtain a m o r p h l s m

is not the trivial mor-

CA9

A

y(A):

then

A ~(A)-->

(A,W)

to

= Lim(X,W)

the .

is not

This uses that both d o m a i n and

are f i n i t e l y generated a b e l l a n groups.

isomorphic

= Homfunct(X,W)

~

is an object such that

is not the zero homomorphism,

that

and a n o n t r l v i a l m o r p h i s m

.

of functors phism,

v> W .

~

W

A

functor But

fO 9

(X9

Thus

Note also

Homfunct(q~1",~)

= 0 , by assumption,

c-

which g i v e s

the

contradiction

sought.

We now state and prove Ed B r o w n ' s in our context

(cf.

a

sentable in Proof.

theorem.

is a finite cohomological

functor,

We shall construct,

Xl = Y2

"""

is repre-

(z) such that

by a d J u n c t l o n of cells of d i m e n s i o n

Ne shall also find compatible m o r p h l s m s where

m

below 9 a sequence

of simply connected finite CW-complexes Yq-1

then

@ .

Yo

from

theorem

[7]).

Representability

If

representability

~q = ( 9

Yq

is built

q 9 and

of functors

q+l .

~q--> ~ ,

.

The m o r p h l s m of functors

~q--> ~

will have the p r o p e r t y

88 -

that Y

~q(S i) --~ ~(S i)

7.14

is an isomorphism for

i ~ q .

denote the direct limit of the CW-complexes

would represent the limit functor category

~ .

~ = lim(~q), were X

Y

of

Y in the

~ , ~(X) =

where the latter group is Hom in the stable

homotopy category of CW-complexes.

Note also that these limits

are actually stationary in the sense that large enough choice of

~(S i) --~ ~(S i)

~(X) = ~N(X)

N , depending of course on

tain a morphism of functors, that

Yq , then

This means that for any object

lim~ ~q(X) = (X,Y)

If we let

~ --~ m

X

X .

We ob-

which now has the property

is an isomorphism for all

from (li) and the fact that every

for a

in

~

i .

It follows

may be built up

from spheres by a finite number of cofibrations that

~ --~

is an isomorphism of f ~ c t o r s . Our assumption (iii) then assures us that homological dimension,

Y

has finite co-

and (i) says that the homotopy groups are

finitely generated in each dimension. allow us to conclude that

Y

These two facts put together

has finitely generated integral homol-

ogy groups, which vanish beyond a certain dimension.

It then fol-

lows that

Y' .

Y

prop. ~.l).)

may be replaced by a finite CW-complex It is here that we use that

This finite CW-complex

y1

Y

([30],

is simply connected.

being, of course, again simply con-

nected, is an object of our category

~ , and represents the functor

So

We shall achieve point.

Suppose

Yq-1

(Z) inductively. given.

Take

Yo = Y1

to be a

For simplicity we let an object of

stand for the contravariant functor which it represents. bl,...,b m

generate the group

a(S q) .

Let

Use the same letters to

89 -

denote the corresponding morphisms Let

7.15

of functors,

Z = Y q - l V (Sq)lv (sq)2 v... v (Sq)m .

nite CW-complex obtained from

Yq-1

Extend the functorial morphlsm by setting it to be follows that

on

Z(S q)

cj

for

j = l,...,k

satisfies

morphism

Yq--> ~ , and moreover

and cells of dimension

for

q+l

giving us also:

i < q .

is the fi-

copies of

to a morphism j = 1,...,m .

f j: S q --> Z and define

(ii), the morphism

Since we have added to

hypothesis,

for

m

Z

Let

Sq .

Z --> a It then

Cl,...,c k

be

which generate the kernel of the above homo-

Since

phism.

by wedging

is surjective.

Take continuous maps

tatives of

That is

Yq-1 -> e

(sq)j

Z(S q) --> m(S q)

elements in morphism.

bj

bj: S q --> ~ .

Z--> ~

which are represenYq = ZUfj(Dq+l)j extends to a

Yq(S q) --> ~(S q) Yq-i

9

is an isomor-

only wedges of q-spheres,

we have not disturbed the inductive Yq(S i) --> ~(S i)

is an isomorphism

q.e.d.

A consequence of the representability

theorem is that T

de-

fined above is representable by a finite simply connected CWcomplex,

Xv .

Proposition (7.lB): type as

(i)

XT

has the same stable prohomotopy

X .

(ii)

c(X,X T) =

(iii) If

Y

is a finite simply connected CW-complex having

the same stable prohomotopy type as then

M

c(X,Y) = T,

Y ~ XT . Proof:

A

X , and such that

(i) follows from cot. (7.7)together with the fact that A iS free of rank one over A . (ii) is clear:

7 916

90-

Hom(X,X T) =

(X) = ( X , X ) % ( x ) M ~

(iii) comes from this fact:

The natural homomorphism

Hom(A,X)@E(x)Hom(X,Y ) Y> Hom(A,Y)

(7.1#)

A , provided

coming from composition is an isomorphism for all X

and

Y

are of the same stable prohomotopy type.

this we need only show that true using cor.

is an isomorphism.

But this is

(7.7).

Corollary (7.14): Let complex.

~

To prove

X

be a finite simply connected CW-

The set of distinct isomorphism classes of finite ~ -

complexes of the same stable prohomotopy type as in a one:one correspondence with the set to every such

T

the class

P(X)

X

may be put

by associating

c(X,Y) E P(X) .

Of course, our methods made use of simple connectivity in only one place, namely to insure that

XT

be a finite CW-complex,

and so we may state more generally: Corollary(7.15):

Let

X

be a finite ~ - c o m p l e x .

distinct stable homotopy types of CW-complexes stable prohomotopy type as dence with the set class

P(X)

X

Y

The set of

of the same

may be put in a one:one correspon-

by associating to every such

Y

the

c(X,Y) E P(X) . Since

P(X)

is a finite set by Jordan-Zassenhaus,

Corollary (7.16): Let

X

be a finite CW-complex.

we obtain: There are

-

91

7.17

-

only a finite number of stably inequivalent same stable prohomotopy Examples: ing a class

1.

type as

n

p , where

must be greater than one, i ~ +J

.

Xi

is simply connected.

(rood p), then

Xi

[!iy] C ~ m ( S n )

~

(7.18)

A:

and to try to recover,

as an invarlant of

considering all coflbrations

Sn - > f

X i --~ S m+l g

is any cofibratlon of the above form, ~n(Xl) = Z .

Xj

are of

f

~

Xi

a priori,

the

We do this by of the form

.

of

Xi .

Now, if

A

must represent a gener-

This may be seen by applying the long exact

stable homotopy sequence to m+l

Xi .

in the category

Note that there is one by the construction

snu(iu

and

The idea is to start with the finite complex

regarded as an object of

ator of

is a prime.

stable homotopy types.

Proof:

classes

p

represent-

i , consider the CW-complex,

Lemma (7.17): If distinct

y: Sm --> S n

Fix a continuous map

X i = snu(iy)D m+l

Since

of the

X .

u E ~m(S n) , of order

For any integer

CW-complexes

A .

Thus, up to homotopy,

f: S n -->

is either the canonical inclusion or the canonical

inclusion composed with the automorphism

of

Sn

of degree -1 .

From this we may conclude that up to isomorphism in a unique cofibratlon

A

of the form

(7 .18).

~

there is

-

Continuing

this cofibration

cofibration induced by

(7.19) where

92

7.18

one stage further,

forming the

g , we obtain:

Sn _ ~ f

h = ~S(iy)

-

X i --~ S m+l _> S n+l g h

, the sign depending on the orientations

for the

Sn

and

Sm+l

classes

+iy E ~n(S

manner,"

concluding

occuring.

TM) ,

Desuspending

h

chosen

we obtain the

and this was obtained in an "a priori

the proof of the lemma.

We state without giving the proof: Lemma (7,20):If i,j are not congruent to zero mod p , then ^ ^ X i ~ Xj . Note that to show the profinite completions of X i and Xj

are the same, it suffices to show that their q-completlons

the same for all

q .

simply connected

(cf.

completions for all

of

i .

p-completlon.

Xi

(This is true since (12.13)).)

split,

Therefore

Xi,X j

But for all

in the sense that

are

are finite and

q ~ p , the q(Xi) q ~ (Sn)qv(sm+l)q

one need only concentrate

on the

- 93 -

w

8.1

Hypercoverlngs.

In this section we review, with some extensions, of Verdier

([3] SGA 4, fascicule l, expos~ 5, appendice).

will be convenient

It

to make the following preliminary definitions,

generalizing Kan's notion of a free simplicial S(n,m)

the theory

group:

denote the set of surJectlve monotone maps

Set

Let

s: An --~ ATM .

n

SCn)

=

I

I sCn, m ) =

m=l Deflnltion cial object objects

(8.1):

K.

I

I SCn,m) .

m=l

A splitting up to level

with values in a category

Nk c Kk

for every

dition: For every

k ~ n

C

n

of a simpli-

consists of sub-

satisfying the following con-

j = O,...,n , the canonical map

J J Ns->K j s S(J) is an isomorphism,

where for

N k , its mapping to object

Nk

Kj

Ns

C

Remark:

K..

denotes a copy of

s: K k --> Kj .

will be called the nondegenerate

composition of

part of

The

Kk

and the

will be called a degeneracy de-

We shall refer to a simplicial

object with

with a given splitting as a split slmplicial object. In the category of sets, any slmpllcial object has

a unique splitting.

More generally,

using

be a locally connected and distributive exists,

,

being induced by

above direct sum decomposition

values in

s E S(j,k)

it is unique.

If

C

(w

below,

category.

let

C

If a splitting

is the category of etale,

separated

94

schemes over a scheme in

C

Let

products, and let

K.

C

(We do not use this fact.)

be a category closed under finite co-

be a simpliclal object of

a splitting up to level Proof:

8.2

X , then every simplicial object with values

has a unique splitting. Lemma (8.2):

-

m .

Then

SkmK.

C

which admits

is representable.

Set m

S(n)Im = I I S(n,j)

.

j=O Then we write

(SkmK")n =

t s Im Nt "

Now for

r: Ai --> An , we seek a definition of the operator

(S~K.)n

' and we do this inductively with respect to

each direct summand

(a)

If

Nt

n

r

on

and on

separately. r

take

to be the identity map

r

is surjective,

r

is not surjective, then construct the following

N t --> Ntr 9 (b)

If

commutative square of monotone maps:

An

t

rI Ai

> Aj

!~ > k U

There is a unique such square with Since

r

is not surjective, we have

u

surjective and k ( n .

v

injective.

Thus by inductive

95 hypothesis, u

is.

the operation of

v

8.3

is defined,

and by

(a), that of

Set

rlN t = u I N ,

and one easily sees that representing

K.

Let

C

be a category closed u n d e r finite direct

a simplicial

splitting up to level ting up to level subobject

so defined is ~ simplicial object

the functor after which it is named.

Lemm.a (8.3): sums, and

SkmK.

object with values

n-1 .

in

C , given a

To extend this splitting to a split-

n , it is n e c e s s a r y and sufficient

Nn c K n

to give a

such that the natural map below is an isomor-

phism:

( S k n - I K " ) n " Nn -> Kn

Proof:

Now let U).

Evident from the c o n s t r u c t i o n

C

be a site

nite products and fibred products,

Skn_lK.

that

C

is a sheaf on

C .

A point of

(Sets) --> C , i.e.,

i.e., C

to some universe

and u n d e r finite

that for

above.

is closed under fi-

we always assume that the topology of

than the canonical topology,"

Ill, 2)

of

([3], 1 . 2 ) ( b e l o n g i n g

We will assume for convenience

Moreover,

"

C

coproducts. is "weaker

x E C , Hom(.,x)

is a m o r p h i s m of sites

a f u n c t o r on the u n d e r l y i n g

([3],

categories

p: C --> (Sets)

which carries coverings

to surjective

families,

and which is exact

-

96

8.4

-

in the sense that it commutes with finite fibred products and arbitrary direct limits. the choice of a point.

A pointed site is one together with A pointed simplicial object

values in a pointed site

C

covering in

C

X.

(8.4):

of

C

Let

C

P(Xo) .

be a (pointed) site.

Let

A hyper-

is a (pointed) slmpllclal object with values

satisfying the following conditions for all (sUrJo)

with

is a slmplicial object together

with a choice of point in the set D eflnltlon

X.

e

be the final object of

X

O

C .

n : The map

--> e

is a covering. (sUrJn)

The canonical morphlsm

Xn+l --~ (C~

is a covering. A truncated hypercoverlng of level cial object satisfying If

X

(surjq) for

is any object of

ing of the site

C/X

n

is a truncated slmpll-

q ~ n .

C , we will refer to a hypercover-

as simply a hypercovering of

X .

Note that a morphism of sites carries hypercoverlngs to hypervcoverings. Enlightenment (8.5):

The reader dismayed by the austerity of

the above should work things out carefully in these particular

-

97

8.5

-

cases:

(a)

Let

C

be the site

(sets), whe're as always coverings

are surJective families of maps. non-empty

A hypercoverlng

slmplicial set, by (sUrJo).

construction

of coskeletons

plained as follows:

(w

the condition

map of the n-slmplex

An

that such a simpliclal

to

to

is then a

Taking into account the (sUrJn) is ex-

For every map of the simplicial

the boundary of the n-slmplex,

cont rac tible.

X.

set

BA n ,

X. , there is at least one

X.

inducing it.

set satisfies

It follows easily

the Kan condition,

In fact hypercoverings

are characterized

and is by these

two properties. (b)

Let

C

be any site,

a covering morphism. cated at level

o

Y

Regarding

U

with values in

This is Just, in plainer language,

It is a hypercoverlng feature,

an object of

C/Y , we may form the canonical <

<

<

Y .

This gives the clue as to the novel

start out w i t h an old-fashioned

covering

the computation of, say, cech cohomology, object above,

determined by the original covering. coverings,

however,

mined.

Kn+l--

(C~

U--> Y

Namely,

if we

and proceed

to

then we must mechanwhose n-simplexes

In the construction

we are given much greater freedom.

choice of n-simpllces to take

COSkoU .

slmpliclal object

and the ultimate use, of hypercoverings.

ically form the simplicial

U--> Y

as a slmplicial object trun-

<__

of

C , and

are of hyper-

In the

Kn+ 1 , the mechanical procedure would be ' when say

K./n

is already deter-

Here we are given the option of "refining"

(Cos~K.)n+l

-

98

8.6

-

further

(see below),

since we require only that condition

(surJn)

holds.

It is this option of refining further in each dimension

that makes hypercoverings useful in cases where the category of coverings is too coarse.

mere

We shall now prove some elementary lemmas about hypercoverings: Lemma 8.6: (~/n___~)> (K./n)

Let

K.

be a hypercovering of

a morphism of truncated

jects,

such that

level

n .

L./n

Then there is a hypercovering

n

is the morphism

L. ~> K.

(L./n)

(pointed) slmplicial ob-

is a hypercovering of

phism of simplicial objects,

C , and

L.

C of

truncated at C , and a mor-

whose truncation at level

~/n .

In fact, we m a y take

- (COSknT.) • (COSkni" )i.

The verification that this works is easy, and we omit it. Lemma

(8.7):

Let

C

covering truncated at level

be a (pointed) site and n , split up to level

is a m o r p h i s m of hypercoverings

truncated at level

which induces an isomorphism on (n-1)-skeleta, is split up to level Proof:

We take

n . L/n-i = K./n-I , and

Ln = ( S ~ _ I K . ) n " K n

.

K.

a hyper-

n-i . n,

There

L.-->K.

and such that

L.

,

-

The boundary operators summand,

99

8.7

-

Ln--> ~n-i

and the degeneracy maps

from the simplicial object

are the natural ones on each Ln_ I --> L n

Skn_iK.

Ln_l = Kn_l A>

:

(Skn_lK.)nr__> Ln

This yields a slmplicial

(the inclusion being as first summand). object which maps to

K.

map on each summand.

So defined,

ditlon sUrJn,

are those coming

by sending

Ln

to

Kn

via the natural

clearly satisfies

L.

the con-

q.e.d.

We therefore obtain: Lemma 8.8:

Let

K.

be a hypercovering,

a slmplicial object up to level K ~. --> K.

of simpliclal objects

n .

There is a morphlsm,

such that

(a)

It induces an isomorphism

on

(b)

K|

object,

ing for

C .

which is split as

is a split slmpliclal

Sk n . which is a hypercover-

We shall call such a thing a split hy2ercoverlng.

The above lemma allows us to replace hypercoverlngs hypercoverings, Lemma level

8.9:

by split

if we wish. Let

K.

be a hypercoverlng

of

C , split up to

n : l

Kn = (S~n-IK")n" Nn '

and let L.

of

M--> N n C

be a covering morphism.

and a morphlsm

L. --> K.

There is a hypercovering

which induces an isomorphism

i00 on

n-i

skeleta,

and such that

the expected way to Proof: fined.

~

Ln =

(Skn-lK")n" M

and maps in

.

The truncation of

Now apply

8.8

L.

at level

n

is taken as de-

(8.?).

The above lemma assures us that we may "refine the nondegenerate part"

of a split hypercovering

a refinement at level

n

entails no change on the

This is a quite convenient Lemma

(8. l0 ) :

Let

arbitrarily. n-1

K.,L.

be hypercoverings,

and let

be a map.

ping to

K|/n = K./n , and an extension of

map with

with

9': KI --> L.

.

skeleton.

technical fact.

f/n: K.In--> L.In K.

Moreover,

There is a hypercovering

Kt ~/n

map to a

In particular,

if

K., L.

are hypercoverir~gs

K./n = L./n , then there is a

K!

dominating both, with

K|/n = K./n = L./n . Proof:

We will construct

suffices

to find

B y lemma

(8.6), it suffices

K.'/ (n+l)

.

Let

N'

K!

so that

K!

by induction on

$/n

n .

extends to dimension

Thus it n+l .

to find a truncated hypercovering

be the object making the following diagram cartesian

h N'

Kn+ 1

The map

f

is a covering,

>

Ln+ 1

> (Cos~L)n+ 1

since

g

is one.

Since compositions

-

i01

of coverings are coverings, the map from

f

q~_n

, and = N',, (SknK)n+l

and define the truncated object and degeneracy maps.

Now

Thus we can take for

I Sn+l

h: N' --> Ln+ 1

X

If

S

KL/n+I

~/n

Set

obtained

K~ = Kq

for

,

with the evident face

extends canonically to

S~K.

.

this canonical map together with the

above.

For the moment, let ucts.

N'--> (COSknK)n+ I

by composition is also a covering.

~+l

map

8.9

-

C

be any category with finite coprod-

is a set and

with itself indexed by

X E C , we denote the direct sum of S

by

X@S .

This is a covariant bifunctor in

X

and

S .

The unit interval will mean the 1-simplex in the category of sets, denoted by

I.

Given a simplicial object

=

K.

A .1

with values in

the simplicial object

K.@I.

whose n-simplex object is Just

.

G , we may form

- i02 ~@I

n =

If

i6in

8.10

Ki n

'

where the superscript denotes the index9 which is an increasing sequence of zeros and ones of length degeneracies

n+l .

Thus faces and

operate on these indices, vlz. dvi ~-i

dr: ~ - - >

Svi

Two m a p s

fo,fl: K. --> L.

of slmpllcial objects will be called strictly homotoplc if there is

J

a map

f: K.@I. --> L.

such that f eo _ fo

where

cv: K. ,> K.@I.

9

f el

(v = 09

fl

are the evident "inclusions."

Two maps are homotopic if they can be related by a chain of strict homotoples 9

in the usual sense.

If

fo 9

are maps of pointed

slmpllclal objects9 then the strict homotoples will be assumed to preserve the base point. This notion of homotopy is functorial in

C .

Proposition

(8.11):

be hypercoverings with maps between them. between

Let K.

103 -

8. i i

C

K.,L.

be a (pointed) site, let

split, and let

fo, fl: K. --> L.

be

Suppose given a truncated strict homotopy

fo fl : f/n: K.|

There is a refinement of

L./n .

K. , i.e., a map of hypercoverings

~: K! --> K. , and an extension of

(~|

= f'/n

to a strict

homotopy f' : Kl|

Moreover,

K!

--> L.

.

can be chosen so as to be equal to

K.

in dlmen-

sions < n-i . Proof: dimension

By induction, it suffices to extend the homotopy to n+l , applying

(8.9).

Now using the splitting of

K. ,

we have in the notation of (8.1)

(K.|

=

I I isI n

Nit

t S(n) from which one sees immediately that from that of

K.

K.|

acquires a splitting

.

The degeneracies (Skn(K.| i with the sum of those N t , i 6 In+ 1 , t 6 S(n+l) can find a degeneracy operator

i = svi'

sv: An+l --> An

,

t = svt'

1

identify

for which we

satisfying

104 for suitable N~

i',t I .

8.12

One checks easily that the non-degenerate

are of the following two types:

any

(a)

t = identity 6 S(n+l,n+l)

(b)

t = sv: An+l --> An

i' .

Thus in this case

, i.e.,

for some

N i = Nn+ 1 .

v , and

for

i ~ Svi'

t E S(n+l,n) 9 and so

N i = Nn .

We have therefore

(K. |

where

Mn+ 1

)n+l = S ~ (K. |

is the sum of the

truncated homotopy

f/n

N%

of types (a), (b).

extends uniquely to

What we have to do is to refine g: Mn+ 1 --> Ln+ 1

)n+l = ~ + l

K.

S~(K.@I.)n+ 1 .

so that there is a map

commuting with the appropriate face operators.

On the other hand 9 we also have a unique extension of map

Now the

K.@I. --> COSknL.

.

Therefore there is a map

(COSknL)n+l , and the requirement on

g-

g

f

to a

Mn+ I -->

is Just that the diagram

"'> /

(COSknL)n+ 1

commute.

Since

r

is a covering by (surJ n)

coverings of the components a map exists. proof.

N n , Nn+ 1

of

for L. , there are Mn+ 1

for which such

Thus two applications of lemma (8.9) complete the

We leave the considerations of base point to the reader.

Corollary (8.12): K./n = L./n , and that

With the notation of (8.11)9 suppose that f~

= fl/n = identity.

Then a refinement

I05

K|

of

K.

can be found with

8.13

-

K|/n = K./n , such that the trun-

cated identity homotopy at level

n

extends to a strict homotopy

fI: K|@I. --> L . . Here a straight application of (8.11) would require a refinement differing from

K.

in dimension > n .

the existence of non-degenerate n+l , a s i n

N~

the proof of (8.11).

homotopy extends trivially to are summands of

of type (b) in dimension

However, in our case the identity

SknK.@I.

(SknK.@I.)n+ 1 .

This comes from

, and the

N~

of type (b)

Thus we do not after all need to

drop back a step in order to extend the homotopy of dimension

n+l .

Now an application of (8.11) yields the result. Combining

(8.11),

Corollary (8.13):

(8.12) with lemma (8.10), we obtain (1)

For a (pointed) site

C , let

denote the category whose objects are hypercoverlngs, maps are homotopy classes of morphisms. HR(C) ~

is filtering.

underlying site.

Let

C

and whose

Then the dual category

be a pointed site, and

The stripping functor

HR(C)

U

HR(C) ~ --> HR(U) ~

its is co-

final. (ii) note by tion

Let

K./n

HR(C,K./n)

be a truncated split hypercovering,

the category of hypercoverings with tranca-

K./n , and with homotopy classes of morphisms.

HR(C,K./n) ~

and de-

Then

is filtering.

Verdier's main result ing proposition:

((8.16) below) is based on the follow-

106

-

Proposition object

(8.1#):

Let

K.

8.14

-

be a hypercovering.

U E C , we denote as in [3] by

~U

For an

the sheaf of abelian

groups characterized by the property that for any abelian sheaf A

we have Hom(~u,A)

Consider the simplicial sheaf

= A(U) .

~K.

with its augmentation

2ZK. --> ~--- = ~e (e of

the final object of ~e

C).

This augmentation is a resolution

"

We will prove the proposition under the extra assumption that the site

C

has "sufficiently many points," so that a sequence

A --> A' --> A" of sheaves is exact when for every point (sets) the sequence

p*A --> p*A' --> p*~' is exact.

enough for our applications,

p: C -->

This will be

and we leave the general case to the

reader. We have

P*( ~ U ) = ~ p ( U )

' where the abelian group on the

right is just the direct sum of set

p(U) .

For, if

~

with itself indexed by the

A E (Ab) , then

Horn( 2~p(u),A ) = A p(U) = p.A(U) = Horn(~u,P.A) = Horn(p* 2Zu, A) .

hence the functor

(*)

p*

applied to the above augmentation yields

mp(x.)

->

-

-

But

p(X.)

(8.5).

is a h y p e r c o v e r i n g

107

-

8.15

in (sets), whence is contractible

T h e r e f o r e its h o m o l o g y is that of the point,

exactness

which is the

of (*).

Corollary(8.15): a b e l i a n sheaf on

Let

C .

K.

be a hypercovering,

and

F

an

There is a spectral sequence

E~q- ~q(~p,F) --> ~P+q(c,;)

where

H

denotes

sheaf cohomology,

This is a standard

site

C .

(8.16):

H'(C,F) = H'(e,F)

spectral sequence of

tion, via the i d e n t i f i c a t i o n Theorem

and

Let

F

Ext

.

for a resolu-

HP(*,F) = ExtP(2Z.,F))

.

be an a b e l i a n sheaf on the

(pointed)

We have a canonical i s o m o r p h i s m

Hq(C,F) - lira Hq(F(K.)) -9 K. where the group

H

F(K.)

on the right denotes

c o h o m o l o g y of the cosimpliclal

, and where the limit is taken over the c a t e g o r y

HR(C) ~ Proof:

We pass to the limit over the terms

spectral sequence

E pq

of the

(8.15) :

E~q(K. ) _- HP (Hq(K.,F)) .

Let

K.

be a split hypercovering,

and

~

a class in

Hq(~,F)

.

-

Write

Kn = ~ I

classes

Nt

as in (8.1).

a t 6 Hq(Nt,F)

there are coverings Applying K!

of

.

N~

such that

-

8.16

Then

a

is the product of

Since cohomology vanishes locally, of

(8.9) repeatedly, K.

I08

Nt

such that

a t ~-> 0

in

Hq(N~,F).

one finds that there is a refinement

a ~> 0

in

Hq(KI,F)

.

This proves that

lim EPq(K.) = 0 , for all

lim Hq(K.,F) = 0 , hence that

q > 0 ,

-9

which yields the theorem. Definition

(8.17):

Let

C

be a class of groups.

of a site is said to have C-dlmension < d constant abelian sheaf stant sheaf site

C

A

for

on

if for every locally

C , locally isomorphic to a con-

A 6 C , Hq(X,F) = 0

for all

q > d .

is said to have local C-dimension < d if for every

there is a covering Theorem K./n

F

An object

X I --> X

(8.18) :

be a truncated

Let

C

such that

X'

all of C-dimension ~ d

C .

for

X 6 C,

has C-dimension < d .

have local C-dimenslon < d , let

split hypercovering,

cally constant C-sheaf on

The

and

F

an abelian lo-

Assume that the objects

q = 0,...,n .

Kq

are

Then the canonical

map --)

Hq(F(K. ))

Hq(C,F)

K. is surJective if K.

q = n+d+l , and bijective if

runs through the category Proof:

By

HR(C,K./n)

HR(C,K./n)

that as

K.

of (8.13).

(8.9), we can refine the non-degenerate parts of

a split hypercovering arbitrarily, gory

q > n+d+l , where

.

in dimension > n , in the cate-

Since cohomology vanishes locally,

runs over

HR(C,K./n)

, we have

it is clear

-

l o 9

lira HqCK.,F)

8.3.7

-

= HqCSknK.,F)

K. for all

q > 0 .

By lemma

(8.19) below, it follows that

lira HP(Hq(K.,F))

= 0

K. if

p > n , and

q > 0 .

Because the

Kq

(q = 0,...,n)

have

G-dimension <_ d , we also have by (8.2)

Hq(S~K.,F)

if

q > d .

quences

Thus if we pass to the limit over the spectral se-

(8.15), we obtain

lira K. if

= 0

or if

q > d

= lira HP (Hq(K.,;)) K.

p > n

and

q > 0 .

o

This yields the theorem

immediately. L emma category

(8.19): C

Let

K.

with finite coproducts,

a functor sending coproducts

Hp(T(S~K.))

Proof: T

be a split simplicial object in a and let

to products.

= 0

if

Consider the inclusion

carries sums to products and

K.

T: C ~ --> (Ab)

be

Then we have

p > n .

S~_IK.~-> is split,

yields an exact sequence of co-simpliclal

S~K..

Since

this inclusion

groups

-

(*)

where

i z o

8.18

-

0--> X.--~ T(SknK.) --~ T(Skn=iK.) --~ 0

X.

can be described as follows:

Let

Sn

,

denote the

minimal simpllcial pointed set representing the n-sphere, i.e., the simpllclal set with two non-degenerate simplices: O-simplex and one n-simplex. part of

~

.

Then

by the pointed set to one).

Thus

Xq Snq

Let

N E C

is the product of

one

denote the non-degenerate N

with itself indexed

(where the base point factor is set equal

Hq(X.) = Hq(s.n,T(N))

pointed n-sphere with values in

is the cohomology of the

T(N) , which is zero for

q ~ n .

The lemma now follows from the exact cohomology sequence of (*) and induction.

-iii-

9.1

w

The Verdier functor.

Let

C

be a category admitting finite fibred products.

say that

C

is distributive

and if the following

if it has an initial object

condition holds:

We

~ ,

For every set of objects

T i 6 C , i E I , such that the coproduct able in X--> S

I I Yi is representi Yi -> S , and for every m~p

C , for every set of maps in

C , the canonical morphism of functors

LiI X• i

x• s(! i I q )

is an isomorphism. E~ery site ([3] II, 4.8b). X 6 C

Let

has an underlying distributive C

be a distributive

is called connected if

and if

X

C

implies that

object.

category

category.

An object

is not the initial object

coproduct decomposition,

Xi = ~

for exactly one

~ ,

so that

i .

The cate-

is called locally connected if every object is a coprod-

uct of connected connected

X

has no non-trivial

X = Xl.. X 2 gory

C

objects.

For brevity,

if it is locally connected, Using the distributivity,

the expression of an object

X

Sects is essentially unique. an object its set of connected

C

it is immediately

seen that

as a coproduct of connected Moreover,

,

ob-

the rule associating to

components is a functor.

(Sets)

is

and has a connected final

note this functor by

~: C - >

we will say that

We de-

-

1 1 2

9.2

-

and call it the connected component functor. Now let functor

~

C

be a locally connected site.

to the category

HR(C)

Then applying the

of hypercoverings (w

we

obtain a pro-object

T-]- c = [~(K. ) ]K. ~ m~(c) in the homotopy category

~

of simplicial sets.

associating to a locally connected site

C

The rule

the pro-object

is functorlal with respect to morphisms of sites.

V-K c

We call it the

Verdier functor. Note that if functor

C

is a connected pointed site, then since the

p: C--~ (sets) is exact, it commutes with coproducts.

Hence an element ponents of

X .

x E p(X)

"lles in" one of the connected com-

Thus a pointed hypercovering yields a pointed

simplicial set via

~o ' and so the Verdier functor

pointed sites to the homotopy category sets.

~o

~-[

of pointed slmpllclal

Via the geometric realization, we can identify - ~ K

an object of

carries

C

with

~o " and thus the theory developed in sections 2-6

is applicable.

In particular, the homology pro-groups of a con-

nected (pointed) site, and the homotopy pro-groups of a connected pointed site are defined, by the rule

Hq(C,A) -- Hq(T-]- C,A) (9.1) 9 q(C) If

A

= ~qC[-]- C) .

is an abelian group, we may form the "constant'' sheaf

-

= Ae

on

Lemma X 6 C .

C , where (9.2):

9.3

-

is the final object,

e

Let

i13

C

as in (8.14).

be a locally connected site, and let

Then there is a canonical isomorphism

We leave the proof as an exercise. is a hypercoverlng,

then

It follows that if

~(K.) = A ~(K')

Hq(_A(K.))

--

A v(x) ~> A(X)

. K.

, therefore that

Hq(?z(K.),A)

where the group on the right is the cohomology of the slmpliclal set.

Applying

(8.16), we find that cohomology with constant co-

efficients factors through the Verdier functor: Corollary A

(9.3):

an abellan group.

Let

C

be a locally connected

site and

There is a canonical isomorphism

Hq(c,A) Hq<]-]-C,A) Example

(9.9):

etale site over schemes

X'

X .

Let

X

be a prescheme,

and let

C

be the

Recall that this site has as objects the

etale over

X , and as coverings surJectlve families.

It is clear that an object

X'

of

if it is a connected prescheme.

C

is connected if and only

We say that

nected if the etale site is, so that every

X X'

is locally conetale on

X

is

a coproduct of connected schemes. proposition connected. cf. EGA I 6.1.9

(9.5) :

A locally n o e t h e r i a n prescheme is locally

-

114

-

9.@

For a locally connected prescheme

(9.6)

X , we write

Xet E p r o - ~

to denote the pro-object We refer to

Xet

-[~-C

for the etale site

as the etale homotopy type of

C

X .

over

X .

By pointed

prescheme we mean a prescheme together with a geometric point. This point then gives a point for the etale site

C , in the

obvious way.

then

Thus if

pointed site, and so

X

is a pointed prescheme,

Xet

is in p r o - ~ o

in this case.

C

is a

Notations

of the type

Hq (Xet, A)

(9.7)

q(Xet) A 6 (Ab)

H q (Xet, A)

are defined as in (2. i), (2.2). Example

(9.8):

Let

X

be a pointed, connected, and locally

connected topological space, and let whose objects are coproducts X .

C

be the ordinary site

(disjoint unions) of open subsets of

Then we obtain a definition of homotopy pro-groups for

X ,

which we will denote by V

~q(X) = ~q(C)

We will discuss the relation of these pro-groups with the usual homotopy groups in section 12. Example

(9.9):

Let

G

be a group, and let

C

be the site

115

of left G-sets, where

always

as

a

-

9.5

covering is

a

surJective map.

This site is canonically pointed by the stripping functor (G-sets) --> (Sets).

Among the hypercoverlngs we have the one

(9.10)

> > GxG

GxGxG

> G=K.

We leave it as an exercise to show that since the G-set

G

is

a projective object, this hypercovering dominates every other. Thus in the category prO-Ko

we have

7r(K.) ~ - [ ~ - C

whence

77-C

is actually in

the orbit space of

K.

Ko "

Moreover,

under the action of

~(K. )

is just

G , and this orbit

space is the standard construction of the Eilenberg-MacLane space K(G,I)

.

Hence C = K(G, 1)

,

and

=l(c)

= G

=q(C) - o Example

(9. ii) :

be the site of finite

Let

G

if

.

be a pro-finite group, and let

continuous G-sets.

one sees that every truncated hypercovering by one of the form

>-G•215

q>O

,. ~

G•

>~

Then via example L./n

C (9.10),

is dominated

-

9.6

116-

m

wh e re

G

is a s u i t a b l e

finite

quotient

of

G .

Thus

we obtain

C ~ K(G,I)

and

~i(c)

- G

~q(c)

= o

if

q > o

.

- I17

The fundamental

w

Let

C

[13]) over site

i0.i

group.

be a site, and C

-

~

a flbred category

which is a category of descent

C , i.e.,

so that for

X E C

and

([15], exp. 190,

(loc.clt.) for the

a,b E $(X) , the functor

f ~-> Hom(f*a, f'b)

for maps

f: Y--> X

is a sheaf on the site

morphisms between objects of

~

"descend"

C/X , or, so that with respect to cover-

ings. Let

K.

be a hypercovering

of

C , and

x ~ ~(K o)

By descent data for

x

(lO.1)

relative to

K. , we mean an isomorphism

(~: doX ---->d~x

between the two pull-backs compatibility

.

of

x

condition that in

(10.2)

we have

"

as in (loc.cit.).

(10.3):

one correspondence

~(K l) , satisfying the

~(K 2)

d~r = % r 1 6 2

with the conventions Proposition

to

The map

K. --> COSkoK.

between descent data for

x

induces a onerelative to

K.

- 118-

and descent data for Proof:

x

relative to

First of all, since

(8.4), the compatibility iff it does on COSklK.

COSkoK.

.

K 2 --> (COSklK.)2

condition

COSklK..

is a coverin~

(10.2) holds for

~

on

K.

Thus descent data depend only on

, and so we may replace

account the description of as the subobject of

10.2

K.

Cosk I

KlXKlXK 1

by

COSklK..

(w

Taking into

we may then write

of triples

(bo,bl,b 2)

satisfy-

ing the identities

dob o -- dob I

(lO.4)

dlb o = dob 2 dlb I = dlb 2

Consider the following diagram:

(lO.5)

K I X (K ~ xK ~ )K I

d

<

0

Ko <

K1

d~

Ko < . . . .

The map

u

I'

K2

<.

do

<. KoXK o < <

is defined as follows:

K o xK o xK o

If

=

K.

I =

COSkkK.

(a,b) E KIX(KoXKo)K I

ii9 -

10.3

(here we are using a shorthand notation in which "elements" supposed to be replaced by maps from a "test object" KlX(KoXKo)K1)

, so that

dla = dib

for

is indeed in

(10.4) are immediately checked,

pull-back to

K.

is a covering

(8.4).

$

.

determines

x

relative to

it uniquely,

since

COSkoK. (do,d I)

K.

comes from one relative to

For this, we need only show that the two pull-backs For, since

the isomorphism

~

, its of (10.5)

Thus it remains to show that any descent

relative to

are equal.

u(a,b)

so that

K2 .

Now given descent data for

data

into

i = 1,2 , then

u(a,b) = (b,a, Sodla ) E K 2 c KI•215 I

The conditions

are

(do,dl)

descends to

is a covering, KoXK o .

COSkoK.. 6o~ , 6[~

this implies that

The fact that it is

descent data follows from the surjectivity of the map

r

of (10.5).

We have

~o = d o U

61 = dlU Sodl61 = d2u

Therefore

(10.2) implies

=

Hence it suffices to show with simplicial identities,

*** (61dlSor

"

d*~*~l~oV= identity. and (10.2),

But, calculating

-

1T

10.4

120-

* * (So%r

=

*d*

or

r = ~ts*d* ) O 2V Since

~

r

* * = dlSo~ * * = id , as required. Sod2~

is an isomorphism,

Thus descent data depend only on the covering the final object category in a here by

of

C , and on

well known way

~

Let

e

.

(~15],

~ E ~(e)

, and put to

ical descent data for

e*~-

x -- r

(r

~13]), which we will denote

.

=

x .

Then

Y(e)

.

d*Xo and

d~x

are

(edl)*~ , which yields canon-

If a descent data

(x,~)

is

to s u c h a canonical descent data, it is said to be

effective descent data. the categories

Y(e)

In case every descent data is effective,

and

are equivalent.

is a full subcategory of As an example,

(cf.

suppose

C

Otherwise,

~(e)

[13 ] for a detailed account).

closed under arbitrary coproducts,

and consider the fibred category of "trivial ~(X)

of

They form a

We think of it as a completion of

canonically isomorphic

isomorphic

x E ~(K o) .

K o --r e

coverings,"

so that

is the category of objects

(sets)] ,

where maps are ones compatible with the projection The category

~(e)

locall~trivlal

X@S--~ X .

is then to be thought of as the category of

coverings

of

e .

etale site of a locally noetherian

It is known that if scheme

C

is the

Y , then every object

Jl.,

of

~(Y)

is represented by an element of

data for a covering

X@S , X--~ Y

C , i.e.,

that descent

an etale surjectlve map, is

121

-

effective in

C .

-

1 0 . 5

This is proved in ([9],X,6).

Now suppose in addition that

C

is locally connected.

it is immediately seen that the group of automorphlsms of over

X

is just the permutation group

connected, and is

P(S) ~(X)

P(S)

in general.

of

S

Thus if

if K.

ing the condition P(S)

Kl@S=

is

Ko|

is satisfy-

(10.2), which in turn is given by an element of K1 .

Obviously this is

nothing but a 1-cocycle of the simplicial set

~(e)

X

do(Ko@S ) = d~(Ko@S ) *

for each connected component of

in the group

X@S

is a hyper-

covering, then descent data for the trivial covering given by a n automorphism of

Then

P(S) .

~(K.)

with values

Two such data yield isomorphic elements of

iff the cocycles are cohomologous. We recall that covering spaces of the geometric realization

of a simplicial set

L.

are in 1-1 correspondence up to canon-

ical isomorphism with simplicial covering spaces, which are maps of simplicial sets

L! --> L.

satisfying the following axiom:

For any simplex

x 6 Lq , denote by

mapping to

Then the face operators

x .

L~

the set of simplexes dv

carry

L'

iso-

X

morphically to

L~v x .

'

"

This fact is well known and elementary.

Moreover, a moment's reflection shows that isomorphism classes of simplicial covering spaces for all

x

L! --> L.

for which

are classified by elements of

card(L~) = card(S)

Hl(L.,P(S))

.

Thus we

obtain the following Corollary (i0.61:

Let

C

be a locally connected site closed

under arbitrary coproducts, and let

K.

be a hypercovering of

The category of locally trivial coverings of the final object of

C

which become

trivial on

Ko

is equivalent with the

C. e

122

-

10.6

category of simpllclal covering spaces of Now suppose that a group, and denote

C

is pointed and connected.

by

~l(c,G)

Let

G

be

the set of isomorphism classes

of locally trivial coverings of G

~(K. ) .

e

given with an operation of

making them into a principal fibre bundle.

By the above

corollary, we have Corollary (10.7):

With the above notation,

~I(c,G) = Hom(grps)(~l(C), G)

Thus the pro-group

~I(C)

has the usual property.

In case

C

is the etale site on a noetherian scheme, this is the "pro-groupe fondamentale enlargi" of ([9]X,6, [21]). As another example, we can let

~

be the fibred category of

locally constant abelian sheaves, i.e., sheaves

F

is an abelian group

such that the

restriction of C

F

A to

and a hypercovering C/K o

K.

is the constant sheaf

for which there

A .

If again

is locally connected, then a reasoning analogous to the above

shows that to give such a sheaf is equivalent with giving a local coefficient group on morphism

~(K. )

(in particular, is given by a homoif

~l(~(K. )) --~ Aut(A)

C

is connected and pointed).

Consequently we have by (9.3) Corollary (10.8):

Let

C

be a locally connected site.

category of locally constant abelian sheaves on

C

with the category of locally coefficient groups on

The

is equivalent ]--KC , and

123

-

10.7

there is a canonical isomorphism

~q(c,;) ~ Hqg[-T c,;) where

F

denotes both the corresponding sheaf and local coef-

ficient group.

124 -

w

A profiniteness

Let

X

ll. 1

theorem.

be a locally noetherian prescheme.

be geometrically unibranch if for every closure of the local ring

~X,x

x 6 X

X

is said to

the integral

is again local

(EGA IV, 6.15.1).

An equivalent

condition is that every

X'

is connected,

is in fact irreducible.

Thus the notion is local

for the etale topology.

etale over

X

which

A normal prescheme is geometrically uni-

branch. Theorem

(ll.1):

ally unibranch, Xet

Let

be a pointed,

noetherian prescheme.

is pro-finite,

~q(Z)

E @

and

@Ko

for all

To prove the theorem, hypercovering

connected,

@

denotes

denotes the category of

the Z E Ko

q .

we remark to begin with that every

K.

of

X

one in which each

Kq

is a noetherian prescheme--we

a hypercovering

geometric-

Then the etale homotopy type

i.e., is in p r o - @ K o , where

class of finite groups, such that

X

for the etale topology is dominated b y

noetherian.

This is trivial.

will call such

Thus the theorem

will result from the following more precise assertion: Theorem noetherian

(11.2):

etale hypercovering

Proof: lent,

X .

([13]VIII, l.1), we may suppose

over

and so

of

Since the etale sites of

irreducible, Y

With the notation of (ii.i), let

X

it has a generic point will cover

Y• P

P .

If

Y

Then X

X

and

~(K.) Xre d

reduced.

K. be a

6 d2(O 9 are equivaSince

X

is

P , and every non-empty etale is connected,

will be a one point scheme.

it is irreducible,

Thus the connected

-

components

of

Y

and of

125

YXxP

-

l l . 2

are in 1-1 correspondence,

whence

we have ~r(K. ) = 7r(K. XxP )

where

K. XxP

,

is the induced etale hypercovering

Now the etale site over

Spec K ,

the site of continuous G-sets,

K

a field,

G = Gal(~/K)

G-sets.

Assume

K. Kq

a hypercoverlng

Truncating quotient

Let

G

be a pro-

q .

Then

~q(~(K.))

q .

K. , and replacing

G

by a suitable finite

~ , we are reduced to the following:

Proposition the pointed

site

covering of

C .

Proof: functor. functor

is equivalent with

of the site of continuous

a finite set for all

are finite groups for all

P = Spec k(P).

([13],VIII,2).

Hence we are reduced to proving the following: finite group, and

of

(ii.3): (w

Let

G

be a finite group,

of finite G-sets,

Then

~q(~(K. ))

The functor

~: C-->

and let

([3],VIII,2), correspond

q .

(sets) is just the orbit space

This functor is a morphism of sites C

be

be a hyper-

i.e.,

the stripping

(sets) sending a G-set to its underlying

the category of sheaves on

C

are finite groups for all

We can also consider the point, p: C-->

K.

let

(sets) --> C .

set.

If we identify

with the category of G-modules,

then the inverse and direct image functors

to the restriction of a G-module

p*,p.

to the trivial sub-

group, and the induction of a module from the trivial subgroup to G .

We have for any sheaf

F

on

C

maps as with G-modules

126

(ll.4)

whose

-

F re_~s > p,p*F

composition Lemma

Let

that the u n d e r l y i n g

L.

that

is a finite

group for every

constant

by the order

be a simplicial

simplicial

> 0 , i.e.,

Proof:

trace.> F

is m u l t i p l i c a t i o n

(ll.5):

i1.3

set

p(L. )

sheaf on

is acyclic Then

G . C

such

in dimension

Hq(v(L.),ZZ)

q > 0 .

Hq(v(L. ),~ ) = Hq(2Z(L.))

C .

of

object in

Hq(p(K.),~- ) = 0 , q > 0 .

We have

n

D e n o t i n g by

~

, where

also the constant

2Z

is the

sheaf on

(sets), we have

Hq(p.2Z (L.)) = H q ( p ( L . ) , ~

Since

p*~=

nihilated rated,

by

~

, it follows n .

from

K.

For,

that

.

Hq(~(K.))

is angene-

the lemma follows.

is a hypercovering,

suppose

this shown.

ing as simplicial in

(11.4)

q > 0

Since these groups are obviously finitely

Now we claim that to prove if

) = 0 ,

set.

C , and a map

(ll.2) it suffices

then Let

Vl(V(K.))

S. --> ~(K.)

We can construct

L. --> K.

such that

to show that

is a finite group. be the u n i v e r s a l

a simplicial

object

coverL.

v(L.) = S. , as follows:

We put symbolically

L. = K.@~(K.)S.

i.e.,

for every connected

the elements

of

Sq

component

mapping

to

x .

x

of Then

Kq , let

Sx

denote

-

Lq =

127

~ K

-

ll.#

) x@S x q

Or, viewing Lq

Kq

as a set with G-operation,

the underlying set of

is p(Lq) = p(Kq)Xv(Kq)S q

and

G

operates on the left factor.

is clear that

p(L.)

simplicial set

,

Via this description,

it

is a simplicial covering of the contractible

p(K. ) .

Therefore

p(L. )

is acyclic in positive

dimensions, and by lemma (ll.5), we find that

Hq(~(L. ),~ ) =

Hq(s.,~ )

Since

are finite groups for all

q > 0 .

S.

connected, it follows from the m o d - ~ Hurewicz theorem that = ~q(~(K.))

are finite groups for all

is simply ~q(S. )

q > 1 , which completes

the proof of the theorem. Thus it remains to show that

~l(~(K.))

is a finite group.

Since this is a finite simplicial set, it will suffice to prove Lemma (ll.6):

Let

G

be a group and let

covering of the site of G-sets.

Then

~(K.)

K.

be a hyper-

satisfies the Kan

condition in dimension 1. For, then Kan's construction of of the finite set

~l

gives it as a quotient

K1 . I

Proof: do~ .

Write

K. = ~(K.) .

We need to find

S E

Let

with

x,y E KI doW = ~

is clear that we may choose representatives ly speaking, in z E K1

with

P(K1) )

with

doX = doY .

satisfy and

x,y

doX =

dlb = ~ . in

K1

It

(strict-

By (8.4), there is a

128

*

11.5

doZ = dlX dlZ = dly

Then the triple c KlXKlXK 1 to

(x,y,z)

(cf.(lO.~)).

(x,y,z) , and

~

9

identifies with an element of By (8.4), there is a

has the required property.

b E

(co,%K.)2 ~ppCn8

129

w

-

12.1

C ompari son theo reins.

We begin with a topological result: pointed topological space. is paracompact, and that every point

x E X

neighborhoods. C

X

S.X

hypercoverlng of

X

Let

of coproducts of open subsets.

the singular complex, i.e., the slmpllcial

C

SqX = [maps Aq --> X] .

With the above notation, let

U.

be a

such that every connected component of every

is contractable.

Then the simpliclal set

ically homotopic to the singular complex 7-I C

X

is locally contractable, i.e., that

Such a space is locally arc-wise connected.

Theorem (12.1):

pro-object

be a connected,

contains arbitrarily small contractable

set of singular simplexes

Uq

X

Assume that every open subset of

be the ordinary site on

We denote by

Let

S.X

~(U.) of

is canon-

X .

Hence the

is canonically isomorphic to the element

S.X

of Ko 9 and in particular, in the notation of (9.8), ~q(X) v ~q(X) . Corollary (12.2) : scheme over

C .

Let

X

be a connected pointed algebraic

With the notation of (12.1), we define

Xcl = - ~

C

.

Then in pro-K o

Xcl ~ S.X

For, the hypotheses of (12.1) on

X

are satisfied for an algebraic

-

130

scheme, since it is triangulable Proof of (12.1): Uq .

Then

S.U.

Let

-

12.2

(cf. [20] ).

S.Uq

denote the singular complex of

is a bisimpliclal set.

We denote by

A.

its

diagonal simplicial set

Aq = S qUq

.

Then we have obvious maps of simplicial sets

s.x

and we claim that these two maps are homotopy equivalences, which will prove the theorem. First of all, to prove that the maps induce an isomorphism on

~l ' it is convenient to do this by showing that the category

of simplicial covering spaces of the three objects are equivalent via

~,8 9

Since

X

is locally arcwise connected, we can identify

simplicial covering spaces of spaces of

X .

X

with locally trivial covering

Now by the discussion of section lO, simplicial

covering spaces of space of

S.X

~(U. )

are given by gluing data for a covering

relative to the open covering

trivial that coverings of

~(U.)

and of

Uo . S.X

Thus it is

correspond.

It

remains to show for instance that every descent data for a simplicial covering of a unique way.

A.

is obtained from descent data on

~(U. )

This is a simple exercise which we leave to the

in

12.3

13I

reader. X' --> X

Now let

is obviously a hypercoverlng

u' = x' •J %

assumptions space set

say connected.

of

X'

corresponding

covering

= A| , it is in fact the diagonal

SqUq ' to

of

S.U ~ . . X'

simplicial

Moreover the covering space of

is Just

S.X'

.

Then

satisfying the

of (12.1) , and if we form the simplicial

~(Ut')• Aq' =

be a covering space,

S.X

Therefore we obtain a

diagram

)

covering

(12.3).

s.x'

To show that

~,8

are homotopy equivalences,

it suffices to show that for every such induce isomorphlsms

X' , the maps

on cohomology with arbitrary coefficients

Thus we may as well drop the By the Eilenberg-Zilber

theorem,

the two spectral sequences

"EPq = H~(H~(AS'U'))

Since

Uq

'E~ q = 0

if

(h = horizontal, v = vertical)

the graded groups of certain filtrations

is component-wise q > 0 .

'E p~ = HP(H~

A .

' .

,EPq = ~ ( ~ ( A S ' U ' ) )

have abutments

~' , 8'

contractable,

we have

of

Hq(Up,A) =

Hence the spectral sequence yields

= HP(~(U.),A)

~ HP(A.,A)

H'(A.,A).

132

This shows that

~

To show that

constant sheaf

8

induces an isomorphism on cohomology,

sheaf A

12.4

is a homotopy equivalence.

use the sheaf of singular cochains The cosimplicial

-

S'(A)

gq(.,A)

we

(cf. [6], p. 18).

is a flabby resolution of the

([16], p. ll9), and it is used to give the

usual identification

of sheaf c o h o m o l o g y w i t h

singular cohomol-

ogy (loc.cit.). Consider the surjective map ([16], p. 19) of bicomplexes

A SpUq

~ > gP(Uq,A)

> 0

.

A consideration of the first spectral sequences shows that associated quences:

~

above

induces an isomorphism on the cohomology of the

total complexes. We have

Consider the second spectral se-

Hq(gP(u.,A))

~ Hq(X,~P(A))

from the spectral sequence

(8.15) since

moreover

if

"E~ q

'E~ q

Hq(X, gP(A)) = 0

q > 0 .

.

gP(A)

This follows is flabby, and

Thus the map

r

on

yields

(12.4)

,,po_2 = Hph

and the groups sequences.

HP(x,A)

(AS.U. )) >

HP (X,A)

,

are the abutments of the two spectral

Now we are interested in the map

A S.X

given by the projection

S.Uq

> A S'U"

-> S.X .

This map induces a map

-

1 3 3

-

12.5

Hp (X,A)

> ,,~po

Clearly the composition of this map with which shows that completes

and

f* D

induces an isomorphism on cohomology,

and

the proof.

Now let back

6

(12.4) is the identity,

f

be a morphism of sites

carries hypercoverings

C --> D .

Then the pull-

to hypercoverings.

are pointed and connected,

and if

f

Thus if

C

is a pointed morphism

then we obtain a map

U-Ff--T We callanf-m_~_~ ~ C

:' I 1

D

c

in p r o - K o

between hypercoverings

L.

of

.

D

and

K.

of

a map

$: L. -> f*K.

.

Such a map induces a map

~(f,r

~ ( L . ) --~ ~(K.)

,

and it is these maps which induce the map of pro-objects Theorem let that

T-F

^

(12.5):

Let

~

denote ~-completion. C,D

class of groups,

With the above notation,

are of local ~-dimension < d

is a ~-isomorphism. Proof:

be a complete

Then

for some

~-~f

Our problem is the following:

~-Kf and

suppose

d , and that

is an isomorphism. Let

W E ~o

be a

.

- 134

-

12.6

"test object," and suppose given a map

b: ~ - [ D --> W .

to show that it factors uniquely through

7--]-f .

We have

Thus we need

to show: (i)

There is an f-map

$: L. --> f*K.

of hypercoverings,

and a homotopy commutative diagram

~(L. )

=(f,r

,

> =(K.)

(m.6) W

in

~o 9 such that (ii)

Given

8

represents

$: L. --> f*K.

is a "refinement" of

b . and two maps

mi

as above, there

~ , i.e., a homotopy commutative diagram

L|,

~,

> f~K|

L.

~

> f~.

such that the pull-back of

~i

to

~(K.)

are equal in

These two problems can be treated analogously.

~'o "

We give a

detailed treatment of (i) in steps (a)-(c) below, leaving (ii)

iS5

12.7

-

to the reader. We shall make repeated use of the "elementary construction" of (4.9), which replaces homotopy-commutative diagrams by actually commutative ones, after a choice of homotopy-commutativity. Our obstruction theory argument below is, in fact, quite similar to the argument of (4.9), except that we must now be a bit more delicate in our method of killing obstructions, to be able to pass to a limit.

so as

In fact we only see how to do

this by an inductive process in the skeleta of

~(L.),

than, as in (4.9), by induction on the coskeleta of

rather

W .

In the special context of schemes, our technique pecially the somewhat convoluted manner of expressing

(and es(12.7))

m~ght be rendered superfluous by a rigidification of the category of hypercoverings.

(a)

Killing Relative Cohomology Classes by Refinement.

We adopt the following conventions:

B

will refer to the

CW-complex which is the realization of the simplicial set ~(L. ) ;

A

will refer to the CW-complex mapping cylinder of

the geometric realization of inclusion complex of

~(~) .

Thus we have the natural

~: B--> A , allowing us to identify A .

If

L! ~ >

f'K|

B

with a sub-

is another morphism of hyper-

coverings, we refer to the CW-pair associated to it by the symbols:

B' ~I> A' .

Lemma (12.7):

Under the hypotheses of (12.5), suppose

-

x E Hq+I(A,B;G) ~-groups.

where

G

136

12.8

-

is a ~-twisted system of abellan

There is a refinement of

~ :

Lr:

> f*K':

~"

L.

> f*K.

which is a homotopy-commutatlve diagram of split hypercoverlngs such that: ; K"Is - K.16

; #"18 - #18

(i)

L"I8 -- L.16

(2)

L" , K'.w

(3)

There is a homotopy-commutativlty rel 6 , F , of the

tl B fl

> A"

B

> A

such that the induced map,

Hq+I(A,B;G) --> Hq+I(A",B '';G)

x

to zero.

8

=

q - d - i

have all objects of d~-dlmenslon < d .

diagram,

sends

where

.

-

Proof:

Hq(A)

12.9

-

Consider the diagram:

-->

Hq(B)

II H q (wK.)

137

,>

> Hq+l(A)

Hq+Z(B,A)

II >

x

in

K' --> K.

L ~q+l(II

c,G)

H q + l ~ - I C,G)

equal to

Then we have a map

(8.10), there is an of this map

K|

to

K.

with

is zero,

in dimensions <_

L|/a = L./a

and by

and an extension

L' --> f*K' .

We obtain a diagram

BI

--.-~~>

At

(~') B

~

>A

which is commutative in dimensions <_ a . L'

in dlmenslons > a-1

conmratatlvity tel(a-l)

By (8.11) we can change

to make the truncated commutative diagram

extend to a homotopy-commutative for

square.

(~') .

Then

Let F'

F'

)

L ,o.~>Hq+~IVD,o)

Hq+l(~K'.,G)

L./a--> f*KI./a = f*K./a

L' --> L.

> Hq§

is zero, hence there is a

such that the image in

and by (8.18) we m a y take a = q-d .

II

Hq+l(~K. )

~qgl-T c,a) i> ~q(I-TD, Q)

refinement

Hq+l(B)

II

H q (TtL.)

The image of

-->

be a homotopy-

induces a map

138

Hq+I(A,B;G) --> Hq+I(A',B';G) image of a class

such that the image of Hq(A",G) (8.18), ment

.

x

l0

is the

We now make a second refinement,

r,

>

f*~:

........

>

f'K|

in

Hq(B",G)

,,

y

12.

such that the image of

y 6 Hq(B ',G) .

T.'

-

is in the image of

This may be done, by our assumptions,

and invoking

(8.11) a second time, we may obtain this with a refine-

$"

which when restricted

$'/6 , and such that

(~")

to 6-skeleta is isomorphic

is homotopy-commutative

ly, for any homotopy-commutativity

F"

rel 6 9

to Clear-

of (~") the induced map

Hq+I(A',B ' ;G) --> Hq+l(A",B ";G)

sends

By = x'

to zero.

To conclude the proof of (12.7) we need

only take for (~) the "composite"

,

L .v .

.

.

.

of the diagrams

>

f*~:

>

f*K~

(~') and (~")

139

-

and take

F

to be

by "composing" F'

the

-

12. ii

homotopy-commutativity rel 8 , obtained

(in the evident sense) the homotopy-commutativity

of (~') with any homotopy-commutativity rel 6 , F" (b)

of

).

The Inductiv e Step.

Let

A,B

be as in (a)

above. Suppose we are given

a

diagram 9o,

B

> A

W

where

W 6 ~o

" such that for a given integer

q > 0 , there is

a homotopy-commutatlve triangle,

Sk qB

-

> Sk qA

W

Lemma 12.8:

There is a refinement

$'

of

~

such that

(i)

r

= @/q-d-i

(2)

LI.,Kt. have all objects of ~-dimenslon < d .

(3)

There is a homotopy-commutative diagram,

Skq+lBt

.> Skq+lAI

Skq+l~

~ q + l W

(~q+l)

140-

where

12.12

m~+l/q-d-i = mq/q-d-i . Proof:

since

Skq~

We may assume that

(~q)

is actually-commutative,

is an inclusion as a subcomplex

([29], Chap. 7.

sec. 6, thm. 12). Let Skq+ 1

Oq+ 1

denote the obstruction to finding an extension to

agreeing with

(~q)

on

a relative cohomology group,

Skq_ 1 .

The obstruction lles in

Hq+l(A,B;G)

coefficient group locally isomorphic to induced by the action of

~l(W)

on

where

G

is the twisted

~q+l(W) , the twist being

~q+l(W)

, whence it is a

~-twisted abelian C-group. We apply (12.?) with ~': L! --> f*Kl

x = Oq+ 1 , to get a refinement

and an actually-commutative

BI

~'

B

~

diagram

> AI

,.> A

satisfying the conclusions of (12.?).

Since

realization of a morphlsm of simpllcial sets, and

~'

cellular.

are clearly cellular.

Hence

tF

We may then consider the diagram

s

is the geometric s

is cellular;

may be assumed (Skq~F)

to obtain

141

SkqB I

-

]2.]3

> SkqA'

1

,[

Sk q ]3

> Sk qA

W

The above being actually commutative gives us an extension,

8~: B t U SkqA t

of

8' , whose obstruction,

>

W

o~+ 1 , is the image of

Oq+ 1

the mapping on relative cohomology induced by

(~F) .

our very choice of refinement

o'q+l = 0

sequently,

Bq_ v I

yields our diagram

(12.7) we have:

may be extended to (~+l)

~q+l ' "

via

Thus, by Con-

This latter map

, and the three properties asserted

in (12.8) come from the similarly numbered conclusions of (12.7). q.e.d. (c)

The Proof of (i):

We find a preliminary refinement

L.(~) ~ (~ f*K.(~)

that the factorization below may be found:

l--Fl(A(~

, ,

T It(w)

> l--[l(B(~

such

142

12.1#

A This is possible since Then for (D1) .

L.(~

r

~l(W)

f.K.(O)

6 @

and

~ f

is a

~-isomorphism.

we may find a diagram of the form

Applying our inductive step (c) we obtain successive re-

finements,

~)(q): L.(q)., > f*K.(q) which admit extensions,

(~+i)

'

for all

q >_ 1

.

Since

(~(q)/q- (d+l) = (~(q+l)/q_ (d+l) we may define the refinement ~(~)/n = $(q)/n

for any

~ ('): L.(")

q l n+d+l .

> f*K.(")

by

Thus admits an extension

of the form (12.6), concluding our proof of (i). Theorem Let

X

(12.9):

(generalized Riemann existence theorem)

be a connected, pointed prescheme of finite type over the

field of complex numbers. >

and

Then there is a canonical map

Xet , and it induces an isomorphism on proflnite Proof:

We introduce in addition to the sites

Cetal e

(9.4) also the site

spaces

X'

its image.

completions. Cclas s

X' --> X

x' 6 X w

(12.2)

whose objects are topological

lying over the underlying topological space of

such that the map every point

C'

e: Xcl

is a local isomorphism,

X ,

i.e., that

has a neighborhood which is isomorphic to

Since an etale map of preschemes

X' --> X

is a local

isomorphism on underlying spaces, and since an open set lles in C t , we have morphlsms of sites

143 -

12.15

C!

Cclass

Cetale

Now it is clear from the definition of local isomorphism that every hypercovering of

Ct

is dominated by a hypercoverlng of

Cclas s .

~7

C' --> Xcl

and so

Thus the map e

yields the map

r .

is a homotopy equivalence,

By the discussion of section lO

and the comparison theorems of ([3], Xl, XVI, 4), this map

r

induces an isomorphism on cohomology with twisted finite coefA

ficients, and on

91 ( ^ =

is a ~-isomorphism,

profinite completion).

by (4.3).

Therefore

A e

Now apply Theorem (12.5) and

([3] X, 4.3) to obtain an actual isomorphism. Corolla r~ (!2.10): addition that

X

With the notation of (12.9), suppose in

is geometrically unibrauch.

morphic to the profinite completion of

Then

Xet

is iso-

Xcl .

Apply (ll.1). The same considerations as used for (12.9), together with assorted comparison results from ([S], XVI) yield the following re suit s: Corollary (12.11) :

Let

k

be a field admitting two imbed-

dings el" r

k -->

into the complex numbers, and let type over

k , Xi

the schemes over

X

be a pointed scheme of finite C

obtained by the imbeddings

i~4 -

ei , i = 1,2 .

12.16

Then

XI, class ~ X2, class A

where

denotes profinite

Corollary Spec(k)

, where

and let

K m k

XK

(12.32): k

completion.

Let

X

be a proper pointed

is a separably algebraically

closed field,

be another separably closed field.

the scheme obtained from

X

scheme over

Denote by

by extension of scalars.

Then

the canonical map

A (XK) et -> Xet

is an isomorphism, Corollary

(12.13):

separably algebraically Spec(R)

A

where

denotes profinite

Let

R

completion.

be a discrete valuation

closed residue field

k , and let

ring with f: X -->

be a smooth proper scheme with connected geometric fibres

Xo,X 1 , both being assumed pointed compatibly with a chosen section of

X/Spec R .

Then there is a canonical isomorphism

XI, et ~

Xo, et

^ where

denotes completion with respect to the class of finite

groups of order prime to the characteristic In what follows, and

p~

let

Up

p

of

k .

denote the class of finite p-groups,

the class of finite groups whose order is not divisible

-

by

p .

If

Z

145

-

12.17

is in pro-K o , we let

pletion with respect to the class p@ A Let denote profinlte completion. Corollary (12.14):

Let

D

pZ

and

and

@p

Z denote comP respectively.

be a dedekind domain, and con-

sider a situation as follows

V

> Spec D Q

where

y

is a geometric point with residue field

the generic point, and where characteristics

p ~ q

proper scheme over Xy, cl

P,Q

C

lying over

are two geometric points of

respectively.

Let

X

be a smooth,

Spec D , with a given section.

is connected and simply connected.

Assume that

Then

A p(Xp, et ) x (XQ, et) p ~ X y, cl

Proof:

We apply (12.12) and (12.9) at the local rings of

the two points

P,Q

respectively,

p(Xp, cl) ~ p(Xy, cl)

,

(XQ,et) p ~ (Xy, cl) p

Thus it suffices to show that if element of

to conclude that

Z

is any simply connected

Ko " then A Z ~ pZXZp

9

.

- 146

In fact, if

Z

is in

-

12.18

the class of finite groups,

@Ko ' @

it is easily s e e n that in fact

Z

Z = p Z X Zp

itself.

of factors in

from a consideration construction

of

Z

@Ko

decomposes

X'

(12.15):

and

elements Z

By the

Z , it is a pro-object of

@Ko ' and hence

splits too.

With the notation of (12.13),

is another scheme satisfying

Xp ~ X'p

This follows for example

for a simply connected

split in the canonical way, whence Corollar~

as a product

of the Postnikov decomposition.

whose members are simply connected

XQ ~ X~ , then

then

the hypotheses

Xy, cl ~ X'y, cl

"

of

X .

suppose If

A.l.1 - 147

APPENDIX

w

Limits.

We recall some facts about limits.

Since the results are

well known and/or routine, we will omit most of the proofs. will work in a fixed universe

U

([3], VI),

gories are assumed to be U-categories, small, i.e., a set of Let

I

meaning),

U

so that all cate-

but not necessarily U-

([3], VI, 2.1).

be an "index" category

(the word index has the empty

and

(1.1)

X: I --> C

a functor, where Xi E C

C

the value of

is another category. X

on

i .

For

i E I

denote b y

A direct limit

lim X = lim X. i is an object of i 6 I

We

C

together with maps

such that for every map

i--> i'

diagram

Xi

"> X i ,

lim X -@

X i --> lim X -@

in

I

the

for each

resulting

- 148-

commutes,

and such that

these properties. morphism,

lim X

Clearly

if it exists.

A.I.2

is universal with respect to

lim X

is unique up to unique iso-

It does always exist if

stance the category of sets, groups,

C

is for in-

or abelian groups and if

I

is a U-small category. The inverse limit

lira X

is defined in an analogous way

([3], A category (1.2) (a)

I

will be called filtering if

Every pair

of objects of

i,i'

I

can be embedded

in a diagram

i i' ~

(b) (essential uniqueness of maps in

I , there is a map

posed maps

i

When (1.2),

then

I

.... > i"

i"

of maps) i' --> i"

set of equivalence

i

, > i'

is a pair

such that the two com-

are equal.

is a U-small filtering llm X

If

category,

and

C = (sets) in

can be described in the familiar way as the classes in

U Xi

for the following equivalence

relation: (1.3)

Let

m 6 X i 9 m' E X i, 9

a diagram

i'

Then

iff there is

- 149

such that the images of

,~ I

A. 1.3

-

in

under the induced maps

X i,,

are equal. For a U-small

(1.4)

I , the functor

lim: Hom(I, (sets)) --> (sets) -9

always

commutes with arbitrary direct limits.

If

I

is filter-

ing, it is also left exact in the sense that it commutes with finite inverse limits,

i.e.,

it preserves monomorphisms

and com-

mutes with finite fibred products. If

X: I-->

(grps)

(or of abelian groups),

is a U-small filtering then the set-theoretic

system of groups limit has an

(abelian) group structure making it into the direct limit in the category of

(ab)

groups.

The functor

lim

is an exact functor

on abelian groups. Now let

I ~> J

We will call (1.5)(a)

be a functor,

if

For every

J E J , there is an

in

I

are two maps in

such that the composed maps

i E I

If

j E J , and

easily that

$

and a map

i E I , and

J , there is a map J ~>> ~(i')

The condition to be coflnal is transitive. verifies

filtering.

#(1) .

(essential uniqueness) > $(i)

I

cofinal if it has the following properties:

j->

(b)

and suppose

cofinal implies

J

i--> i'

are equal. Moreover,

filtering

one

(where we

-

always assume as above that

A.I.#

1 5 0 -

I

is filtering in any case).

dition

(b) can be restated in several equivalent ways,

if

is filtering.

J

even if both

I

Example object

But note that

and

(1.6):

J

subcategory

Let

ff = f . I

(b) does not follow from

J

(a)

be the category consisting of one

f: J --> J Then

J

other than the identity.

is filtering.

consisting of the object

map is also filtering,

especially

are filtering:

j , with one map

Suppose that

Con-

j

The one point

and the identity

and the inclusion satisfies

(a) but not

(b). A functor

X: J-->

a projection map

f

(sets)

from

X

is Just a set to a subset

X

together with

y c X .

Clearly

lim X = Y -@

J while lim Xll = X .

I (1.7)

One case in which

is filtering and that gory

I

I

(b) follows from

(a) is that

is a full subcategory of

J

The cate-

J .

is then automatically filtering.

Proposition ing, and let Denote by

X~

(1.8) :

Let

X: J --> (sets)

If-> J

be cofinal, with

be a functor.

the composed functor

i ~

nonical map lim X

I

--> lim X

J

Suppose Xr

.

I l,J

filterU-small.

Then the ca-

-

151

-

A.1.5

is biJective. Proof:

Apply (1.3).

The surjectlvity of the map follows

immediately from (l.5)(a). u' E X~(i, )

To prove inJectivlty,

and suppose their images are equal in

This means that there is a diagram in

let

~ 6 X$(i),

lim~ X .

J

(1.9)

r such that ~(i")

u = u'

in

(cf. (1.5)(a)).

that there are maps maps

Xj .

Choose

Changing

i--> i"

and

i" 6 I

i"

~(i'" )

i I --> i" .

reasoning to

Then we obtain two

By (1.5)(b), --> such that the composed maps ~(i) _>

i" --> i"'

are equal.

J -->

if necessary, we may assume

~(i) -> $(i") , one factoring through

there is a map

and a map

Replacing

i"

by

i'"

J .

and applying the same

i' , it follows that there exists a diagram

i

in

I

whose image in

J

fits into a commutative diagram

)

152

with (1.9).

Hence the images of

hence equal in

J

Let

in

J

m,~'

I

be the category of maps

f--> g

X@(i,,) '

be a filtering category, and i f> i'

f

in

I , where a morphism

> i'

i"

I .

are equal in

is a commutative diagram

i

in

A.I.6

lira X

Proposition (1.10): let

-

> i "'

Then

Ca) (b)

J

is filtering.

The functors "domain" and "range" from

J

to

I ,

taking

i

f > il

~

i i'

are cofinal. (c)

The functor "identity map"

I --> J

taking-

i ~ i = i

is cofinal. Propositlon (i.ii): and let (a)

J

Let

I

be a U-small filtering category

be the category of maps of For fixed

I

above.

i E I , the sub-category

Ji

of

J

consisting

- 153

of maps from (b) functor

Let

i

to varying X:

J --> (sets)

I --> (sets)

-

i' E I

A.1.7

is filtering.

be a functor.

sending

,l I

and a canonical isomorphism

~ . (~m xl%) z> alm. x . i i Ji J

There is a natural

154-

A.2.1

w

Pro-objects and pro-representable

Let

C

A pro-object

be a category.

(contravariant)

functors.

[15]

in

C

is a

to

C .

functor

X:

I ~

--> C

from some U-small filtering index category

I

We will

often use the notation

x = [x i]i

One thinks of a pro-object of

X

I

as an inverse

system of objects

C , and the point is that the pro-objects

of

C

can be made

in a good way into a category

pro-C

by the rule

(2.1)

Hom(X,Y) = lim(lim Hom(Xi,Yj))

j when

X = [Xi] i 6 I

reader to elucidate

and

i

Y = [YJ]J E J

' where we leave it to the

the maps involved in the directed

Note that the index categories is important to understand much more information

are not assumed equal.

systems. Also,

right away that the pro-object

than the inverse limit

lim X

it

contains

even if the

155 -

latter exists in

A.2.2

C , which we do not assume.

Clearly, a functor

F: C --> C'

induces in the obvious way

a functor

(2.2)

pro-F: pro-C--> pro-C'

C

The objects of

.

themselves are pro-objects if the index C

category is taken to be the one point category, and by (2.1), forms a full subcategory of pro-C .

If

X = IXi ] E

pro-C

and

Y E C , then

(2.3)

Hom(X,Y) = lim Hom(Xi,Y) -@ i

In this way every

X E pro-C

gives rise to a functor

Horn(X,. ): C --> (sets)

It can be shown ([25]) that morphisms between the functors associated to pro-objects elements of

Hom(X,Y)

equivalent via

X,Y

are in 1-1 correspondence with

defined as in (2.1).

(2.4) to a full subcategory of

The functors which are isomorphic to object

X

I--~> J

Horn(X,. )

is

Hom(C,(sets))

.

for some pro-

are called pro-representable functors.

Corollary let

Thus pro-C

(2.5):

Let

be a cofinal functor, with

Then the pro-object

be a pro-object and

X = [Xj]j E J

X~ = [ X ~ ( i ) ] i E i

I

filtering and U-small.

is isomorphic with

X .

- 156-

A.2.3

In fact, it follows from (2.1) that the functors represented by

X

and

X~ --> X

X$

are isomorphic.

Of course, the isomorphism

is given by the obvious element of

We will refer to the index category

X$

as obtained from I

via

X

lim(limw ~ Hom(X$(i),Xj)).

j i by "re-lndexlng" with

$ .

The conditions on a functor to be pro-representable are easily understood.

Recall that if

F: C--> (sets) is a functor,

then the morphisms of functors

Hom(Z,.) --> F

are in 1-1 correspondence with elements of seen.

Suppose we denote by

(2.6)

J

J = [(Z,~) I Z E C

F(Z).

This is easily

the category of pairs

and

~ E F(Z)]

,

where a morphism

(z,r

is a map

Z <-- Z'

is

The functor

~

.

-> (z',r

such that the induced image of

jo _> C

sending

z

~'

in

F(Z)

-

gives rise to a functor

1 5 7

Y: C - >

~(Y) =

A.2.4

-

by

(sets)

llm

Hom(Z,Y)

-@

.

J

The index category

J

is in general not a U-small category,

this limit is a priori in the next u~iverse gives a map

Hom(Z;.) --> F

U+ .

But since

as above, we obtain a map

and it is well known that this morphism is bijective particular that the limit can be taken in Proposition and let above.

J

(2.7):

(i)

Let

There is a ~kunctor

I ~> J

i ~

whe re

~i

This

is cofinal (ii)

J

is the image in (hence

Conversely,

let

(hence in

be a pro-object,

[(Z,~)I~ E Hom(X,Z)}

as

given by

(Xi,~i)

Hom(X,X i) J

of

idxi E Hom(Xi,X i) 9

is filtering).

F: C --> (sets)

be a functor and let

be the category of pairs defined as in (2.6).

pro-representable

~--> F ,

U).

X = [Xi} i E I

be the category of pairs

so

Then

F

is

iff there is a filtering U-small category

and a cofinal functor

I I> j .

pro- representable iff

J

Equivalently

(cf. [15]),

I F

is

is filtering and contains a U-small

cofinal sub-category. we omit this routine verification. Corollary

(2.8):

Suppose

c

is itself a u-small category

158-

A.2.5

which is closed under finite inverse limits, i.e., under finite products and finite fibred products.

Then a functor

F: C-->

(sets) is pro-representable iff it is left exact, i.e., commutes with finite inverse limits.

- 159

-

A.3.1

w

Morphisms of pro-objects.

Let

X = {Xi] i E I ' x = [Xj]j 6 J

be in pro-C .

If

I=J

and if i 6I

fi: Xi -> Yi

is a compatible system of maps, i.e., a morphism of functors f: X --> Y , then pro-objects

f

X --> Y

determines in an obvious way a morphism of which we denote by the same letter

f .

Of

course, a general morphism of pro-objects will not be of this form even if the index categories are equal.

However,

one can

put it into such a form by re-indexing as follows: A morphism of maps

f: X --> Y

{fj: X --> Yj]j 6 J

lim~ Hom(Xi,Yj) i

.

is by (2.1) a compatible collection " and each

fj

is an element of

We will say that a map

: X i --> Yj

represents

f

if the image of

~

in

lim_~Hom(X i,Yj) i This is the same as saying that the diagram in pro-C

iS

fj .

f

commutes.

A morphism

X

>

Xi

-> Yj

r --> r

Y

between maps representing

f

- 160 consists of a map

i --> i'

in

I

A.3.2

and a map

j --> j'

in

J

such that the diagram

I

Xil

> Yj,

Xi

.> Yj

commutes. Proposition ing a map

(3.1):

f: X --> Y

The category

M

of maps

~

represent-

of pro-objects is U-small and filterlng,

and the functors

M

,,,> I

sending

$: X i --> Yj ~

i

M

> J

sending

$: X i

j

and

> Yj ~

are cofinal. Corollary

(3.2):

A map

f: X --> Y

of pro-objects of

C

can be represented, up to isomorphism, by a (U-small) filtering inverse system of maps

[fi: Xi -> Yi]i 6 I ' i.e., by a pro-

object in the category of maps of

C .

The above results can be generalized as follows. Proppsition

(3.3):

("uniform approximation")

Let

A

be

a finite diagram with commutation relations, and suppose that

- 161 A

has no loops, i.e.,

that the beginning and end of a chain of

arrows are always distinct. the type of

Let

D

Remark

(3.4):

following type: .

X,Y

A

[Di]

Let

Let

F: C --> C'

f: F(X) --> Y

follows immediately from

A

be a functor,

C

which represent

tially to show that

i.e., of

D .

I

I

Xi

and

Then we may re-

so that

f

is repre-

fi: (xi) -> Yi"

I

This

of diagrams

D , i.e., D .

Di

each of whose maps

The problem is essen-

is filtering and has a U-small cofinal

and that for every object

that the

such that

(3.3).

represents the corresponding map of

subcategory,

C

X E pro-C

be a morphism.

(3.3) consider the category

in

There is a

is isomorphic to

by a single index category

To prove

of

One can also make related assertions of the

sented by a compatible system of maps

of type

to pro-C.

of diagrams of

{Di}i E I

the diagram in pro-C determined by

index

be a diagram in pro-C

A , i.e., a morphism of

filtering inverse sysiem

Y E pro-C'

A.3.3

X

of

D ,

X X

are coflnal among all maps of

~tXi~i E I r to objects

C . Induction on the number of vertices of

true for

(n-l)

vertices,

an "initial" v e r t e x ing to it.

v

and that

of

A

has

A . n

Suppose it

vertices.

Choose

A , i.e., one having no arrows lead-

This is possible since

A

has no loops.

A

leading out of

be the corresponding diagram in

pro-C .

Let

By induction,

{D~}s E J "

Let

D' D'

v

A'

be the diagram obtained from v .

by removing

Let

and all arrows

can be approximated uniformly,

X = [Xi) i E I

be the object of

D

say by

corresponding

'

162 -

to

v .

Let

K

pair of indices

is a map

be t h e c a t e g o r y o f o b j e c t s

(i,j)

made up out of

A.3.#

and a diagram

Dk

k

c o n s i s t i n g of a

representing

X.,D'. and some maps, where a morphism l D

(i,j) --> (i',j')

morphism of diagrams

D

k--> k'

such that the induced maps give a

Dk. --> D k .

It remains to show that

is filtering,

and that the obvious functors

are cofinal.

We leave this as an exercise for the reader.

Scholie

(3.5):

and

K

K --> I , K --> J

Suppose given a U-small filtering inverse

system of pairs of maps fi >

>Yi

Xi gi

Then by (2.1) the induced maps of pro-objects equal iff for each

i

the maps

equal, i.e., iff for each

i

X --> Yi

f,g: X --> Y

induced by

there is an

i'

are

f,g

are

such that the

composed maps fi X i, --> X i

> Yi gi

are equal.

In particular,

Then the maps is an

f: X --> Y

i --> i'

the zero map.

suppose

C

has a zero object

is the zero map iff for every

such that the composed map If we set

ject is the zero object

i--> i'

there is

Xi = Yi ' fi = id , we find that an ob(which is equivalent with the assertion

such that the structure map Xi, --> X i

is the zero map.

i

Xi' --> Xi --> Yi

that the identity map is the zero map) iff for every is an

0 .

i E I

there

163 -

w

Exactness properties of the pro-category.

Proposition and

C

(4.1):

any category.

limits.

Let

I

C

has finite direct

(inverse)

Then the functor

associating with an

--> pro-C

X: I ~ --> C

commutes with finite direct Proof: I ~ --> C .

be a U-small filtering category

Suppose that

Hom(I~

Let

We will denote by

X = [Xi} i E I

D

the corresponding pro-object

(inverse) limits.

D = [Di] i E I

making up the diagram Let

A.4.1

be a finite diagram of functors X~ =[X~i]i 6 I

the various objects

(we do not bother to label the maps).

be the object

lim D = [lim Di] i E I -@ -@

"

Then in

pro-C , Hom (X, Y) = llm lim Horn(Xi, Yj )

7 =

llm lim lim Hom -@

-@

J

i

@-

(X~i, Yj )

= lim lim lim Hom(X~i,Y ) @-

@-

commutes with finite inverse limits

--~

([3],

j

j

i

commutes with inverse limits

= llm Hom(X~,Y) @(Z

lira

because

lim (-.

-~

I, 2.8)

= lira lim lim Hom(X~i,Yj) @(-. -~ a

because

j

164 -

This shows that

X

A.#.2

has the property required of a direct limit

in the category pro-C .

The proof for an inverse l~m~t goes the

same way. Proposition (4.2):

If

C

is closed under finite direct

(inverse) limits, so is pro-C . Proof:

To show pro-C closed under arbitrary flnite direct

limits, it suffices to show it closed under coproducts and amalgamated coproducts

([3], I, 2)

Xa Y

and

X ~Z Y .

To show

for instance the second, we suppose given a diagram

X

in pro-C.

By (App. 3.3) we may represent it up to isomorphism

by a filtering system

z/xi i6I l ~

.

Yi

Now apply (#,i). Proposltion

(4,5):

If

c

is a U-small category which is

closed under finite inverse limits, then pro-C is closed under arbitrary (U-small) direct limits. Proof:

If

IX~]

is a system of objects of pro-C , the

165

-

A.#.3

-

functor lim (--

Horn(Xa, Y)

is a left exact functor from is left exact.

C

Y6C

to sets, since each

Hom(Xa,Y)

Hence the functor is pro-representable

Proposition

(4.4):

For any

(2.8).

C , pro-C is closed under

(U-small) filtering inverse limits. Proof:

Let

pro-objects of

[xJ]j 6 J

be a filtering inverse system of

C , and say

X j = [X~]

where

i 6 Ij .

be the category whose objects are pairs of indices j E J

and

of a map

i EIj j --> j'

, and where a map and a map

One verifies easily that

K

(J,i)

(j,i) --> (j',i')

X , --> X

Let

representing

K

with

consists

x J' _> x j.

is filtering (and U-small), and we

claim that the pro-object

is an inverse limit of the filtering system is

clear

that

if

Z 6 pro-C

IX j] .

, then an element

In fact, it

l i m Hom(Z,X j )

J determines a unique element of

Lim Hom(Z,X~)

, and conversely.

(j,i) Proposition gory.

from those of

C

Let

A

be an additive

Then pro-A is again additive Proof:

if

(#.5):

(abelian) cate-

(abelian).

The axioms of ([ll]) for pro-A follow immediately A

using (4.1),

(4.2).

Note that

is a U-small category, then pro-A is equivalent by (2.7)

-

with the category

Sex(A, (ab))

Proposition (4.6): monomorphism

Let

-

A.4.4

considered by Gabriel ([ll]). A

be an abelian category.

A

(epimorphism) in pro-A can be represented by an in-

verse system of monomorphisms Proof:

166

(epimorphisms).

In an abelian category, a monomorphism is a kernel

Of a map, and an epimorphism is a cokernel. finite limits, we can apply (4.1). pro-(grps) as an exercise.

Since these are

We leave the assertion for

-

167

-

R 1

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Offsetdruck: Julius Beltz, Weinheim/Bergstr.

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