Noncommutative Groebner bases and filtered-graded transfer

Lecture Notes in Mathematics Editors: J.-M. Morel, Cachan F. Takens, Groningen B. Teissier, Paris 1795 3 Berlin Heide...

Author: Huishi Li

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Lecture Notes in Mathematics Editors: J.-M. Morel, Cachan F. Takens, Groningen B. Teissier, Paris

1795

3 Berlin Heidelberg New York Hong Kong London Milan Paris Tokyo

Huishi Li

Noncommutative Gröbner Bases and Filtered-Graded Transfer

13

Author Huishi LI Department of Mathematics Bilkent University P.O. Box 217 06533 Ankara Turkey e-mail: [email protected] http://www.fen.bilkent.edu.tr/˜huishi

Cataloging-in-Publication Data applied for Bibliograhpic information published by Die Deutsche Bibliothek Die Deutsche Bibliothek lists this publication in the Deutsche Nationalbibliografie; detailed bibliographic data is available in the Internet at http://dnb.ddb.de

Mathematics Subject Classification (2000): 16Z05, 68W30, 16W70, 16S99 ISSN 0075-8434 ISBN 3-540-44196-4 Springer-Verlag Berlin Heidelberg New York This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specif ically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microf ilm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer-Verlag. Violations are liable for prosecution under the German Copyright Law. Springer-Verlag Berlin Heidelberg New York a member of BertelsmannSpringer Science + Business Media GmbH http://www.springer.de c Springer-Verlag Berlin Heidelberg 2002 Printed in Germany The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specif ic statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Typesetting: Camera-ready TEX output by the author SPIN: 10891005

41/3142/ du - 543210 - Printed on acid-free paper

Acknowledgement

On April 5th, 1995, on the way from Bielefeld to Xi’an, I stopped at Beijing and met my old friend Dr. Luo Yunlun (Lao Luo). Drinking Chinese green tea, Lao Luo introduced Computational Algebra to me. It was the ﬁrst time I had heard of the notion of a Gr¨ obner basis. Soon after our meeting, Lao Luo sent me some basic material on commutative computational algebra and a tutorial book introducing Macaulay that enabled me to start my computational algebra seminar combined with my elementary algebraic geometry course given at the Shaanxi Normal University in the spring of 1996. Without this unforgettable story, I would not have had the idea of working on the algebraic-algorithmic aspect. Thank you, Lao Luo. I wish to express my thanks to the referees for their valuable remarks on improving the notes. I am grateful to the LNM editors for their helpful suggestions while I was preparing the manuscript.

Huishi Li

Contents

Introduction CHAPTER I Basic Structural Tricks and Examples 1. 2. 3. 4. 5.

Algebras and their Deﬁning Relations Skew Polynomial Rings ZZ-ﬁltrations and their Associated Graded Structures Homogenization and Dehomogenization of ZZ-graded Rings Some Algebras: Classical and Modern

CHAPTER II Gr¨ obner Bases in Associative Algebras 1. 2. 3. 4. 5. 6. 7. 8.

(Left) Monomial Orderings and (Left) Admissible Systems (Left) Quasi-zero Elements and a (Left) Division Algorithm (Left) Gr¨ obner Bases in a (Left) Admissible System (Left) Gr¨ obner Bases in Various Contexts (Left) S-elements and Buchberger Theorem (Left) Dickson Systems and (Left) G-Noetherian Algebras Solvable Polynomial Algebras No (Left) Monomial Ordering Existing on ∆(k[x1 , ..., xn ]) with chark > 0

CHAPTER III Gr¨ obner Bases and Basic Algebraic-Algorithmic Structures 1. 2. 3. 4.

PBW Bases of Finitely Generated Algebras Quadric Solvable Polynomial Algebras Associated Homogeneous Deﬁning Relations of Algebras A Remark on Recognizable Properties of Algebras via Gr¨ obner Bases

1 5 5 9 13 25 28 33 34 38 42 45 48 54 58 61

67 68 73 81 89

viii

Contents

CHAPTER IV Filtered-Graded Transfer of Gr¨ obner Bases 1. 2. 3. 4.

Filtered-Graded Transfer Filtered-Graded Transfer Filtered-Graded Transfer Filtered-Graded Transfer Polynomial Algebras

of (Left) Admissible Systems of (Left) Gr¨ obner Bases of (Left) Dickson Systems Applied to Quadric Solvable

CHAPTER V GK-dimension of Modules over Quadric Solvable Polynomial Algebras and Elimination of Variables 1. 2. 3. 4. 5. 6. 7. 8.

Gr¨ obner Bases in Homogeneous Solvable Polynomial Algebras The Hilbert Function of A/L The Hilbert Polynomial of A/L GK-dimension Computation and Elimination of Variables (Homogeneous Case ) GK-dimension Computation and Elimination of Variables (Linear Case) The gr -ﬁltration on a Quadric Solvable Polynomial Algebra GK-dimension Computation and Elimination of Variables (General Quadric Case) Finite Dimensional Cyclic Modules

CHAPTER VI Multiplicity Computation of Modules over Quadric Solvable Polynomial Algebras 1. 2. 3. 4.

The Multiplicity e(M ) of a Module M Computation of e(M ) Computation of GK.dim(M ⊗k N ) and e(M ⊗k N ) An Application to An (q1 , ..., qn )

CHAPTER VII (∂-)Holonomic Modules and Functions over Quadric Solvable Polynomial Algebras 1. 2. 3. 4. 5.

Some Operator Algebras Holonomic Functions Automatic Proving of Holonomic Function Identities Extension/Contraction of the ∂-ﬁniteness The ∂-holonomicity

91 92 97 100 103

107 108 110 112 115 119 124 126 130

133 134 135 144 148

153 154 156 162 164 168

Contents

CHAPTER VIII Regularity and K0 -group of Quadric Solvable Polynomial Algebras 1. 2. 3. 4.

Tame Case: A is Auslander Regular with K0 (A) ∼ = ZZ The gr -ﬁltration on Modules General Case: gl.dimA ≤ n General Case: K0 (A) ∼ = ZZ

ix

175 176 177 181 186

References

187

Index

195

Introduction

It is well-known that because of the successful implementation of Buchberger’s algorithm on computer (1965, 1985), the commutative Gr¨ obner basis theory has been very powerful in both pure and applied commutative mathematics. It is equally well-known that Buchberger’s algorithm has been generalized to the noncommutative area via noncommutative Gr¨obner bases in various contexts, for instance, • the Gr¨ obner bases for one-sided ideals in algebras of linear partial diﬀerential operators with polynomial coeﬃcients (or Weyl algebras) introduced in ([Gal] 1985), the Gr¨ obner bases for one-sided ideals in the enveloping algebras of ﬁnite dimensional Lie algebras introduced in ([AL] 1988), and more generally, the Gr¨ obner basis theory developed in ([K-RW] 1990) for a large class of noncommutative polynomial-like algebras, i.e., the class of solvable polynomial algebras, • the Gr¨ obner basis theory for two-sided ideals in a noncommutative free algebra developed in ([Berg] 1978) and in ([Mor1-2] 1986, 1994), • and the Gr¨ obner basis theory introduced in ([Gr] 1993) for two-sided ideals in an algebra which possibly has no identity but possibly has divisors of zero. From now on in this monograph, all strings of references will be listed alphabetically. Supported by several noncommutative computer algebra systems (see an incomplete list of websites at the end of this introduction for the references), noncommutative Gr¨ obner bases have been eﬀectively used in many contexts of pure and applied algebra, such as the algorithmic study of PDE’s, the automatic proof of functional identities, the representation of algebras etc. (see [ACG1–2], [CS], [Gr], [LC1–2], [Oak1–2], [SST], [Tak1–3]).

H. Li: LNM 1795, pp. 1–4, 2002. c Springer-Verlag Berlin Heidelberg 2002

2

Introduction

The development and applications of noncommutative Gr¨ obner bases shows that while noncommutativity forces algorithms to become more sophisticated and varied, it also forces a stronger interaction between algorithmic and structural methods. In particular, since Gr¨ obner bases are not only algorithmic but also structural, eﬀective use of noncommutative Gr¨obner bases often relies on eﬀective algorithmic analysis for the algebraic structures concerned. At this point, deep structure theory is essentially needed. Typical examples are those concerning diﬀerential operators using Gr¨ obner bases in the algebra of linear partial diﬀerential operators with polynomial coeﬃcients (or the Weyl algebra, see [ACG1–2], [Oak1–2], [OT], [SST], [Tak1–3]), in which the D-module structure theory (in the sense of Bernstein and Kashiwara) has been the soul. But this does not mean that algorithmic methods are conﬁned to a passive position with respect to the algebraic structures; indeed, one may observe that Gr¨ obner basis theory proposes and realizes algorithmically a stronger concept of noetherianity in both commutative and noncommutative associative algebras (see [BW], [CLO’], [KRW]), that certain structural properties of associative algebras are recognized via Gr¨ obner bases (see [G-I1–3], [G-IL], [Ufn1–2]), and that a ∂-holonomic module theory stems from an algorithmic approach to the automatic proof of functional identities (see [CS], [Zei1–2]). Therefore, to develop and use a noncommutative Gr¨ obner basis theory is to open an eﬀective structural-algorithmic channel in noncommutative computational algebra, as the extension/contraction problem proposed to us in [CS]. As its title shows, this self-contained monograph is not written with the aim of developing algorithms. Instead it tries to explore the interaction between the structure theory of noncommutative associative algebras and the algorithmic aspect of Gr¨ obner basis theory through several applications of ﬁltered-graded transfer of Gr¨ obner bases. Thus, if we say in the text that a certain problem (or quantity) can be solved (or computed) algorithmically, it means that, after using a certain kind of Gr¨ obner basis, the problem (or quantity) considered can be solved (or computed in a ﬁnite number of steps). And it also means obviously that the monograph will not trace a history of noncommutative Gr¨obner basis theory, or survey others work concerning noncommutative Gr¨obner bases (that is far beyond the author’s goal and ability). So, the author apologizes for the possible omission of some excellent references dealing with noncommutative Gr¨ obner bases. The content of this monograph consists of two parts. Part I (CH.I–CH.IV): This part forms the working basis of this book, which includes some basic structural tricks in dealing with associative k-algebras, some well-known classical and modern examples of k-algebras, an introduction to Gr¨ obner basis in associative algebra, some basic algebraic-algorithmic structures obtained in terms of Gr¨ obner bases, and an introduction to the ﬁltered-graded transfer of Gr¨ obner bases in associative algebras. The Gr¨ obner basis theory

Introduction

3

presented is mostly based on [Gr], [K-RW] and [Mor1–2]. Part II (CH.V–CH.VIII): This part focuses on solving some structuralcomputational problems in quadric solvable polynomial algebras by transfering, between ﬁltered and the associated graded structures, the relevant algorithmic data (determined in terms of Gr¨ obner bases) and structural properties, where solvable polynomial algebra is in the sense of Kandri-Rody and Weispfenning [K-RW]. More precisely, after the preparatory CH.I, we introduce in CH.II a division algorithm and the basic theory of (left and two-sided ) Gr¨obner basis for a (noncommutative) associative algebra that may have divisors of zero. This is realized by distinguishing the quasi-zero elements and introducing the (left) division leading monomials . Such a Gr¨ obner basis theory is compatible with all well-known Gr¨ obner basis theories in the literature. With such a theory, the (left) G-Noetherian structural property of an algebra (which is more stronger than the classical (left) Noetherian property) is algorithmically characterized. As a result, the notion of (left) Dickson system is quite naturally introduced in a wide context. After presenting the solvable polynomial algebra as a perfect example, in the last section of CH.II we prove the impossibility of having a Gr¨ obner basis theory (in the sense of §3) for the algebra of linear partial diﬀerential operators with polynomial coeﬃcients over a ﬁeld of characteristic p > 0. We point out, however, the possibility of applying some well-deﬁned Gr¨ obner bases to this algebra. Starting from CH.III, we highlight the interaction between the noncommutative algebraic structure theory and the algorithmic method of using Gr¨ obner bases. To this end, the applications of very noncommutative Gr¨ obner bases (in the sense of [Mor1–2]) to the noncommutative structure theory given in CH.III motivate a general ﬁltered-graded transfer of Gr¨obner bases in CH.IV (from which one may also see how eﬀective the algorithmic methods are in transfering the ﬁniteness property between algebraic structures at diﬀerent levels). In CH.V–CH.VI, a double ﬁltered-graded transfer of Gr¨ obner bases is used to realize the computation of Gelfand-Kirillov dimension and multiplicity of ﬁnitely generated modules over a quadric solvable polynomial algebra. At this stage, after the theoretic proof is established through a structural channel, the computational aspect becomes surprisingly simple. More clearly, let L be a left ideal of a quadric solvable polynomial algebra A = k[a1 , ..., an ] and G = {g1 , ..., gs } a left Gr¨ obner basis for L. If the leading monomials of G are given by the standard monomials αin i1 αi2 LM(gi ) = aα 1 a2 · · · an , i = 1, ..., s, then all computations concerned can be realized by manipulating only the ﬁnite set of n-tuples of nonnegative integers α(G) = Ri = (αi1 , αi2 , ..., αin ) i = 1, ..., s . As a consequence, an elimination (of variables) lemma for quadric solvable polynomial algebras is obtained, and this lemma is used in CH.VII to formulate a

4

Introduction

∂-holonomicity for both functions and modules with respect to a speciﬁc class of quadric solvable polynomial algebras. In addition to its interesting structural property in module theory, this ∂-holonomicity provides a way of dealing with the extension/contraction problem in the automatic proof of functional identities by using Gr¨ obner bases in a more general context (comparing with [CS] and [Zei1]). In the ﬁnal CH.VIII, a double ﬁltered-graded transfer of data is used again to show that quadric solvable polynomial algebras are regular in the classical sense, i.e., they have ﬁnite global dimension and K0 -group ZZ. As shown in CH.II–CH.VIII, all topics discussed in this monograph are valid for many popular algebras such as the algebra of linear partial diﬀerential operators (or the Weyl algebra) and its many deformations, enveloping algebras of ﬁnite dimensional Lie algebras and many of their deformations, many quantum algebras, and some other popular operator algebras, as well as their associated (quadratic) graded structures (with respect to the standard ﬁltration). Hence, the author hopes that the material covered in this book may be helpful for the researchers and graduate students who are interested in developing and using Gr¨ obner bases in various contexts of mathematics. Finally, although the aim of this monograph is not to develop algorithms related to the topics included, all examples of application are made on a strong algorithmic basis of using noncommutative Gr¨ obner bases, i.e., all what we need in the computation involved is to have a Gr¨ obner basis for the problem considered. Thanks to the algorithms developed in ([FG], [Mor2]) and the computer algebra systems developed at http://www.fmi.uni-passau.de/algebra/projects/ http://felix.hgb-leipzig.de/ http://www.math.vt.edu/people/green/index.html http://www.cis.upenn.edu/ wilf/progs.html http://algo.inria.fr http://www.math.kobe-u.ac.jp/KAN/, those readers interested in verifying the methods introduced in this book may ﬁnd available references.

CHAPTER I Basic Structural Tricks and Examples

We assume that the reader is familiar with basic ring theory and module theory. Throughout this book, unless otherwise stated, k will denote a (commutative) ﬁeld of arbitrary characteristic. All rings considered are associative rings with identity 1. If A and B are rings, then A ⊂ B means that A is a subring of B with the same 1. Furthermore, if ψ: A → B is a ring homomorphism, then we insist that ψ(1) = 1, and write Kerψ and Imψ for the kernel and image of ψ, respectively. We also assume that all modules considered are unitary left modules. Let A be a ring and F = {fi }i∈J a nonempty subset of A. To emphasize the generators of the two-sided ideal generated by F in a formula, we sometimes use fi i∈J to indicate this ideal without making confusion, and similarly write fi ] for the left ideal of A generated by F . Conventionally, we use IN , ZZ, Q , IR, and C for the sets of non-negative integers, integers, rational numbers, real numbers, and complex numbers, respectively.

1. Algebras and their Deﬁning Relations Since both commutative and noncommutative Gr¨obner bases are constructed on the existence of a division algorithm, while a division algorithm essentially manipulates “ordered bases” of algebras which are determined by their deﬁning relations, we start this preparatory chapter by recalling some basic elements

H. Li: LNM 1795, pp. 5–32, 2002. c Springer-Verlag Berlin Heidelberg 2002

6

I. Basic Structural Tricks

concerning algebras. Let k be a ﬁeld. A k-algebra is a ring A together with a nonzero ring homomorphism ηA : k → A satisfying (A1) the image of ηA , denoted ImηA , is contained in the center of A, and (A2) the map k × A −→ A (λ, a) → ηA (λ)a equips A with a vector space structure over k and the multiplicative map µA : A × A → A is bilinear: µA (a + b, c) = µA (a, c) + µA (b, c), a, b, c ∈ A, µA (a, b + c) = µA (a, b) + µA (a, c), a, b, c ∈ A, µA (λa, b) = µA (a, λb) = λµ(a, b), λ ∈ k, a, b ∈ A. If A is a commutative ring, we say that A is a commutative k-algebra. It is clear that any A-module M also has a k-vector space structure. If M is a k-vector space, then the set of all linear operators on M , denoted Endk M , forms a k-algebra with respect to the operator composite and the scalar multiplication of operators, which is known as the k-algebra of linear operators of M . Let A and B be k-algebras. A k-algebra homomorphism from A to B is a ring homomorphism α: A → B such that α ◦ ηA = ηB , i.e., we have the following commutative diagram: ηA k >A ηB

α

∨ B (This also means that α is a linear transformation form the k-vector space A to the k-vector space B.) If : A → B is an injective algebra homomorphism, we say that A is a subalgebra of B. (Compare this with the convention we made for subring in the beginning of this chapter.) Remark Ignoring the conventions we made for ring homomorphisms and for subrings, from the deﬁnitions given above we may also derive the following facts: (i) ηA is nonzero. Hence ηA is injective, k ∼ = ImηA and ηA (1) = 1. (ii) k may be viewed as a subalgebra of A, and we may write λa for ηA (λ)a (the scalar multiplication of the k-vector space A). (iii) α preserves the units, i.e., α(1) = 1.

1. Algebras and Deﬁning Relations

7

Let A be a k-algebra and I a two-sided ideal of A. Then the quotient ring (or factor ring) A/I of A has the k-algebra structure induced by A, called the quotient algebra (or factor algebra) of A, and the natural map π: A → A/I is a k-algebra homomorphism. Let X = {Xi }i∈Λ be a set and S = X the (multiplicative) free semigroup on X. Then S consists of all words in the alphabet X plus the empty word ∅, i.e., S = Xi1 Xi2 · · · Xip | Xij ∈ X, p ≥ 1 ∪ {∅}, and the multiplication on S is deﬁned as the concatenation of words, that is, if u = Xi1 · · · Xip and w = Xip+1 · · · Xin then (∗)

∅u = u∅ = u, and . uw = Xi1 · · · Xip Xip+1 · · · Xin

Consider the k-vector space kX with basis the set S. Then formula (∗) equips kX with an algebra structure, called the free k-algebra on the set X. If X = {X1 , ..., Xn } is ﬁnite, we also denote kX by kX1 , ..., Xn . Free k-algebras have the following universal property. 1.1. Theorem Let X = {Xi }i∈Λ . Given a k-algebra A and a set-theoretic map ϕ: X → A, there exists a unique algebra homomorphism ϕ: kX → A such that ϕ(Xi ) = ϕ(Xi ) for all Xi ∈ X, i.e., we have the following commutative diagram: > kX X ϕ

ϕ

∨ A where is the inclusion map. Proof It is suﬃcient to deﬁne ϕ on any word of S. For the empty word we set ϕ(∅) = 1. Otherwise, if u = Xi1 · · · Xip ∈ S, we deﬁne ϕ(u) = ϕ(Xi1 ) · · · ϕ(Xip ). The remained proof follows easily.

2

Let A be a k-algebra and T = {ai }i∈Λ ⊂ A a nonempty subset of A. Write k[T ] for the subalgebra of A generated by T and k, i.e., k[T ] = λi1 i2 ···is ai1 ai2 · · · ais λi1 i2 ···is ∈ k, aij ∈ T, s ∈ IN .

8

I. Basic Structural Tricks

If k[T ] = A, we say that A is generated by T and that T is a generating set of A (or a set of generators of A). If T = {a1 , ..., an } is ﬁnite and A = k[T ], we say that A is a ﬁnitely generated k-algebra and write A = k[a1 , a2 , ..., an ]. Since every k-algebra A has a generating set T (for instance T = A), say T = {ai }i∈Λ , if we take the set of symbols X = {Xi }i∈Λ and consider the free algebra kX on X and the set-theoretic map ϕ: X → T with ϕ(Xi ) = ai , it follows from Theorem 1.1 that there is a unique k-algebra homomorphism ϕ: kX → A = k[T ], which is clearly onto, such that (∗∗)

A∼ = kX/I

where I = Kerϕ. Thus, the following holds. 1.2. Proposition Any k-algebra A = k[T ] with T = {ai }i∈Λ is the quotient of a free k-algebra kX on a set X. 2 1.3. Deﬁnition In the above (∗∗), if the two-sided ideal I of kX is generated by F = fj = λij X1j X2j · · · Xsj j ∈ J , we say that F is a set of deﬁning relations of the k-algebra A. In this case we also say that the k-algebra A is generated by T = {ai }i∈Λ subject to the relations: fj (a1j a2j · · · asj ) = λij a1j a2j · · · asj = 0, j ∈ J. Example (i) As the ﬁrst example, let X = {Xi }i∈Λ and kX be as before. Let I be the two-sided ideal of kX generated by all elements of the form X j Xi − Xi Xj ,

j, i ∈ Λ, j = i.

Then the algebra A = kX/I = k[xi ]i∈Λ is a commutative algebra generated by {xi }i∈Λ , where each xi denotes the image of Xi in A, subject to the relations: xj xi = xi xj ,

j, i ∈ Λ, j = i,

and is called the symmetric algebra on X. Using Theorem 1.1 and the fundamental theorem of homomorphism, one may easily show that A has the following universal property: • Given a commutative algebra B and a set-theoretic map ϕ: {xi }i∈Λ → B, then there exists a unique algebra homomorphism ϕ: A → B such that ϕ(xi ) = ϕ(xi ) for all xi ∈ {xi }i∈Λ , i.e., we have the following commutative

2. Skew Polynomial Rings

9

diagram: {xi }i∈Λ

>A

ϕ

ϕ ∨ B Hence, every commutative algebra is the quotient of a symmetric algebra. If X = {X1 , ..., Xn } is ﬁnite, then A = k[x1 , ..., xn ] is nothing but the commutative polynomial k-algebra in n variables. And every ﬁnitely generated commutative algebra is the quotient of a polynomial algebra k[t1 , t2 , ..., tn ] in variables t1 , ..., tn for some n.

2. Skew Polynomial Rings An important class of noncommutative algebras, for which a Gr¨ obner basis theory may exist, is the class of skew polynomial algebras. We review the basic structure of a skew polynomial ring (algebra) by following [MR]. Skew polynomial rings are polynomial rings in noncommutative setting. More n precisely, we want to have polynomials over a ring R in a variable x, i=1 ri xi , which is not assumed to commute with the elements of R, but is desired that (S1) each polynomial should be expressed uniquely in the form ri xi for some ri ∈ R, and (S2) xr ∈ Rx + R, i.e., xr = σ(r)x + δ(r) for some σ(r), δ(r) ∈ R. n The degree of a polynomial f (x) = i=1 ri xi is deﬁned to be n provided rn = 0. If rn = 0 we call rn the leading coeﬃcient of f (x) and write degf (x) = n. If all the coeﬃcients ri of f (x) are zero, we say that f (x) is the zero polynomial and write f (x) = 0. Conventionally the zero polynomial has degree −∞ and leading coeﬃcient 0. Thus, if we apply the condition (S1) to xr, r ∈ R, then the condition (S2) guarantees that the degrees behave appropriately:  degf (x) ≥ 0,    degf (x) = −∞ if and only if f (x) = 0, (∗) deg(f (x) − g(x)) ≤ max{degf (x), degg(x)},    deg(f (x)g(x)) ≤ degf (x) + degg(x). And under these conditions, it is clear that σ, δ are ZZ-operators on the additive group of R, ie., σ, δ ∈ EndZZ R, the ring of automorphisms of the additive group

10

I. Basic Structural Tricks

of R. Moreover, for r, s ∈ R, x(rs) = σ(rs)x + δ(rs) (xr)s = σ(r)σ(s)x + σ(r)δ(s) + δ(r)s. Thus σ is a ring endomorphism of R, and δ satisﬁes (∗∗)

δ(rs) = σ(r)δ(s) + δ(r)s,

r, s ∈ R.

Note in particular that σ(1) = 1 and δ(1) = 0. 2.1. Deﬁnition Let σ be a ring endomorphism of R and δ ∈ EndZZ R. δ is said to be a σ-derivation if it satisﬁes the above (∗∗). Conversely, let R be a ring, σ a ring endomorphism of R and δ a σ-derivation. We now construct a noncommutative polynomial ring over R which has exactly the properties (S1)–(S2) listed above. Let E = EndZZ RIN be the ring of ZZ-operators on RIN , where RIN = i∈IN Ri is the direct product of the additive groups Ri = R. Then R → E, acting by left multiplication. Deﬁne x : RIN (ri )

−→

RIN

→

(σ(ri−1 ) + δ(ri )) ,

where (ri ) = (r0 , r1 , ...) ∈ RIN and r−1 = 0. Let R[x; σ, δ] denote the subring of E generated by R and x. Then, viewing r ∈ R as ZZ-operator on RIN , we have xr = σ(r)x + δ(r), r ∈ R, and consequently f (x) = ri xi for any f (x) ∈ R[x; σ, δ]. Since ri xi (1, 0, 0, ...) = (ri ), it follows that the expression for each f (x) ∈ R[x; σ, δ] is unique. Thus we have constructed the noncommutative polynomial ring with the desired properties. 2.2. Deﬁnition The ring R[x; σ, δ] is called the skew polynomial ring over R. Example (i) Skew polynomial algebras on k[x]. Let k[x] be the polynomial algebra over a ﬁeld k in one variable x, and let σ be a k-algebra endomorphism of k[x]. Suppose σ(x) = g(x), and that δ is a σ-derivation on k[x] with δ(x) = f (x). If we construct the skew polynomial ring k[x][y; σ, δ], then yx = σ(x)y + δ(x) = g(x)y + f (x).

2. Skew Polynomial Rings

11

It follows that δ(xi )

= g(x)i−1 f (x) + g(x)i−2 f (x)x + g(x)i−3 f (x)x2 + · · · + g(x)f (x)xi−2 + f (x)xi−1 .

Conversely, given any f (x) ∈ k[x] we can deﬁne a σ-derivation as above and such that δ(x) = f (x). Now the following is clear: ∂ (a) If σ(x) = g(x) = x is the identity endomorphism of k[x], then δ = f (x) ∂x . kX,Y ∼ (b) k[x][y; σ, δ] = Y X−g(X)Y −f (X) , where kX, Y is the free algebra generated by X and Y , i.e., k[x][y; σ, δ] is the k-algebra generated by two elements x and y subject to the relation: yx = g(x)y + f (x). Remark (i) In the literature R[x; σ, δ] is also called an Ore extension of R because of the remarkable historical literature [Ore]. (ii) R[x; σ, δ] can also be described as being the ring T generated freely over R by i an element X, as follows. Consider the free R-module T = ⊕∞ i=0 RX and, then i j deﬁne the multiplication on T by putting Xr = σ(r)X + δ(r), X X = X i+j . Thus there is an obvious onto ring homomorphism T → R[x; σ, δ]. Since the xi are R-independent in R[x; σ, δ], they are also R-independent in T . Hence T ∼ = R[x; σ, δ]. From the deﬁnition it is clear that if σ = 1 and δ = 0 then R[x; σ, δ] is nothing but the usual commutative polynomial ring over R in one variable x. Moreover, we have the following easily veriﬁed facts: a. There is an obvious embedding of R into R[x; σ, δ] with r → rx0 . b. R[x; σ, δ]/xR[x; σ, δ] ∼ = R as left R-modules. If δ = 0, then xR[x; σ, δ] becomes a two-sided ideal of R[x; σ] and the foregoing isomorphism becomes a ring isomorphism. c. R[x; σ, δ]/(x − 1)R[x; σ, δ] ∼ = R as left R-modules. Next, we list some basic properties of R[x; σ, δ], and we refer to ([MR] CH.I §§2, 4, [Kass] CH.I 1.7) for the detailed proofs. 2.3. Theorem (universal property) If ψ: R → S is a ring homomorphism and y ∈ S has the property that yψ(r) = ψ(σ(r))y + ψ(δ(r)), then there exists a unique ring homomorphism

for all r ∈ R, χ: R[x; σ, δ] → S

such

that

12

I. Basic Structural Tricks

χ(x) = y and the following diagram commutes: R

⊂

ψ

> R[x; σ, δ]

χ

∨ S 2 n As before, we say that the polynomial f (x) = i=1 ri xi has degree n provided rn = 0, denoted degf (x) = n; and then rn is called the leading coeﬃcient of f (x). If f (x) = 0 we put degf (x) = −∞ and we say that f (x) has leading coeﬃcient 0. Thus the degree function d :

R[x; σ, δ] f (x)

−→ N ∪ {−∞} → degf (x)

is well deﬁned on R[x; σ, δ] in the sense of the foregoing (∗). 2.4. Proposition If R = k is a ﬁeld, then k[x; σ, δ] satisﬁes the (left) Euclidean division algorithm relative to the degree function d: • for f, g ∈ k[x; σ, δ] with g = 0, there exists q, r ∈ k[x; σ, δ] such that f = qg+r, d(r) < d(g). It follows that in this case k[x; σ, δ] is a principal left ideal ring. 2 2.5. Proposition Let A = R[x; σ, δ]. (i) If σ is injective and R is a domain (i.e., R does not have nontrivial divisors of zero), then A is a domain. (ii) If σ is injective and R is a division ring (i.e., R is a noncommutative ﬁeld), then A is a principal left ideal domain. (iii) If σ is a ring automorphism of R, and R is a prime ring, then A is a prime ring. (iv) If σ is a ring automorphism of R and R is a left (right) Noetherian ring, then A is a left (right) Noetherian ring. 2 Note that, for r ∈ R, xn r = αn (r)xn + αn−1 (r)xn−1 + · · · + α0 (r), where each αi ∈ EndZZ R. So if αn (r) = 0 then xn r has degree n and leading coeﬃcient αn (r). The general formula for αi in terms of σ, δ and n are rather

3. ZZ-ﬁltrations and Gradations

13

complicated, but for R[x; σ] and R[x; δ] they are simple. For R[x; σ], αn = σ n and αi = 0 for i = n; and for R[x; δ],

n δ n−i . αi = i

n n n (n) r(i) s(n−i) .) (Compare Leibnitz’s formula for δ (rs): (rs) = i=0 i In the case where σ is injective, it is possible to give a general formula for n m i i the product in R[x; σ, δ]. Consider P = i=0 ri x and Q = i=0 si x . Set n+m i P Q = i=0 ui x . Let Sn,k be the element in EndZZ R deﬁned as the sum of all

n possible compositions of k copies of δ and of n − k copies of σ. k 2.6. Proposition With notation as above, if σ is injective, then for all i with 0 ≤ i ≤ m + n we have p i ui = rp Sp,k (si−p+k ) p=0

k=0

and for all r ∈ R and n ∈ IN we have in R[x; σ, δ] xn r =

n

Sn,k (r)xn−k .

k=0

2

3. ZZ-ﬁltrations and their Associated Graded Structures This section and the next are devoted to the notions and basic tricks concerning ZZ-ﬁltrations and their associated graded structures that will be used to develop the later chapters. We refer to [NVO] and [LVO4] for a general theory on ZZﬁltered and ZZ-graded rings. The gr -ﬁltration on a quadric solvable polynomial algebra is introduced and discussed in CH.V and CH.VIII. First look at graded rings. 3.1. Deﬁnition A ring A is said to be ZZ-graded if A = ⊕p∈ZZ Ap , where the Ap are subgroups of the additive group of A, such that Ap Aq ⊂ Ap+q

for all p, q ∈ ZZ.

14

I. Basic Structural Tricks

For p ∈ ZZ, Ap is called the pth homogeneous part (or the part of degree p) of A. An element ap ∈ Ap is called a homogeneous element of degree p. If Ap = 0 for all p < 0, i.e., A = ⊕p∈IN Ap , then A is called a positively graded ring. If A is a k-algebra over a ﬁeld k, we require that each Ap is a subspace of the k-vector space A. Note that if A = ⊕ZZ Ap is a ZZ-graded ring (k-algebra), then it follows from the deﬁnition that A0 is automatically a subring (subalgebra) of A, and 1A ∈ A0 is an element of degree 0. 3.2. Deﬁnition Let A be a ZZ-graded ring. An A-module M is said to be a graded A-module if M = ⊕p∈ZZ Mp , where the Mp are subgroups of the additive group of M , such that Ap Mq ⊂ Mp+q

for all p, q ∈ ZZ.

For p ∈ ZZ, Mp is called the pth homogeneous part (or the part of degree p) of M . An element mp ∈ Mp is called a homogeneous element of degree p. If A is a k-algebra over a ﬁeld k, we usually require that each Mp is a subspace of the k-vector space M . If A is a ZZ-graded ring and M is a graded A-module, then it follows from Deﬁnition 3.1–3.2 that every element m ∈ M has a unique expression as a sum of homogeneous elements: m = mp + mp−1 + · · · + m0 ,

mi ∈ Mi .

3.3. Proposition Let M be a graded A-module and N a submodule of M . Then the following are equivalent. (i) N = ⊕p∈ZZ (Mp ∩ N ), (ii) If u ∈ N , then all homogeneous components of u are contained in N . (iii) N is generated by homogeneous elements. (iv) The quotient module M/N = p∈Z Z (Mp + N )/N is graded as M/N = ⊕p∈ZZ (M/N )p with (M/N )p = (Mp + N )/N . 2 3.4. Deﬁnition Let A be a graded ring and M a graded A-module. A submodule N of M satisfying one of the equivalent conditions of Proposition 3.3 is called a graded submodule of M . If submodules are replaced by left (two-sided) ideals of A in Proposition 3.3, we arrive at the deﬁnition of a graded left ideal (graded ideal).

3. ZZ-ﬁltrations and Gradations

15

3.5. Deﬁnition (i) Let A = ⊕p∈ZZ Ap and B = ⊕p∈ZZ Bp be graded rings (graded algebras). A ring homomorphism (algebra homomorphism) ϕ: A → B is called a graded ring homomorphism (graded algebra homomorphism) if ϕ(Ap ) ⊂ Bp for all p ∈ ZZ. (ii) Let M = ⊕p∈ZZ Mp and N = ⊕p∈ZZ Np be graded modules over a graded ring (graded algebra) A = ⊕p∈ZZ Ap . An A-homomorphism ϕ: M → N is called a graded A-homomorphism if ϕ(Mp ) ⊂ Np for all p ∈ ZZ. As a consequence of Proposition 3.3 and the above deﬁnitions, the following holds. 3.6. Corollary (i) If ϕ: ⊕p∈ZZ Ap = A → B = ⊕p∈ZZ Bp is a graded ring homomorphism, then Imϕ is a graded subring of B in the sense that Imϕ = ⊕p∈ZZ (Bp ∩ Imϕ), and Kerϕ is a graded ideal of A. (ii) If ψ: ⊕p∈ZZ Mp = M → N = ⊕p∈ZZ Np is a graded A-homomorphism, then Imψ is a graded submodule of N and Kerψ is a graded submodule of M . 2 Example (i) Let kX be the free k-algebra on X = {Xi }i∈Λ , and let S be the free semigroup generated by X (see §2). If w = Xi1 Xi2 · · · Xip is a word of S, then the length p of w is called the degree of w and is denoted by d(w). We write d(1) = 0. For each p ∈ IN , consider the k-subspace of kX:     kXp = cw w cw ∈ k, w ∈ S .   d(w)=p

Then kX = ⊕p∈IN kXp is a positively graded k-algebra with kX0 = k. If I is an ideal of kX generated by homogeneous elements, then by Proposition 3.3, the k-algebra A = kX/I is a positively graded algebra, i.e., A = ⊕p∈IN Ap with Ap = (kXp + I)/I. The gradation given on A in such a way is called the natural gradation on A. For instance, the commutative polynomial algebra A = k[x1 , ..., xn ] in n variables has the natural gradation αn 1 Ap = λ λα xα · · · x ∈ k, α + · · · + α = p , p ∈ IN , α 1 n n 1 i.e., each Ap consists of all homogeneous polynomial of degree p. (ii) Let A = ⊕p∈ZZ Ap be a graded ring. Consider the polynomial ring A[t] over A in one commuting variable t. Then A[t] may be graded by the mixed gradation: A[t] = ⊕p∈ZZ A[t]p , where     A[t]p = ai tj ai ∈ Ai , j ∈ IN , p ∈ ZZ.   i+j=p

16

I. Basic Structural Tricks

Now let us turn to ﬁltered rings. 3.7. Deﬁnition A ring A is called a ZZ-ﬁltered ring with a ﬁltration F A if there is a sequence F A = {Fp A}p∈ZZ of subgroups of the additive group of A such that (F1) Fp A = A, p∈Z Z

(F2) 1 ∈ F0 A, (F3) Fp−1 A ⊂ Fp A for all p ∈ ZZ, (F4) (Fp A)(Fq A) ⊂ Fp+q A for all p, q ∈ ZZ, We say that F A is separated if it also satisﬁes: (F5) Fp A = {0}. p∈Z Z

If F−1 A = 0 then A is called a positively ﬁltered ring and F A is called a positive ﬁltration on A. Obviously positive ﬁltrations are separated. If A is a k-algebra over a ﬁeld k, we usually require that each Fp A is a subspace of the k-vector space A. Let A be a ZZ-ﬁltered ring with ﬁltration F A. The associated graded ring of A with respect to F A, denoted G(A), is deﬁned to be the ZZ-graded ring G(A) = ⊕p∈ZZ G(A)p with G(A)p = Fp A/Fp−1 A and the multiplication Fq A Fp A × Fp−1A Fq−1 A

−→

Fp+q A Fp+q−1 A

(a + Fp−1 A, b + Fq−1 A)

→

ab + Fp+q−1 A

is deﬁned to be the ZZ-graded The Rees ring of A with respect to F A, denoted A, p = Fp A and the multiplication = ⊕p∈ZZ A p with A ring A q p × A A

−→

p+q A

(h(a)p , h(b)q )

→

h(ab)p+q

p represented by a ∈ Fn A, where h(a)p denotes the homogeneous element in A n ≤ p. 1 = F1 A represented by Let X denote the homogeneous element of degree 1 in A 1. To be convenient, we call X the canonical element of A. 3.8. Proposition With notation as above, the following holds. and is not a divisor (i) The canonical element X is contained in the center of A of zero. Hence we have the two-sided ideals X = X A, 1 − X = (1 − X)A. Moreover, (1 − X)A does not contain any nonzero homogeneous element of A.

3. ZZ-ﬁltrations and Gradations

17

(ii) A ∼ − X)A. = A/(1 ∼ p + X A)/X as graded rings. A (iii) G(A) = A/X A = ⊕p∈ZZ (A Proof (i) This may be directly veriﬁed. p is the homogeneous element of degree pi repre(ii) Note that if h(ai )pi ∈ A i sented by ai ∈ Fn A, n ≤ pi , and if ≥ pi , then X −pi h(ai )pi = h(ai ) ∈ A is the homogeneous element of degree represented by ai ∈ Fn A. Thus, if with h(ai )p ∈ A p and ≥ pi , then h(ai )pi ∈ A i i h(ai )pi = X −pi h(ai )pi − X −pi h(ai )pi + h(ai )pi = h ai + h(ai )pi 1 − X −pi .

Now consider the ring homomorphism A

ψ:

−→

h(ai )pi

→

A

ai

Hence (ii) is Then it is easy to see that ψ is surjective and Kerψ = (1 − X)A. proved. A) p = (A p + X A)/X with (iii) Note that since G(A)p = Fp A/Fp−1 A and (A/X A Ap = Fp A. It is not hard to construct the desired graded ring isomorphism by using the formula Xh(a)p−1 = h(a)p for a ∈ Fp−1 A. 2 is to identify it with the graded Another way to understand the Rees ring A n −1 subring A = ⊕n∈ZZ Fn At of A[t, t ], where A[t, t−1 ] is the ring of Laurent poly i nomials (i.e., the formal polynomials ai t ai ∈ A, i ∈ ZZ) in the commuting variable t, and the addition and multiplication on A[t, t−1 ] are deﬁned as in the usual polynomial ring. The ZZ-gradation on A[t, t−1 ] is given by the degree of to A is deﬁned as: polynomials, and the graded ring isomorphism from A −→ A = Fn A = A Fn Atn n∈Z Z

n∈Z Z

h(ai )pi

→

ai tpi

Obviously, t is a homogeneous central element of degree 1 in A which is also not a divisor of zero, and X → t. (This may be easily recaptured from Proposition 3.8.) of a ﬁltered ring A constructed with respect to a given ﬁltration The Rees ring A F A has been widely used in the literature, e.g., [AVO], [Gin], [Kass] and [LVO4]. Also see CH.III §3 for some exposition.

18

I. Basic Structural Tricks

3.9. Deﬁnition Let A be a ZZ-ﬁltered ring with ﬁltration F A, and M an Amodule. If there is a sequence F M = {Fp M }p∈ZZ of subgroups of the additive group of M such that (FM1) Fp M = M , p∈Z Z

(FM2) Fp−1 M ⊂ Fp M for all p ∈ ZZ, (FM3) (Fp A)(Fq M ) ⊂ Fp+q M for all p, q ∈ ZZ, then M is called a ﬁltered A-module with ﬁltration F M . If Fn M = 0 for some n ∈ ZZ, then we say that F M is a discrete ﬁltration on M . (Obviously the positive ﬁltration on a ﬁltered ring is discrete.) We say that F M is separated if it also satisﬁes: (FM4) Fp M = {0}. p∈Z Z

If A is a k-algebra over a ﬁeld k, we require that each Fp M is a subspace of the k-vector space M . Let M be a ﬁltered A-module with ﬁltration F M . The associated graded module of M with respect to F M , denoted G(M ), is deﬁned to be the graded G(A)module G(M ) = ⊕p∈ZZ G(M )p with G(M )p = Fp M/Fp−1 M , where the module action of G(A) on G(M ) is given by Fq M Fp A × Fp−1 A Fq−1 M

−→

Fp+q M Fp+q−1 M

(a + Fp−1 A, m + Fq−1 M )

→

am + Fp+q−1 M

, is deﬁned to be the The Rees module of M with respect to F M , denoted M graded A-module M = ⊕p∈ZZ Mp with Mp = Fp M and the module action q p × M A

−→

p+q M

(h(a)p , h(m)q )

→

h (am)p+q

p represented by a ∈ Fn A where h(a)p denotes the homogeneous element in A q represented by with n ≤ p, and h(m)q denotes the homogeneous element in M m ∈ Fs M with s ≤ q. , M , and G(M ) are mentioned as follows. Relations between M 3.10. Proposition With notation as above, let X be the canonical element of Then the following holds. A. , and (1 − X)M does not (i) X does not annihilate any nonzero element of M contain any nonzero homogeneous element of M .

3. ZZ-ﬁltrations and Gradations

19

/(1 − X)M . (ii) M ∼ =M ∼ (iii) G(M ) = M /X M . 2 Any ﬁltration F M on an A-module M deﬁnes an order function v: M → ZZ as follows:   −∞, if m ∈ ∩n∈ZZ Fn M, v(m) =  p, if m ∈ Fp M − Fp−1 M. Important notation If m ∈ M with v(m) = p, we write σ(m) for the nonzero for the homogeneous element image of m in G(M )p = Fp M/Fp−1 M , and write m h(m)p in Mp . 3.11. Lemma With notation as before, the following holds. represented by m (i) If m ∈ ∩n∈ZZ Fn M , then any homogeneous element in M can be uniquely written as X sm for some s ≥ 0. (ii) If a ∈ A with v(a) = p, and m ∈ M with v(m) = q such that am ∈ ∩n∈ZZ Fn M , then am = X s am, where s = p + q − v(am). (iii) Let a ∈ A, m ∈ M . Then σ(a)σ(m) = 0 if and only if v(a) + v(m) = v(am) if and only if σ(a)σ(m) = σ(am) if and only if am = am. 2 3.12. Proposition Let A be a ZZ-ﬁltered ring with ﬁltration F A. is a domain. (i) A is a domain if and only if A (ii) Suppose that F A is separated. If G(A) is a domain then so are A and A. 2 3.13. Proposition Let A be a ZZ-ﬁltered ring with ﬁltration F A, and M a ﬁltered A-module with a discrete ﬁltration F M (i.e., Fn M = 0 for some n ∈ ZZ). . If G(M ) is Noetherian then so are M and M 2 If A is a positively ﬁltered ring such that G(A) is Noetherian, then the above are Noetherian. This is a special case of the proposition entails that A and A so called Zariskian ﬁltered ring in the sense of ([Li1], [LVO4]). For a Zariskian ﬁltered ring A, many algebraic structure properties of G(A) may be lifted to A.

20

I. Basic Structural Tricks

This shows on one hand the advantage of using the associated graded structures. We refer to [LVO4] for a general theory of Zariskian ﬁltrations. Let A be a ﬁltered ring with ﬁltration F A, and let M , N be ﬁltered A-modules with ﬁltrations F M , F N , respectively. An A-homomorphism ϕ: M → N is said to be a ﬁltered A-homomorphism if ϕ(Fp M ) ⊂ Fp N for all p ∈ ZZ. If ϕ is an A-homomorphism such that ϕ(Fp M ) = Fp N ∩ Imϕ for all p ∈ ZZ, we say that ϕ is a strict ﬁltered A-homomorphism. Let ϕ: M → N be a ﬁltered A-homomorphism. Then ϕ induces two graded module homomorphisms in a natural way: G(ϕ) ϕ −→ G(M ) −→ G(N ), M N. Let M be a ﬁltered A-module with ﬁltration F M , and N a submodule of M . Then the induced ﬁltration on N is deﬁned as: Fp N = Fp M ∩ N , p ∈ ZZ. The induced quotient ﬁltration on M/N is deﬁned as: Fp (M/N ) = (Fp M + N )/N , p ∈ ZZ. 3.14. Proposition Let ϕ: M → N be an onto strict ﬁltered A-homomorphism with Kerϕ = K. Consider the induced ﬁltration on K and let : K → M be the inclusion map. Then the exact sequence of A-modules

ϕ

0 → K −→ M −→ N → 0 induces two exact sequences of graded modules: G()

G(ϕ)

0 → G(K) −→ G(M ) −→ G(N ) → 0 ϕ −→ −→ 0→K M N →0 2 3.15. Corollary Let M be a ﬁltered A-module with ﬁltration F M , and N a submodule of M . Consider the ﬁltrations on N and M/N induced by F M , respectively. The following holds. is a graded submodule of M . (i) G(N ) is a graded submodule of G(M ), and N ∼ /N . (ii) G(M/N ) ∼ = G(M )/G(N ), M/N =M 2 Example (iii) Let R = ⊕n∈ZZ Rn be a ZZ-graded ring. Then the grading ﬁltration F R on R is deﬁned as Fp R = Rn , p ∈ ZZ. n≤p

It is easy to see that G(R) ∼ = R as graded rings. In Example (vi) below, we will ∼ prove that R = R[t] as graded rings, where R[t] has the mixed gradation (see the foregoing Example (ii)).

3. ZZ-ﬁltrations and Gradations

21

(iv) Let A = k[ai ]i∈Λ be a k-algebra generated by {ai }i∈Λ over k. Then the naturally deﬁned positive ﬁltration F A on A consisting of k-subspaces     αn 1 Fp A = cα aα , p ∈ IN , i1 · · · ain cα ∈ k, αj ∈ IN   α1 +···+αn ≤p

is called the standard ﬁltration on A. We note the following basic facts for the standard ﬁltration F A. a. If A = k[a1 , ..., an ] is a ﬁnitely generated algebra over k with the ﬁnite generating set {a1 , .., an }, then every Fp A is a ﬁnite dimensional subspace of the k-vector space A. In particular, F0 A = k. If A = k[ai ]i∈Λ = ⊕n∈IN An is also positively graded such that ai ∈ A1 , i ∈ Λ, i.e., A is generated by homogeneous elements of degree 1, then it is easy to see that b. A0 = k, A1 = i kai , An = An1 and the standard ﬁltration F A on A is given by the grading ﬁltration on A as deﬁned in (iii) above. with respect to the standard ﬁltration Concerning the generators of G(A) and A on A, where A = k[ai ]i∈Λ , we have the following fact. 3.16. Proposition (i) G(A) = k[σ(ai )]i∈Λ , i.e., G(A) is generated by homogeneous elements of degree 1. = k[X, i.e., A is also (ii) A ai ]i∈Λ , where X is the canonical element of A, generated by homogeneous elements of degree 1. Proof (i) Since F A is the standard ﬁltration, it is separated. If a ∈ A is nonzero, then a ∈ Fh A − Fh−1 A for some h ≥ 0, i.e., v(a) = h, in particular, every σ(ai ) has degree 1. For a ∈ A with order v(a) = h, writing αn 1 cα aα |α| = α1 + · · · + αn , a= i1 · · · ain , |α|≤h

α1 αn We may assume that every coset σ(a) = |α|=h cα ai1 · · · ain + Fh−1 A . αn 1 aα · · · a + F A = 0 in the above expression is nonzero. Then by Lemma h−1 i1 in 3.11 we have cα σ(ai1 )α1 · · · σ(ain )αn . σ(a) =

|α|=h

p be a nonzero homogeneous (ii) That every ai is of degree 1 is clear. Let F ∈ A element of degree p. By Lemma 3.11, F = X s a for some s ≥ 0, a ∈ A. Writing αn 1 a= ci1 ···in aα i1 · · · ain and using the order function as deﬁned before Lemma αn 1 3.11, we may assume that v(a) = h and v(aα i1 · · · ain ) = pα . Put |α| = α1 +

22

I. Basic Structural Tricks

· · · + αn . Then since F A is the standard ﬁltration we have |α| ≤ h, pα ≤ h, and by Lemma 3.11 αn 1 = a X |α|−pα (aα i1 i1 · · · ain )

α1

· · · a in

αn

.

Thus a =

αn 1 ci1 ···in X h−pα (aα i1 · · · ain )

αn 1 ci1 ···in X h−|α|+|α|−pα (aα i1 · · · ain ) α1 αn = ci1 ···in X h−|α| a · · · a i1 in

=

and consequently a= F = X s

ci1 ···in X h−|α|+s a i1

α1

· · · a in

αn

. 2

This ﬁnishes the proof.

Now let kX be the free k-algebra generated by X = {Xi }i∈Λ . Then from the foregoing Example (i) we know that kX is a positively graded ring, i.e., kX = ⊕p∈IN kXp , and Xi ∈ kX1 , i ∈ Λ. Hence, the standard ﬁltration F kX on kX is given by the grading ﬁltration on kX: Fp kX = ⊕q≤p kXq , p ∈ IN . If we consider the natural onto algebra homomorphism ϕ: kX → A = k[ai ]i∈Λ with ϕ(Xi ) = ai , i ∈ Λ, then it is also clear that ϕ(Fp kX) = Fp A, p ∈ IN , i.e., ϕ is a strict ﬁltered kX-homomorphism. If furthermore we consider the ﬁltration on I = Kerϕ induced by F kX, then by Proposition 3.11, the exact sequence of kX-modules

ϕ

0 → I −→ kX −→ A → 0 induces two exact sequences of graded modules: G()

G(ϕ)

0 → G(I) −→ G(kX) −→ G(A) → 0 ϕ −→ 0 → I −→ kX A→0 From the above Example (iii) we know that G(kX) ∼ = kX. In Example (vi) ∼ below it will be shown that kX = kX[t] as graded rings, where kX[t] is the polynomial ring over kX in one commuting variable t, and the gradation used on kX[t] is the mixed gradation as deﬁned in the foregoing Example (ii). Later in CH.III we will see that if the the deﬁning relations of A are given, namely, a generating set of the above I is given, then it is possible to algorithmically determine the generating sets for both G(I) and I. Example (v) This example may be viewed as the dual process of constructing the Rees ring of a ﬁltered ring with a given ﬁltration. The ﬁrst general study of this process and some applications were given in [LVO3].

3. ZZ-ﬁltrations and Gradations

23

Let R = ⊕n∈ZZ Rn be a ZZ-graded ring. Suppose that R contains a homogeneous element Y of degree 1, i.e., Y ∈ R1 , which is contained in the centre of R and is not a divisor of zero. Then it is clear that both Y = Y R and 1−Y = (1−Y )R are two-sided ideals in R, in particular, Y is a graded ideal of R by Proposition 3.3. Put A = R/1 − Y . Note that if rn ∈ Rn is a homogeneous element of degree n, then Y rn ∈ Rn+1 because Y is of degree 1. Thus rn = Y rn + (1 − Y )rn implies Rn + 1 − Y ⊂ Rn+1 + 1 − Y . Consequently we have Rn+1 + 1 − Y Rn + 1 − Y ⊂ , 1 − Y 1 − Y

n ∈ ZZ,

Rn + 1 − Y R = = A. 1 − Y 1 − Y

n∈Z Z

Hence the ZZ-gradation on R naturally induces a ZZ-ﬁltration F A on A, i.e., Fn A =

Rn + 1 − Y , 1 − Y

n ∈ ZZ.

3.17. Proposition With notation as above, the following holds. (i) 1 − Y does not contain any nonzero homogeneous element of R. (ii) G(A) ∼ = R/Y as graded rings. ∼ (iii) A = R as graded rings, where Y corresponds to the canonical element X in (see the deﬁnition given before Proposition 3.8). A Proof (i) Since Y is a homogeneous element of degree 1 in the center of R and it is not a divisor of zero, this may be veriﬁed directly. (ii) By deﬁnition Rn + 1 − Y G(A)n ∼ , n ∈ ZZ. = Rn−1 + 1 − Y Note that since R/Y = ⊕n∈ZZ (R/Y )n is a graded ring (see Proposition 3.3) with Rn + Y (R/Y )n = , n ∈ ZZ, Y If for each n ∈ ZZ we deﬁne the map ϕn :

Rn + Y Y

−→

Rn + 1 − Y Rn−1 + 1 − Y

rn + Y

→

rn + (Rn−1 + 1 − Y )

24

I. Basic Structural Tricks

then it is not hard to see that these maps yield a graded ring isomorphism ϕ = ⊕ϕn : R/Y → G(A). (iii) By deﬁnition = ⊕n∈ZZ Fn A = A

Rn + 1 − Y n∈Z Z

1 − Y

.

It follows from (i) above that for each n ∈ ZZ there is a natural group isomorphism: Rn + 1 − Y ψn : Rn −→ 1 − Y rn

→

rn + 1 − Y

It is easy to verify that these ψn yield a graded ring isomorphism ψ = ⊕ψn : in particular, ψ(Y ) = X where the latter is the canonical element of A. R → A, 2 Example (vi) Let R = ⊕n∈ZZ Rn be a ZZ-graded ring, and consider the grading ﬁltration F R on R as deﬁned in Example (iii) above, i.e., Fn R = ⊕i≤n Ri , n ∈ ZZ. We have seen that G(R) ∼ = R. The aim of this example is to determine R. Let R[t] be the polynomial ring over R in one commuting variable t, and consider the mixed gradation on R[t] as deﬁned in the foregoing Example (ii), i.e., R[t] = ⊕n∈ZZ R[t]n with     ai tj ai ∈ Ri , j ∈ IN , n ∈ ZZ. R[t]n =   i+j=n Note that t is a homogeneous element of degree 1 contained in the centre of R[t] and it is not a divisor of zero. It follows from Example (v) above that the mixed gradation on R[t] induces a ﬁltration F A on A = R[t]/1 − t, i.e., Fn A =

R[t]n + 1 − t , 1 − t

n ∈ ZZ.

Now we have the following easily veriﬁed facts: c. The onto ring homomorphism φ: R[t] → R with φ(t) = 1 has Kerφ = 1−t, hence A = R[t]/1 − t ∼ = R as rings, R[t]n + 1 − t ∼ d. Fn A = = R[t]n as additive groups by Proposition 3.17(i), 1 − t and R[t]n + 1 − t e. ⊕i≤n Ri = Fn R ∼ as additive groups, where each = Fn A = 1 − t n−i ai t + 1 − t. i≤n ai ∈ Fn R is sent to Hence, it follows from Proposition 3.17(ii)–(iii) that the following holds.

4. Homogenization and Dehomogenization

25

3.18. Proposition Let R = ⊕n∈ZZ Rn be a ZZ-graded ring and F R the grading ﬁltration on R. Then (i) G(R) ∼ = R as graded rings, ∼ ∼ (ii) R =A = R[t] as graded rings. 2

4. Homogenization and Dehomogenization of ZZ-graded Rings Let k be an algebraically closed ﬁeld of chark = 0, and k[x1 , ..., xn ] the commutative polynomial k-algebra in n variables. If I is an ideal of k[x1 , ..., xn ] and I ∗ is the homogenization ideal of I in k[x0 , x1 , ..., xn ] with respect to x0 , then a well known fact from algebraic geometry is that I ∗ deﬁnes the projective closure of the aﬃne algebraic set V (I) in the projective n-space Pnk . In this section we will see that there is a nice relation between the algebra k[x1 , ..., xn ]/I and the algebra k[x0 , x1 , ..., xn ]/I ∗ , which has a very noncommutative version, and the latter indeed yields more elegant structural properties describing the associated graded structures of a ZZ-ﬁltered ring. Let R = ⊕n∈ZZ Rn be a ZZ-graded ring and R[t] the polynomial ring over R in one commuting variable t. Consider the familiar onto ring homomorphism φ:

R[t] −→ R

with φ(t) = 1. We know that Kerφ = 1 − t and hence R ∼ = R[t]/1 − t. Considering the mixed gradation on R[t] as deﬁned in §4 Example (ii), i.e., R[t] = ⊕p∈ZZ R[t]p with     R[t]p = Fi tj Fi ∈ Ri , j ≥ 0 , p ∈ ZZ,   i+j=p we also observe that • For every f ∈ R, there exists a homogeneous element F ∈ R[t]p , for some p, such that φ(F ) = f . More precisely, if f = F0 + F1 + · · · + Fp where Fi ∈ Ri , Fp = 0, then f ∗ = tp F0 + tp−1 F1 + · · · + tFp−1 + Fp is a homogeneous element in R[t]p satisfying φ(f ∗ ) = f . 4.1. Deﬁnition (i) For any F ∈ R[t], we write F∗ = φ(F ). F∗ is called the dehomogenization of F with respect to t.

26

I. Basic Structural Tricks

(ii) For any f ∈ R, if f = F0 + F1 + · · · + Fp with Fp = 0, then the homogeneous element f ∗ = tp F0 +tp−1 F1 +· · ·+tFp−1 +Fp in R[t]p is called the homogenization of f with respect to t. (iii) If I is a two-sided ideal of R, then we let I ∗ stand for the graded two-sided ideal of R[t] generated by {f ∗ | f ∈ I}. I ∗ is called the homogenization ideal of I with respect to t. For f ∈ R, if f = F0 + F1 + · · · + Fp is the homogeneous decomposition of f and Fp = 0, we write d(f ) = p. Note that since t is a commuting variable, the following lemma can be easily veriﬁed. 4.2. Lemma (i) For F, G ∈ R[t], (F + G)∗ = F∗ + G∗ , (F G)∗ = F∗ G∗ . (ii) For f, g ∈ R, (f g)∗ = f ∗ g ∗ , ts (f + g)∗ = tr f ∗ + th g ∗ , where r = d(g), h = d(f ), and s = r + h − d(f + g). (iii) For any f ∈ R, (f ∗ )∗ = f . (iv) If F is a homogeneous element of degree p in R[t], and if (F∗ )∗ is of degree q, then tr (F∗ )∗ = F , where r = p − q. (v) If I is a two-sided ideal of R, then each homogeneous element F ∈ I ∗ is of the form tr f ∗ for some r ∈ IN and f ∈ I. 2 4.3. Proposition Let I be a proper two-sided ideal of R. Then there is an onto ring homomorphism α: R[t]/I ∗ → R/I with Kerα = 1 − t, where t denotes the coset of t in R[t]/I ∗ . Moreover, t is not a divisor of zero in R[t]/I ∗ , and hence 1 − t does not contain any nonzero homogeneous element of R[t]/I ∗ . Proof If we deﬁne α by putting α : R[t]/I ∗ F + I∗

−→ →

R/I F ∈ R[t], F∗ + I,

then by Lemma 4.2 we easily see that α is a ring epimorphism with Kerα = 1−t. For any homogeneous element F ∈ R[t], if tF ∈ I ∗ , then F∗ = (tF )∗ ∈ (I ∗ )∗ ⊂ I by Lemma 4.2. Again by Lemma 4.2 we have F = tr (F∗ )∗ ∈ I ∗ . Hence t is not a divisor of zero in R[t]/I ∗ . The fact that 1 − t does not contain any nonzero homogeneous element of R[t]/I ∗ now follows easily. 2 Now let kX be the free algebra over a ﬁeld k with generating set X = {Xi }i∈Λ . From §4 we retain that kX = ⊕p∈IN kXp has the natural gradation deﬁned by the degree of words in the free semigroup S = X and the standard ﬁltration F kX on kX is given by the k-subspaces: Fp kX = ⊕i≤p kXi ,

p ∈ IN .

4. Homogenization and Dehomogenization

27

If I is any two-sided ideal of kX and we put A = kX/I, then F kX induces the quotient ﬁltration F A on A: Fp A = (Fp kX + I)/I,

p ∈ IN ,

which indeed coincides with the standard ﬁltration on the k-algebra A = kX/I = k[xi ]i∈Λ where xi is the coset of Xi in kX/I. Let kX[t] be the polynomial ring over kX in one commuting variable t and consider the mixed gradation on kX[t]. If we consider the associated graded = ⊕p≥0 Fp A of A, then ring G(A) = ⊕p≥0 (Fp A/Fp−1 A) of A and the Rees ring A the following proposition shows that G(A) and A are completely determined by I ∗ , where I ∗ is the homogenization ideal of I in kX[t] (in the sense of Deﬁnition 4.1). 4.4. Proposition (compare with Proposition 3.16) With notation as above, there are graded k-algebra isomorphisms: ∼ (i) A = kX[t]/I ∗ , and (ii) G(A) ∼ = kX[t]/(t + I ∗ ), where t denotes the ideal of kX[t] generated by t. Proof First note that the ring homomorphism α: kX[t]/I ∗ → kX/I = A deﬁned in Proposition 4.3 yields the isomorphism of k-subspaces: kX[t]p + 1 − t ⊕i≤p kXi + I = Fp A, −→ I 1 − t

p ∈ IN .

Since we know from Proposition 4.3 that t is a homogeneous element of degree 1 in kX[t]/I ∗ and it is not a divisor of zero, it follows from Proposition 3.8 that (i) and (ii) hold. 2 Remark Let us go back to the commutative case and consider A = k[x1 , ..., xn ]/I, where I is an ideal of the polynomial algebra k[x1 , ..., xn ]. Now it is clear that, with respect to the standard ﬁltration F A on A (which is in fact induced by the natural grading ﬁltration on k[x1 , ..., xn ]), G(A) ∼ = k[x0 , x1 , ..., xn ]/(x0 + I ∗ ) ∗ ∗ ∼ and A = k[x0 , x1 , ..., xn ]/I , where I is the homogenization ideal of I in k[x0 , x1 , ..., xn ] with respect to x0 . This implies that the deﬁning relations of of A correspond to the deﬁning equations of the projective the Rees algebra A ∗ closure V (I ) of the aﬃne algebraic set V (I), and the deﬁning relations of the associated graded ring G(A) correspond to the deﬁning equations of the part of the projective closure V (I ∗ ) at inﬁnity.

28

I. Basic Structural Tricks

5. Some Algebras: Classical and Modern In this section, we list some well-known algebras as the basic examples in developing later chapters. More examples are constructed in CH.III. (i) Weyl algebra The ﬁrst Weyl algebra A1 (k) over a ﬁeld k is deﬁned to be the k-algebra generated by x and y subject to the relation yx = xy + 1. Let k[t] be the polynomial algebra over k in t and δ = ∂/∂t. It follows from §2 Example (i) that A1 (k) ∼ = k[t][z; δ]. The nth Weyl algebra An (k) over a ﬁeld k is deﬁned to be the k-algebra generated by 2n generators x1 , ..., xn , y1 , ..., yn subject to the relations: xi xj = xj xi , yi yj = yj yi , yj xi = xi yj + δij the Kronecker delta,

1 ≤ i < j ≤ n, 1 ≤ i, j ≤ n.

If we write R = k[t1 , ..., tn ] for the commutative polynomial algebra in n variables, δi for ∂/∂ti , and consider the sequence of skew polynomial algebras: R0 = R,

Ri+1 = Ri [zi+1 ; δi+1 ], i = 0, ..., n − 1,

then the k-algebra Rn has generators which satisfy the relations that deﬁne the Weyl algebra (note that since for each i, βi n+i αn β1 1 (α · · · t z · · · z , ..., α , β , ..., β ) ∈ I N tα 1 n 1 i n 1 1 i is a k-basis of Ri and ti+1 is in the centre of Ri , it is easy to see that ∂/∂ti+1 deﬁnes a derivation on Ri ). On the other hand, the generators of An (k) satisfy the deﬁning relations for the Ri . Thus, one easily construct an algebra isomorphism An (k) ∼ = Rn , i.e., An (k) is indeed an iterated skew polynomial ring (or an iterated Ore extension) on k[t1 , ..., tn ]. Historically, the Weyl algebra is the ﬁrst “quantum algebra” ([Dir] 1926, [Wey] 1928). In the literature the standard ﬁltration (see §3) on An (k) is customarily called the Bernstein ﬁltration because of the celebrated paper of I.N Bernstein [Bern]. It is well-known that, with respect to the standard ﬁltration on An (k), the associated graded ring G(An (k)) of An (k) is isomorphic to the commutative polynomial ring over k in 2n variables (see, e.g., [Bj]). In CH.III §3 we will recapture this structure in an algorithmic way, and we will also give an algorithmic determination of the deﬁning relations of the associated Rees algebra A n (k). Another well-known fact about An (k) is that, if chark = 0, then An (k) coincides with the algebra of linear partial diﬀerential operators of the polynomial ring

5. Some Algebras

29

k[tx , ..., tn ] (or the ring of polynomial functions of the aﬃne n-space Ank ). We will come back to this point in CH.II §8 and CH.VII. (ii) Additive analogue of the Weyl algebra This algebra was introduced in Quantum Physics in ([Kur] 1980) and studied in ([JBS] 1981), that is, the algebra An (q1 , ..., qn ) generated over a ﬁeld k by x1 , ..., xn , y1 , ..., yn subject to the relations: xi xj = xj xi , yi yj = yj yi , yi xi = qi xi yi + 1, xj yi = yi xj ,

1 ≤ i < j ≤ n, 1 ≤ i ≤ n, i = j,

where qi ∈ k − {0}. It is not hard to see that An (q1 , ..., qn ) is isomorphic to the iterated skew polynomial ring on the commutative polynomial ring k[t1 , ..., tn ]: k[t1 , ..., tn ][z1 ; σ1 , δ1 ] · · · [zn ; σn , δn ], where zj zi = zi zj , 1 ≤ i < j ≤ n, zi ti = σi (ti )zi + δi (ti ) = qi ti zi + 1, 1 ≤ i ≤ n, zi tj = tj zi , i = j. If qi = q = 0, i = 1, ..., n, then this algebra becomes the algebra of q-diﬀerential operators. In CH.VII we will come back to this point. (iii) Multiplicative analogue of the Weyl algebra This is the algebra stemming from ([Jat] 1984 and [MP] 1988, where one may see why this algebra deserves its title), that is, the algebra On (λji ) generated over a ﬁeld k by x1 , ..., xn subject to the relations: xj xi = λji xi xj ,

1 ≤ i < j ≤ n,

where λji ∈ k − {0}. It is easy to see that On (λji ) is isomorphic to the iterated skew polynomial ring k[z1 ][z2 ; σ2 ] · · · [zn ; σn ], where zj zi = σj (zi )zj = λji zi zj , 1 ≤ i < j ≤ n. If n = 2, then O2 (λ21 ) is the quantum plane in the sense of Manin [Man]. If λji = q −2 = 0 for some q ∈ k − {0} and all 1 ≤ i < j ≤ n, then On (λji ) becomes the well-known coordinate ring of the so called quantum aﬃne n-space (e.g., see [Sm1]). Algebras of type On (λji ) will play a key role in the structural-algorithmic study of quadric solvable polynomial algebras (CH.III–CH.VIII), and for the sake of using a uniﬁed notion, these algebras will be called homogeneous solvable polynomial algebras (CH.III deﬁnition 2.1).

30

I. Basic Structural Tricks

(iv) Enveloping algebra of a ﬁnite dimensional Lie algebra Let g be a ﬁnite dimensional vector space over the ﬁeld k with basis {x1 , ..., xn } where n = dimk g. If there is a binary operation on g, called the bracket product and denoted [ , ], which is bilinear, i.e., for a, b, c ∈ g, λ ∈ k, [a + b, c] = [a, c] + [b, c] [a, b + c] = [a, b] + [a, c] λ[a, b] = [λa, b] = [a, λb], and satisﬁes: [a, b] = −[b, a], a, b ∈ g, [[a, b], c] + [[c, a], b] + [[b, c], a] = 0

Jacobi identity, a, b, c ∈ g,

then g is called a ﬁnite dimensional Lie algebra over k. Note that [ , ] need not satify the associative low. If [a, b] = [b, a] for every a, b in a Lie algebra g, g is called abelian. The enveloping algebra of g, denoted U (g), is deﬁned to be the associative kalgebra generated by x1 , ..., xn subject to the relations: xj xi − xi xj = [xj , xi ],

1 ≤ i < j ≤ n.

For example, the Heisenberg Lie algebra h has the k-basis {xi , yj , z | i, j = 1, ..., n} and the bracket product is given by 1 ≤ i ≤ n, [xi , yi ] = z, j, [xi , xj ] = [xi , yj ] = [yi , yj ] = 0, i = [z, xi ] = [z, yi ] = 0, 1 ≤ i ≤ n. In Example (v) below, one will see that the enveloping algebra U (h) of the Heisenberg Lie algebra h is an iterated skew polynomial ring. If we consider the standard ﬁltration on U (g), then it follows from the famous Poincar´e-Birkhoﬀ-Witt theorem (abbreviated PBW theorem hereafter) that G(U (g)) is (as a graded ring) isomorphic to the commutative polynomial ring k[t1 , ..., tn ] (see, e.g., [Dix]). In CH.III §3 we will recapture this structure in an algorithmic way, and we will also give an algorithmic determination of the deﬁning relations of the associated Rees algebra U (g). (v) q-Heisenberg algebra This is the algebra stemming from ([Ber] 1992, [Ros] 1995) which has its root in q-calculus (e.g., [Wal] 1985), that is, the algebra hn (q) generated over a ﬁeld k by the set of elements {xi , yi , zi | i = 1, ..., n} subject to the relations: xi xj = xj xi , yi yj = yj yi , zj zi = zi zj , 1 ≤ i < j ≤ n, xi zi = qzi xi , 1 ≤ i ≤ n, 1 ≤ i ≤ n, zi yi = qyi zi , xi yi = q −1 yi xi + zi , 1 ≤ i ≤ n, xi yj = yj xi , xi zj = zj xi , yi zj = zj yi , i = j,

5. Some Algebras

31

where q ∈ k − {0}. It is not hard to see that hn (q) is isomorphic to the iterated skew polynomial ring on the commutative polynomial ring k[t1 , ..., tn ]: k[t1 , ..., tn ][u1 ; σ1 ] · · · [un ; σn ][v1 ; θ1 , δ1 ] · · · [vn ; θn , δn ], where vi vj = vj vi , ui uj = uj ui , ui ti = σ(ti )ui = qti ui , vi ui = θ(ui )vi = qui vi , vi ti = θ(ti )vi + δi (ti ) = q −1 ti vi + ui , uj ti = ti uj , vj ti = ti vj , vj ui = ui vj ,

i = j, 1 ≤ i ≤ n, 1 ≤ i ≤ n, 1 ≤ i ≤ n, i = j.

(vi) Manin algebra of 2 × 2 quantum matrices See ([Man] 1988). This is the algebra Mq (2, k) generated over a ﬁeld k by a, b, c, d subject to the relations: ba = qab, ca = qac, dc = qcd, db = qbd, cb = bc, da − ad = (q − q −1 )bc, where q ∈ k − {0}. Note that this is a positively graded quadratic algebra with the natural gradation (see §3), and it is is also an iterated skew polynomial algebra starting with the ground ﬁeld k (see CH.III Example (vi)). (vii) Hayashi algebra In order to get bosonic representations for the types of An and Cn of the Drinﬁeld-Jimbo quantum algebras, Hayashi introduced in ([Hay] 1990) the qWeyl algebra A− q , which is constructed as follows, by following [Ber]. Let U be the algebra generated over the ﬁeld k = C by the set of elements {xi , yi , zi | i = 1, ..., n} subject to the relations: xj xi = xi xj , yj yi = yi yj , zj zi = zi zj , 1 ≤ i < j ≤ n, 1 ≤ i, j ≤ n, xj yi = q −δij yi xj , zj xi = q −δij xi zj , zj yi = yi zj , i= j, zi yi − q 2 yi zi = −q 2 x2i , 1 ≤ i ≤ n, −1 U, the localization of U at the multiplicative where q ∈ k −{0}. Then A− q =S monoid S generated by x1 , ..., xn . Note that U is a positively graded quadratic algebra with the natural gradation (see §3), and it is also an iterated skew polynomial algebra starting with the ground ﬁeld k (see CH.III Example (vi)).

(viii) Grassmann Algebra The Grassmann algebra (or exterior algebra, see, e.g., [Vdw]) is the algebra G(x) generated over a ﬁeld k by the set of elements x = {xi }i∈Λ subject to the relations: x2i = 0, i ∈ Λ, xj xi = −xi xj , j > i,

32

I. Basic Structural Tricks

where a well-ordering is assumed on Λ. Obviously, G(x) has nontrivial divisors of zero; and moreover, if |x| = n is ﬁnite, then G(x) is isomorphic to On (λji )/x2i ni=1 , where On (λji ) is the algebra given in Example (iii) above but with all λji = −1, a root of unity. (A proof of the latter assertion follows from Example (ii) of CH.II §5, also see the proof given in CH.II §7 Example (i).) (ix) Down-up Algebra Based on the study of algebras generated by the down and up operators on a diﬀerential or uniform partially ordered set (poset), Benkart and Roby introduced in ([Ben], [BR], 1998) the down-up algebra A(α, β, γ), which is the algebra generated by {u, d} over a ﬁeld k subject to the relations: d2 u du2 where α, β, γ ∈ k.

= αdud + βud2 + γd = αudu + βu2 d + γu,

CHAPTER II Gr¨ obner Bases in Associative Algebras

Throughout the present chapter–CH.VIII, most notation and notions adopt those commonly used in the quoted literature concerning Gr¨ obner bases and related computation, so that the reader may refer to similar texts more conveniently. In particular, to better understand this chapter, we refer the reader to [AdL], [BW], and [CLO ] for a commutative Gr¨ obner basis theory, and to [Gr], [K-RW], and [Mor1-2] for typical noncommutative Gr¨ obner basis theories. The ﬁrst three sections of this chapter introduces Gr¨obner bases for (both twosided and left) ideals in a (not necessarily commutative) associative k-algebra A = k[ai ]i∈Λ over the ﬁxed ﬁeld k of arbitrary characteristic. This is reached by developing a division algorithm (§2) that is eﬀective for any (left) admissible system (A, B, ) including the case where A has divisors of zero. The technical path is to have a suitable “monomial ordering”, distinguish the “quasi-zero” elements in A, and implement the division by using the (left) “division leading monomial” of an element f ∈ A. In §4 we indicate the main diﬀerences of (left) Gr¨ obner bases in various contexts, and in §5 we discuss the possibility of having a version of Buchberger’s algorithm for a given (left) admissible system. With the theory developed in §§2–3, we mimic the commutative case using Dickson basis ([BW] Ch.5) to characterize the (left) admissible system (A, B, ) in which ﬁnite (left) Gr¨obner bases always exist for (left) ideals of A. From an algebraic structure point of view, such algebras A will be called (“left G-Noetherian”) “G-Noetherian” algebras; from

H. Li: LNM 1795, pp. 33–65, 2002. c Springer-Verlag Berlin Heidelberg 2002

34

II. Gr¨ obner Bases in Algebras

an algorithmic point of view (though we are not really dealing with algorithms), we name such (left) admissible systems as (left) “Dickson systems” (§6). §7 is for presenting the class of solvable polynomial algebras (in the sense of KandriRody and Weispfenning) that holds a successful Gr¨ obner basis theory in the noncommutative setting. The chapter is closed by a closer look at the algebra of linear partial diﬀerential operators with polynomial coeﬃcients over a ﬁeld k with chark = p > 0, on which a Gr¨ obner basis theory (in the sense of §3) does not exist.

1. (Left) Monomial Orderings and (Left) Admissible Systems Let k be a ﬁeld and A = k[ai ]i∈Λ a k-algebra with generating set {ai }i∈Λ . Then every element f ∈ A is of the form αn 1 f= ci1 ···in aα ci1 ···in ∈ k, aij ∈ {ai }i∈Λ , αj ∈ IN . i1 · · · ain , Learning how to “input” and “output” polynomials of k[t] in implementing the familiar Euclidean division algorithm on computer, we see that the ﬁrst step to have a feasible division algorithm for A, in which the data of each “input” and “output” is ordered “compatibly” with the multiplication of A, is to deﬁne a suitable “monomial ordering” on a k-basis of A. That is the goal of this section. αn 1 By abusing language, elements of the form aα i1 · · · ain in A are called monomials. Let B be a ﬁxed k-basis of A consisting of 1 and monomials, that is, αn 1 B = aα i1 · · · ain n ≥ 1, αi ∈ IN .

To be convenient, from now on we use characters s, t, u, v, w, ... to denote the monomials in B. αn 1 If w ∈ B, w = aα i1 · · · ain , then we write |α| = α1 + · · · + αn , and call |α| the degree of w, denoted d(w) = |α|. Let denote a ﬁxed well-ordering on B (of course, any set can be well-ordered). Then any nonzero element f ∈ A has a unique ordered linear expression in terms of wi ∈ B: f=

n i=1

ci wi , with ci ∈ k − {0}, and w1 w2 · · · wn .

1. Monomial Orderings and Admissible Systems

35

1.1. Deﬁnition Let f be a nonzero element in A as above. (i) The degree of f is deﬁned as d(f ) = max d(wi ) i = 1, .., n . (ii) With respect to we write LM (f ) = w1 , for the leading monomial of f, LC (f ) = c1 , for the leading coeﬃcient of f, LT (f ) = c1 w1 , for the leading term of f. If there is no confusion possible for the monomial ordering be of used, we will also just use LM(f ), LC(f ), and LT(f ) for simplicity. 1.2. Deﬁnition Let be a well-ordering on B. (i) is said to be a monomial ordering on A if the following conditions are satisﬁed. (MO1) If w, u, v, s ∈ B with w ≺ u, LM(vws) = 0 and LM(vus) = 0, then LM(vws) ≺ LM(vus). (MO2) For w, u ∈ B, if u = LM(vws) for some v, s ∈ B with v = 1 or s = 1, then w ≺ u. (Hence 1 ≺ u for all u ∈ B with u = 1.) (ii) is said to be a left monomial ordering on A if the following conditions are satisﬁed. (LMO1) If w, u, v ∈ B with w ≺ u, LM(vw) = 0 and LM(vu) = 0, then LM(vw) ≺ LM(vu). (LMO2) For w, u ∈ B, if u = LM(vw) for some v ∈ B with v = 1, then w ≺ u. (Hence 1 ≺ u for all u ∈ B with u = 1.) (iii) A (left) monomial ordering on A is called a graded monomial ordering, denoted gr , if for u, w ∈ B the following holds: u gr w if and only if d(u) > d(w) or d(u) = d(w) and u w. (iv) The triple (A, B, ) is called an admissible system if is a monomial ordering on A, and is called a left admissible system if is a left monomial ordering on A. From the deﬁnition it is clear that if (A, B, ) is an admissible system, then it is a left admissible system as well. But if (A, B, ) is a left admissible system, it seems not necessarily an admissible system since we have the following simple but important fact which is based on the fact that A may have divisors of zero. (Nevertheless, we do have many examples of left admissible systems that are admissible systems, e.g., see Example (ii), (iii) below, and later §7.)

36

II. Gr¨ obner Bases in Algebras

1.3. Lemma Let (A, B, ) be an admissible system, f ∈ A, where 0 = f = n i=1 λi wi such that LM(f ) = w1 w2 · · · wn . For any u, v ∈ B, if uf v = 0 then there is some i such that LM(uf v) = LM(uwi v) = 0 and LM(uwj v) ≺ LM(uwi v) = LM(uf v) for all j = i with LM(uwj v) = 0. (Indeed, all possible nonzero LM(uwj v) with j = i are among LM(uwi+ v), where 1 ≤ ≤ n − i.) Putting v = 1 in the above statement we get the similar result for a left admissible system. 2 The monomial wi appeared in the above lemma will be called the (left) division leading monomial of f after we distinguish the (left) quasi-zero elements of A in the next section. Let IN n be the set of all n-tuples α = (α1 , ..., αn ) with αi ∈ IN . Write |α| = α1 + · · · + αn . Recall that the lexicographic ordering on IN n in natural order, respectively in reverse order, denoted ≥lex , is deﬁned as: α = (α1 , ..., αn ), β = (β1 , ..., βn ) ∈ IN n , β >lex α if and only if the left-most nonzero entry of β − α = (β1 − α1 , ..., βn − αn ) is positive, respectively if and only if the rightmost nonzero entry of β − α is positive; and the graded lexicographic ordering on IN n in either natural order or reverse order, denoted ≥grlex , is deﬁned as: α = (α1 , ..., αn ), β = (β1 , ..., βn ) ∈ IN n , β >grlex α if and only if |β| > |α|, or |β| = |α| and β >lex α, where ≥lex is the lexicographic ordering on IN n in either natural order or reverse order. It is well-known (e.g., [BW]) that ≥lex and ≥grlex are well-orderings on IN n . If A = k[a1 , ..., an ] is a ﬁnitely generated k-algebra with n generators a1 , ..., an , αn 1 then we call the monomial of the form aα 1 · · · an a standard monomial. Suppose that n αn 1 B = aα , 1 · · · an (α1 , ..., αn ) ∈ IN the set of all standard monomials, forms a k-basis for A. Then it is easy to see that every well-ordering on IN n induces a well-ordering on B, and if ≥lex is the lexicographic ordering on IN n in either natural order or reverse order, then it induces on B either the monomial ordering a1 >lex a2 >lex · · · >lex an or the monomial ordering an >lex an−1 >lex · · · >lex a1 . Similarly, the graded lexicographic ordering on IN n induces a graded monomial ordering on B. It follows from the above remark and CH.I §§3–6 that the following examples are not hard to be veriﬁed. Example (i) Let A = k[x1 , ..., xn ] be the commutative polynomial algebra in n variables and n αn 1 . (α B = xα · · · x , ..., α ) ∈ I N 1 n n 1

1. Monomial Orderings and Admissible Systems

37

Then (A, B, ≥lex ) and (A, B, ≥grlex ) are (left) admissible systems. (ii) Let k[x1 , ..., xn ] be the commutative polynomial algebra in n variables and A = k[x1 , ..., xn ][xn+1 ; σn+1 , δn+1 ][xn+2 ; σn+2 , δn+2 ] · · · [xn+m ; σn+m , δn+m ] an iterated skew polynomial k-algebra over k[x1 , ..., xn ], where the σn+j (1 ≤ j ≤ m) are algebra endomorphisms and each δn+j (1 ≤ j ≤ m) is a σn+j -derivation, that is, for f, g ∈ k[x1 , ..., xn ][xn+1 ; σn+1 , δn+1 ] · · · [xn+j−1 ; σn+j−1 , δn+j−1 ], δn+j (f g) = σn+j (f )δn+j (g) + δn+j (f )g. Then

βm n+m αn β1 1 B = xα 1 · · · xn xn+1 · · · xn+m (α1 , ..., αn , β1 , ..., βm ) ∈ IN

forms a k-basis for A. Suppose that for 1 ≤ i < j = n + , 1 ≤ ≤ m we have σj (xi ) = λij xi , λij ∈ k − {0}, δj (xi ) = Pij ∈ k[x1 , ..., xn , xn+1 , ..., xn+h ], n + h < j = n + . Then (A, B, ≥lex ) is a left admissible system, where ≥lex is the lexicographic ordering on IN n+m in the reverse order. If degPij ≤ 2, then (A, B, ≥grlex ) is a left admissible system, where ≥grlex is the graded lexicographic ordering on IN n+m in the reverse order. If degPij ≤ 1, then (A, B, ≥grlex ) is a left admissible system, where ≥grlex is the graded lexicographic ordering on IN n+m in either natural order or reverse order. Example (ii) above may be applied to CH.I §5 Examples (i), (ii), (iii), (v), (vi), and (vii). (iii) Let A = U (g) be the enveloping algebra of an n-dimensional k-Lie algebra g = kx1 ⊕ kx2 ⊕ · · · ⊕ kxn with the bracket multiplication given by (∗)

[xj , xi ] =

n

λij x ,

j > i.

=1

(See CH.I §5.) Then by the PBW theorem, n αn 1 B = xα · · · x n (α1 , ..., αn ) ∈ IN 1 forms a k-basis of A. From the above (∗) we may derive that (A, B, ≥grlex ) is an admissible system and hence a left admissible system, where ≥grlex is the graded lexicographic ordering on IN n in either natural order or reverse order. (iv) Let A = kX be the free k-algebra on the set X = {Xi }i∈Λ (see CH.I §1). Then A has the k-basis {1}. B = w = Xi1 Xi2 · · · Xip Xij ∈ X, p ≥ 1

38

II. Gr¨ obner Bases in Algebras

Recall that the degree of an element w = Xi1 · · · Xip in B is deﬁned to be the length p of w, i.e., d(w) = p. In particular, d(1) = 0. Suppose that Λ is totally ordered by <. Recall from [Mor2] that the graded lexicographic ordering on B is deﬁned as: For u, v ∈ B, u >grlex v if and only if either d(u) > d(v) or d(u) = d(v) and v is lexicographically less than u, where v is lexicographically less than u if and only if either there is r ∈ B such that u = vr, or there are Xj1 , Xj2 ∈ X with j1 < j2 such that v = lXj1 r1 , u = lXj2 r2 for some l, r1 , r2 ∈ B. Note that ≥grlex induces on X the ordering Xj2 >grlex Xj1 whenever j2 > j1 in Λ. Since ≥grlex is a well-ordering on B, we conclude that (A, B, ≥grlex ) is an admissible system and hence a left admissible system. Observe that all algebras considered above do not have divisors of zero. In §7 we will give examples containing divisors of zero. Finally, we point out, for a k-algebra A and the ﬁxed k-basis B, that there are many possible monomial orderings on B and that diﬀerent orderings lead to diﬀerent (left) admissible systems.

2. (Left) Quasi-zero Elements and a (Left) Division Algorithm Let A = k[ai ]i∈Λ be a k-algebra with the ﬁxed k-basis αn 1 B = aα n ≥ 1, α · · · a ∈ I N i i1 in consisting of monomials and 1, and let be a well-ordering on B. In this section, we develop a division algorithm in A, in case (A, B, ) is a (left) admissible system in the sense of Deﬁnition 1.2(iv). Let f, g ∈ A be two nonzero elements, where  m    ci ui , ci ∈ k − {0}, ui ∈ B, u1 u2 · · · um , f =     i=1 (∗)  n     g = λj wj , λj ∈ k − {0}, wj ∈ B, w1 w2 · · · wn .   j=1

2. Quasi-zero Elements and a Division Algorithm

39

If we recall any existed division algorithm from the literature, then roughly speaking, an eﬀective division on f by g should be to “eliminate” the ui by using certain “monomial multiples” of w1 . Since the k-algebra A may contain divisors of zero, in order to have a feasible division algorithm on f by g, it is natural to introduce the following notion and notation. 2.1. Deﬁnition Let f be as in (∗) above. (i) If vf s = 0 for all v, s ∈ B but not both v and s are equal to 1, we say that f is a quasi-zero element of A. If vf = 0 for all v = 1 in B, we say that f is a left quasi-zero element of A. (ii) Put M(f ) = {ui | i = 1, ..., m}. If f is not a quasi-zero element in the sense of part (i) above, then, with respect to , we set there are v, s ∈ B such that vui s = 0 d LM (f ) = max ui ∈ M(f ) where either v = 1 or s = 1 and call LMd (f ) the division leading monomial of f . If f is not a left quasi-zero element in the sense of part (i) above, then, with respect to , we put LMd (f ) = max ui ∈ M(f ) there is v = 1 in B such that vui = 0 and call LMd (f ) the left division leading monomial of f . n Let v, w, s ∈ B and suppose vws = i=1 ci ui , where ci ∈ k − {0}, ui ∈ B. We may assume that u1 u2 · · · un . Then u1 = LM(vws), and   (A, B, ) is an admissible system and v = 1 or s = 1, w ≺ u1 where or  (A, B, ) is a left admissible system and v = 1, s = 1. This leads to the divisibility in an admissible system, respectively in a left admissible system. 2.2. Deﬁnition Let (A, B, ) be an admissible system, respectively a left admissible system. For w, u ∈ B, we say that u is divisible by w if u = LM(vws) for some v, s ∈ B, respectively u is divisible by w on left if u = LM(vw) for some v ∈ B. (Note that the above deﬁnition excludes the case where vws, respectively vw, is equal to 0.)

40

II. Gr¨ obner Bases in Algebras

Before having a division algorithm in A with respect to a (left) admissible system, several basic facts concerning quasi-zero and non-quasi-zero elements of A are summarized in the following lemma. 2.3. Lemma Let (A, B, ) be an admissible system (or a left admissible system). (i) Let w, u ∈ B. If w is a quasi-zero element, then u is divisible by w if and only if w = u. n (ii) If g is a (left) quasi-zero element, g = i=1 λi wi , then wi = 1, i = 1, ..., n. (iii) If G = {g1 , ..., gs } is any subset of quasi-zero elements of A, then the left and two-sided ideals of A generated by G, respectively, are actually the same. Moreover, the set of all quasi-zero elements of A forms an ideal of A with the zero multiplication on it. (iv) (a new version of Lemma 1.3) Let f be as in (∗) above. If vf s = 0 for some v, s ∈ B, where either v = 1 or s = 1, then LM(vf s) = LM(vLMd (f )s). If vf = 0 for some v = 1 in B, then LM(vf ) = LM(vLMd (f )). In the case where B is closed under multiplication or A is a domain we have LM(f ) = LMd (f ) = LMd (f ) and hence LM(vf s) = LM(vLM(f )s) and LM(vf ) = LM(vLM(f )). 2 Given an admissible system (A, B, ), we now proceed to establish a division algorithm in diﬀerent cases. Division on f by g n m Input: f = i=1 ci ui with u1 u1 · · · um , and g = j=1 λj wj with w1 w2 · · · wn . Start: If g is a quasi-zero element, then clearly none of ui in M(f ) (Deﬁnition 2.1) is divisible by any wj , j = 1, ..., n. In this case we conventionally have Output 1: write f = 0 + r with r = f , and say that the remainder of f for division by g is r which has the property that none of ui ∈ M(r) = M(f ) is divisible by any wj ∈ M(g). If g is not a quasi-zero element, we may assume that LMd (g) = wi for some 1 ≤ i ≤ n. Note that LM(f ) = u1 . If u1 is divisible by wi , then (0)

u1 = LM(vwi s), and wi ≺ u1 .

v, s ∈ B,

Note that since wi is the division leading monomial of g and LM(vwi s) = 0, we have vgs = λi vwi s + λi+1 vwi+1 s + · · · + λn vwn s

2. Quasi-zero Elements and a Division Algorithm

41

and by Lemma 2.3 LM(vwj s) ≺ LM(vwi s) = u1 , whenever j = i and vwj s = 0, and c1 vgs = c1 u1 + g with g ∈ A, LM(g ) ≺ u1 = LM(f ). λi Hence f= (1)

c1 c1 vgs + f1 with f1 = f − vgs λi λi

and LM(f1 ) ≺ u1 = LM(f ).

In the case that u1 is not divisible by wi = LMd (g), we write (2)

f = f1 + r1 with r1 = c1 u1

and consider the divisibility of LM(f1 ) = u2 by wi = LMd (g). Since is a well-ordering on B, after a ﬁnite number of repetition of the foregoing process (1)–(2) we arrive at p Output 2: f = =1 b v gs + r, where b ∈ k − {0}, v , s ∈ B, LM(v gs ) q LM(f ) whenever v gs = 0, and r = 0 or r = k=1 ck tk , ck ∈ k − {0}, tk ∈ B, with the property that LM(r) LM(f ) and none of the tk is divisible by w1 , w2 , ..., wi−1 , wi = LMd (g). Division on f by G m Input: f = i=1 ci ui with u1 u2 · · · um , G = {g1 , ..., gm } ⊂ A, where ni λij wij with wi1 wi2 · · · wini , i = 1, ..., m. By the 0 = gi = j=1 foregoing discussion we may assume that all the gi are not quasi-zero elements and that LMd (gi ) = wij for some 1 ≤ j ≤ ni , i = 1, ..., m. Start: with f0 = f , r0 = 0. If LM(f0 ) is divisible by some LMd (gi ), then put r1 = 0, c1 f1 = f0 − vi gi si , where λi ∈ k − {0} and LM(f1 ) ≺ LM(f ), λi and consider the divisibility of LM(f1 ) by G. If LM(f0 ) cannot be divided by any LMd (gi ), i = 1, ..., m, then put r1 = LT(f0 ), f1 = f0 − r1 , where LM(f1 ) ≺ LM(f0 ). Continue the above procedure for f1 until d Output 3: f = h=1 µh vh gh sh + r, where µh ∈ k − {0}, vh , sh ∈ B, gh ∈ G, d ≤ m, LM(vh gh sh ) LM(f ) whenever vh gh sh = 0, and r = 0 or r= cj tj , cj ∈ k − {0}, tj ∈ B, with the property that LM(r)

42

II. Gr¨ obner Bases in Algebras LM(f ) and none of the tj is divisible by any wi1 , ..., wij−1 , wij = LMd (gi ), i = 1, ..., m.

It is not diﬃcult to see that if (A, B, ) is a left admissible system then a similar division algorithm exists and we have the following d Output 4: Let f and G be as above. Then f = h=1 fh gh + r, where fh ∈ A, fh = 0, gh ∈ G, d ≤ m, LM(fh gh ) LM(f ) whenever fh gh = 0, and r = 0 or r = cj tj , cj ∈ k − {0}, tj ∈ B, with the property that LM(r) LM(f ) and none of the tj is divisible by any wi1 , ..., wij−1 , wij = LMd (gi ) (Deﬁnition 2.2) from left hand side, i = 1, ..., m. 2.4. Deﬁnition The r appeared in both Output 3 and Output 4 above is G called a remainder of f for division by G, denoted f . G

Note that the above f depends on the order of the gi in G and hence it is usually not unique. In the next section we will see that if G is a (left) Gr¨ obner basis in G A, then f is unique and is independent of the order of gi s in G.

3. (Left) Gr¨ obner Bases in a (Left) Admissible System Let (A, B, ) be a (left) admissible system in the sense of §1. Based on the division algorithm developed in §2, in this section we introduce the notion of a (left) Gr¨ obner basis for a (left) ideal in A. We start with an admissible system (A, B, ) and a two-sided ideal I of A. For f ∈ A, we may also consider the division on f by I. Maintaining the notation as in §2, after a ﬁnite number of division steps on f by the elements in I we arrive at d Output 5: f = i=1 λi ui fi vi + r, where λi ∈ k − {0}, ui , vi ∈ B, fi ∈ I, LM(ui fi vi ) LM(f ) whenever ui fi vi = 0, and r = 0 or r= µj tj , µj ∈ k − {0}, tj ∈ B, with the property that LM(r) LM(f ) and none of the tj is divisible by any w1 , ..., wi−1 , wi = n LMd (g), where g = i=1 λj wj is an arbitrary non-quasi-zero element in I with LMd (g) = wi . I

Write f for the remainder r of f for division by I in the above Output 3, i.e., I I f = r. Then it is easy to see that f is unique.

3. Gr¨ obner Bases

43

If we start with a left admissible system (A, B, ), replace I by a left ideal L of A, and use the division on left, then a similar argumentation as given above works for L. For a subset T in A we write

LM(T ) = LM(f ) ∈ B f ∈ T O(T ) = B − LM(T ).

3.1. Theorem With notation as above, let T represent an ideal or a left ideal of A. The following holds. (i) As a k-space, A = T ⊕ Spank O(T ), where the latter is the k-subspace in A spanned by O(T ). (ii) There is a k-space isomorphism between A/T and Spank O(T ). T (iii) For each f ∈ A, there is a unique r = f ∈ Spank O(T ) such that f − r ∈ T . T (iv) For f, g ∈ A, f = g T if and only if f − g ∈ T . T (v) For f ∈ A, f ∈ T if and only if f = 0. Proof We prove the theorem for the two-sided ideal T because a similar argumentation works for left ideals. (i) Let f be an element of A. By the division algorithm we have f = d i=1 λi vi fi si + r, where λi ∈ k − {0}, vi , si ∈ B, fi ∈ T , LM(vi fi si ) LM(f ) T whenever vi fi si = 0, and r = f . If r = 0, we may assume that r = b t , where b ∈ k − {0} and t ∈ B. Since r has the property that none of the m t is divisible by any w1 , ..., wi−1 , wi = LMd (g), where g = h=1 λh wh is an arbitrary non-quasi-zero element in T , it follows that none of the t belongs to LM(T ). This proves that r ∈ Spank O(T ) and hence f ∈ T + Spank O(T ). That T ∩Spank O(T ) = {0} is clear by the deﬁnition of O(T ). Therefore, A = T ⊕ Spank O(T ). (ii) By (i) above, this is clear. (iii) Take f ∈ A and consider the division of f by T . If f = λi vi fi si + r = λj wj fj uj + r , namely there are two remainders r and r for the division of f by T , then it follows from the proof of (i) that r − r ∈ T ∩ Spank O(T ) = {0}. Hence r = r , as desired. (iv) and (v) follow from the foregoing (i)–(iii). 2 Considering a generating set for an ideal I, respectively for a left ideal L, the above discussion leads the deﬁnition of a Gr¨ obner basis for I, respectively a left Gr¨ obner basis for L. 3.2. Deﬁnition Let (A, B, ) be a (left) admissible system, and G = {gi }i∈J ⊂ A.

44

II. Gr¨ obner Bases in Algebras

(i) For the two-sided ideal of A generated by G, denoted I = G, if every f ∈ I has a presentation λi ui gi vi , λi ∈ k − {0}, ui , vi ∈ B, gi ∈ G f= i

such that LM(ui gi vi ) LM(f ) whenever ui gi vi = 0, then G is called a Gr¨ obner basis of I. (ii) For the left ideal of A generated by G, denoted L = G], if every f ∈ L has a presentation f= hj gj , λj ∈ k − {0}, hj ∈ A, gj ∈ G j

such that LM(hj gj ) LM(f ) whenever hj gj = 0, then G is called a left Gr¨ obner basis of L. The presentation for f in (i), respectively in (ii) above, is called a Gr¨ obner representation of f , respectively a left Gr¨ obner representatin of f by G. (iii) A (left) Gr¨ obner basis G is called reduced if, for all j, LC(gj ) = 1 and no nonzero term in gj is divisible by any LM(gi ) for any j = i. Remark (i) In the above deﬁnition the reason that we do not reqiure G to be ﬁnite is obvious, namely, not every A is (left) Noetherian, and consequently, not every (left) ideal in A is ﬁnitely generated. Moreover, from [Mor2] or [Ufn] (or the example given in the next section) one may see that, even if we start with a single element, a noncommutative version of Buchberger’s algorithm may produce an inﬁnite Gr¨ obner basis. (ii) At this stage, one might be wondering the case where G = {gi }i∈J consists of quasi-zero elements. Indeed, it easily follows from Lemma 2.3 that G is a (left) Gr¨ obner basis for the (left) ideal of A generated by G. Moreover, let I, respectively L, be the ideal, respectively the left ideal of A, generated by the subset G = {gi }i∈J ∪ {gk }k∈T where {gi }i∈J consists of quasi-zero elements and {gk }k∈T does not contain quasi-zero element. Suppose that G is a Gr¨obner basis of I, respectively a left Gr¨obner basis of L. Then for any f ∈ I, respectively obner representation of f is completely determined f ∈ L but f ∈ {gi }i∈J , a Gr¨ by {gk }k∈T . Theorem 3.1 now makes the algorithmic feature of a (left) Gr¨ obner basis more clearer. obner basis for an ideal I, or a left 3.3. Proposition Let G = {gi }i∈J be a Gr¨ G Gr¨ obner basis for a left ideal L. Let f ∈ A and f the remainder of f for division by G. G G (i) f is independent of the order of the gi ∈ G, and f is unique in A.

4. Gr¨ obner Bases in Various Contexts

45

G

(ii) f ∈ I or f ∈ L if and only if f = 0.

2

3.4. Proposition Let G = {gi }i∈J be a subset of A and I, respectively L, the two-sided ideal of A, respectively the left ideal of A, generated by G. Then G is G a Gr¨ obner basis of I, respectively a left Gr¨obner basis of L if and only if f = 0 for every f ∈ I, respectively for every f ∈ L. 2 It is easy to see that Deﬁnition 3.2 includes all well-known deﬁnitions of (left) Gr¨ obner bases in the literature. Historically, the successful development of commutative Gr¨obner basis theory was motivated by a study of the problem in determining the k-basis of the vector space k[x1 , ..., xn ]/I by using a (ﬁnite) generating set of I, where I is an ideal of the commutative polynomial algebra k[x1 , ..., xn ] over a ﬁeld k (see [Eis] 15.6, History of Gr¨ obner Bases). Theorem 3.1 shows, not exceptionally, that various noncommutative Gr¨ obner basis theories fully share this feature as a starting point. The algorithmic and structural aspects of Theorem 3.1 will become more apparent in CH.III when we apply the very noncommutative Gr¨ obner bases of ([Berg], [Mor1–2]) to algebras deﬁned by relations.

4. (Left) Gr¨ obner Bases in Various Contexts In the last section, (left) Gr¨ obner bases are deﬁned in a quite general extent. Since various noncommutativities lead to various noncommutative algebraic structures, from a practical point of view, it is then necessary to indicate the main diﬀerences of (left) Gr¨obner bases in various contexts. Let I be a two-sided ideal in one of the following algebras: A = k[x1 , ..., xn ], the commutative polynomial algebra in x1 , ..., xn , A = kX1 , ..., Xn , the noncommutative free algebra in X1 , ..., Xn . Suppose that (A, B, ) is an admissible system with B the standard k-basis. Then the following results are well known. (i) A subset G = {gi }i∈J is a Gr¨ obner basis of I if and only if LM(f ) | f ∈ I = LM(gi ) | gi ∈ G. (Indeed the latter property has been used as the deﬁnition of a Gr¨ obner basis for I, e.g., [BW], [CLO ], [Mor2]. In [Gr] this is also used as the deﬁnition of a Gr¨ obner basis for a two-sided ideal in an algebra, possibly with no 1 but possibly with divisors of zero.) (ii) If G = {wi }i∈J ⊂ B and I is the ideal generated by G, then G is a Gr¨ obner basis of I.

46

II. Gr¨ obner Bases in Algebras

Note that the above (i)–(ii) are based on the fact that • the k-basis B used in those cases is closed under multiplication. For an arbitrary associative algebra A, however, the above property (•) no longer necessarily exists. The following 4.1–4.2 brings us both good and bad news on the impact of this fact more precisely in a general setting. 4.1. Proposition Let (A, B, ) be an admissible system, and let I, respectively L, be an ideal, respectively a left ideal, generated by G = {wi }i∈J ⊂ B. Suppose B satisﬁes: (∗)

w, v ∈ B implies λwv ∈ B for some λ ∈ k.

Then G is a Gr¨ obner basis for I, respectively a left Gr¨obner basis for L. A similar statement holds for a left admissible system and any left ideal L. If B does not satisfy the condition (∗) above, then G is not necessarily a Gr¨ obner basis for I, respectively not necessarily a left Gr¨obner basis for L. Proof Let I = G with G = {wi }i∈J . If f ∈ I we may assume that f = s i=1 hi wi gi for some hi , gi ∈ A and that w1 w2 · · · ws , ni hi = λij tij , ti1 ti2 · · · tini , j

gi =

mi

λid vid , vi1 vi2 · · · vimi .

d

Thus, LM(hi ) = ti1 and LM(gi ) = vi1 . By the assumption (∗) on B, there are λi ∈ k, i = 1, ..., s, such that LM(hi wi gi ) = λi ti1 wi vi1 ,

i = 1, ..., s.

Since for i = 1, ..., s we have ti1 wi vi1 ti1 wi+ vi1 t(i+)1 wi+ vi1 t(i+)1 wi+ v(i+)1 , it follows that ti1 wi vi1 = t(i+)1 wi+ v(i+)1 , i = 1, ..., s − 1, = 1, ..., s − i. Thus, LM(hi wi gi ) LM(f ) whenever hi wi gi = 0. This shows that f has a Gr¨ obner representation. Hence, {wi }i∈J is a Gr¨obner basis for I. A similar argumentation works for L. Now let us consider the ﬁrst Weyl algebra A1 (k) = k[x, ∂] (CH.I §5) for ﬁnding a counterexample in the case where B does not have the property (∗). First note that A is a solvable polynomial algebra with respect to x >grlex ∂ (see later §7). Let L = g1 , g2 ] be the left ideal generated by the standard monomials g1 = x, g2 = ∂ 2 in A. An easy calculation shows that the S-element of g1 and g2

4. Gr¨ obner Bases in Various Contexts

47

in A, denoted S(g1 , g2 ) (see §7 for the deﬁnition), equals 2∂ which can neither be divided by LM(g1 ) = x nor be divided by LM(g2 ) = ∂ 2 . Hence {g1 , g2 } is not a left Gr¨ obner basis for L (see §5 Theorem 5.1 below). We leave the counterexample for a two-sided ideal in A to the reader. (Hint: Consider I = x, ∂.) 2 In the above proposition, if (A, B, ) is only a left admissible system and B does not have the property (∗), then we are sure that G is not necessarily a left Gr¨ obner basis for L = G]. But the author failed to have a counterexample. 4.2. Proposition (i) Let (A, B, ) be an admissible system, and let I be an ideal of A. If a subset G = {gi }i∈J ⊂ I is such that LM(f ) | f ∈ I = LM(gi ) | gi ∈ G, then G is a Gr¨ obner basis for I. (ii) Let (A, B, ) be a left admissible system, and let L be a left ideal of A. If a subset G = {gi }i∈J ⊂ L is such that LM(f ) | f ∈ L] = LM(gi ) | gi ∈ G], then G is a left Gr¨ obner basis for L. The converse of the above (i) and (ii) is not necessarily true. Proof We only give the proof for (i) and a similar argumentation will do for (ii). We prove that under the assumption of (i) every element f ∈ I has a Gr¨obner representation by G. To see this, consider the division of f by G. Then we have f = vi gi si + r, where vi , si ∈ B, LM(vi gi si ) LM(f )i whenever vi gi si = 0, G and r = f . If r = 0, then since r ∈ I and 0 = LM(r) ∈ LM(f ) | f ∈ I = LM(gi ) | gi ∈ G. Thus, LM(r) = i,j wij LMd (gi )uij with wij , uij ∈ B and hence LM(r) = LM(wij LMd (gi )uij ) for some gi . But this contradicts that G

r = f . Therefore r = 0, i.e., f has a Gr¨obner representation by G, as desired. To see why the converse of (i) and (ii) are not necessarily true, again, let us consider the ﬁrst Weyl algebra A1 (k) = k[x, ∂] for ﬁnding a counterexample for (ii) and leave the counterexample for (i) to the reader. Let L = g] be the left ideal generated by g = x + 1 in A. Using the monomial ordering x >grlex ∂, it follows from later §5 Theorem 5.1 and §7 Proposition 7.4 that G = {g = x + 1} is a left Gr¨ obner basis for L and LM(g) = x. Now ∂g = x∂ + ∂ + 1 = h ∈ L implies that LM(h) = x∂ ∈ LM(L). Suppose x∂ = f x for some f ∈ A. Then since it is easy to see that f ◦ x = 0, i.e., the action of the operator f on x is zero, we have f = βi ≥2 λi xαi ∂ βi . Thus   x∂ = f x =  λi xαi ∂ βi  x βi ≥2

=

λi xαi (x∂ βi + βi ∂ βi −1 )

βi ≥2

=

βi ≥2

λi xαi +1 ∂ βi +

βi ≥2

λi βi xαi ∂ βi −1 .

48

II. Gr¨ obner Bases in Algebras

Note that {xα ∂ β | (α, β) ∈ IN 2 } is a k-basis for A. It follows that f = 0. This proves that LM(h) = x∂ ∈ LM(g)] = x]. 2 Another typical feature of Gr¨ obner bases in a commutative polynomial algebra is that any single polynomial forms a Gr¨ obner basis for the principal ideal generated by that polynomial; and this is even true for (noncommutative) solvable polynomial algebras, as one may easily verify by deﬁnitions (see §7). The example given below shows that this nice property does not exist in general (also see [Ufn]). Example (i) Let A = kX, Y be the free algebra in two variables X and Y , and let (A, B, ≥grlex ) be the admissible system as given in §1, where X >grlex Y . Consider the ideal I = g = X 2 − XY generated by the single element g = X 2 − XY . Then we claim that (a) X 2 − XY = I = XY n X − XY n+1 n≥0 . obner basis for I. (b) {g = X 2 − XY } is not a Gr¨ (c) G = {gn = XY n X − XY n+1 }n≥0 is an inﬁnite Gr¨ obner basis for I. Proof (a) We do induction on n. If n = 0, then g0 = XY 0 X − XY 0+1 = X 2 − XY = g ∈ I. Hence I ⊂ XY n X − XY n+1 . Suppose XY n−1 X − XY n = gn−1 ∈ I for n ≥ 1. We prove that XY n X − XY n+1 = gn ∈ I. Note that X 2 − XY = g ∈ I implies both X 2 Y n − XY n+1 and −X 2 Y n−1 X + XY n X are in I. It follows that −X(XY n−1 X − XY n ) + (XY n X − XY n+1 ) ∈ I. By the induction hypothesis, XY n X − XY n+1 = gn ∈ I, as desired. (b) From part (a) we know that XY X − XY 2 = g1 ∈ I with LM(g1 ) = XY X. Note that LM(g) = X 2 . It is clear that g1 cannot have a Gr¨ obner representation by g. (c) This can be veriﬁed either by a programme of [FG], or by an algorithm of [Mor2], or by directly checking that every S-element S(gi , gj ) has a Gr¨obner representation by G (see §5 blow). 2 Remark It is an interesting exercise to check what the result will be in the above example if the ordering Y >grlex X is used. (Also see CH.III Corollary 1.3.)

5. (Left) S-elements and Buchberger Theorem After giving the deﬁnition and deriving some basic properties for (left) Gr¨ obner bases in (left) admissible systems, one may also expect to see the possibility

5. Buchberger Theorem

49

of having a version of Buchberger’s algorithm in order to produce a (left) Gr¨ obner basis in a given (left) admissible system. Since our deﬁnition of (left) Gr¨ obner basis is compatible with all existed deﬁnitions, knowing from the literature, this is only a matter of learning the mathematics principle behind Buchberger’s algorithm in the commutative case, carefully. Let A = k[x1 , ..., xn ] be the commutative polynomial k-algebra in n variables and (A, B ) an admissible system as given in §1 Example (i). Let α = αn 1 (α1 , ..., αn ), β = (β1 , ..., βn ) ∈ IN n and f, g ∈ A with LM(f ) = xα = xα 1 · · · xn , β1 β βn LM(g) = x = x1 · · · xn . Put γ = (γ1 , ..., γn ) with γi = max{αi , βi }. Then xγ−α xα = xγ−β xβ

(∗)

and the S-polynomial of f and g is deﬁned as: S(f, g) =

xγ−α xγ−β f− g. LC(f ) LC(g)

Theorem (Buchberger) Let F = {f1 , ..., fs } be a generating set of the ideal F

I ⊂ A. Then F is a Gr¨obner basis for I if and only if S(fi , fj ) fi = fj ∈ F.

= 0 for all

Proof We adopt the proof of this theorem given in [AdL], and ﬁrst prove a preliminary result. αn 1 • Let f1 , ..., fs ∈ k[x1 , ..., xn ] be such that LM(fi ) = xα = xα 1 · · · xn = 0 for s α all i = 1, ..., s. Let f = i=1 ci fi with ci ∈ k. If LM(f ) ≺ x , then f is a k-linear combination of S(fi , fj ), 1 ≤ i < j ≤ s.

Write fi = ai xα + lower terms, ai ∈ k. Then the assumption yields that s 1 1 i=1 ci ai = 0. Since LM(fi ) = LM(fj ), it follows that S(fi , fj ) = ai fi − aj fj . Thus f

c1 f1 + · · · + cs fs

1 1 = c1 a1 f1 + · · · + cs as fs a1 as

1 1 1 1 = c1 a1 f1 − f2 + (c1 a1 + c2 a2 ) f2 − f3 + · · · a1 a2 a2 a3

1 1 1 +(c1 a1 + · · · + cs−1 as−1 ) fs−1 − fs + (c1 a1 + · · · + cs as ) fs as−1 as as = c1 a1 S(f1 , f2 ) + (c1 a1 + c2 a2 )S(f2 , f3 ) + · · ·

=

+(c1 a1 + · · · + cs−1 as−1 )S(fs−1 , fs ),

50

II. Gr¨ obner Bases in Algebras

because c1 a1 + · · · + cs as = 0. This proves the conclusion (•). We are now ready to prove Buchberger theorem. F If F is a Gr¨ obner basis for I, then S(fi , fj ) = 0 for all i = j by Proposition 3.4 since S(fi , fj ) ∈ I. F

Conversely, let us assume that S(fi , fj ) = 0 for all i = j. We will prove that every f ∈ I has a Gr¨ obner representation with respect to F. Write f = s h f and put i i i=1 αn 1 xα = xα 1 · · · xn = max LM(hi fi ) 1 ≤ i ≤ s . If xα LM(f ), we are done. Otherwise, suppose LM(f ) ≺ xα and let T = i LM(hi fi ) = xα . For i ∈ T , write hi = ci LM(hi )+ lower terms. Set g = i∈T ci LM(hi )fi . Then LM(LM(hi )fi ) = xα for all i ∈ T , but LM(g) ≺ xα . By the preliminary result obtained above, there exists dij ∈ k such that dij S(LM(hi )fi , LM(hj )fj ). g= i,j∈T,i =j

Now, xα = lcm (LM(LM(hi )fi ), LM(LM(hj )fj )), so S (LM(hi )fi , LM(hj )fj )

=

= =

xα LM(hi )fi − LT(LM(hi )fi ) xα LM(hj )fj − LT(LM(hj )fj ) xα xα fi − fj LT(fi ) LT(fj ) α x S(fi , fj ), xβ F

where xβ = lcm(LM(fi ), LM(fj )). By the assumption, S(fi , fj ) hence it is not hard to see from this last equation that

= 0, and

F

S(LM(hi )fi , LM(hj )fj ) = 0. This yields a representation S(LM(hi )fi , LM(hj )fj ) =

s

hij f ,

=1

where by the division algorithm max LM(hij f ) 1 ≤ ≤ s

=

LM(S(LM(hi )fi , LM(hj )fj ))

≺

max {LM(LM(hi fi ), LM(LM(hj fj )}

=

xα .

5. Buchberger Theorem

51

Substituting these expressions into g above, and g into f , we get f = with max{LM(hi fi ) | 1 ≤ i ≤ s} < xα . This is a contradiction.

s i=1

hi fi , 2

Remark From a structural point of view, the above proof indeed tells us that • Buchberger theorem is equivalent to: F is a Gr¨ obner basis if and only if every obner representation by F. S(fi , fj ) has a Gr¨ The following algorithm tests whether F is a Gr¨obner basis or not; if not, it produces a ﬁnite Gr¨ obner basis containing F. Buchberger’s algorithm Input: F = {f1 , ..., fs } Output: G = {g1 , ..., gm }, a Gr¨obner basis for I Initialization: G := F, S := {{fi , fj } | fi = fj ∈ F} While S = ∅ Do Choose any {fi , fj } ∈ S S := S − {{fi , fj }} G

S(fi , fj ) = r If r = 0 Then S := S ∪ {{u, r} | for all u ∈ G} G := G ∪ {r} Although it is a well-know fact in the commutative case, it is necessary to see why this algorithm terminates in a ﬁnite steps and then the output result gives us a ﬁnite Gr¨ obner basis for I. First suppose to the contrary that the algorithm does not terminate. Then, as the algorithm progresses, we construct a set Gi strictly larger than Gi−1 and obtain a strictly increasing inﬁnite sequence G1 ⊂ G2 ⊂ G3 ⊂ · · · . Each Gi is obtained from Gi−1 by adding some r ∈ I to Gi−1 , where r is the nonzero remainder, with respect to Gi−1 , of an S-polynomial of two elements of Gi−1 . By the property of the remainder we know that LM(r) cannot be divided by any of the LM(f ) with f ∈ Gi−1 . Writing LM(Gi ) for the ideal generated by the LM(f ) with f ∈ Gi , we get LM(G1 ) ⊂ LM(G2 ) ⊂ LM(G3 ) ⊂ · · · . This is a strictly asccending chain of ideals in A which contradicts the Noetherian property of A. Now we have F ⊆ G ⊆ I, and hence I = f1 , ..., fs ⊆ g1 , ..., gt ⊆ I. Thus G is a generating set for the ideal I. Moreover, if gi , gj are polynomials in G, G

then S(gi , gj ) = 0 by construction. Therefore G is a Gr¨ obner basis for I by Buchberger theorem. 2

52

II. Gr¨ obner Bases in Algebras

What did we learn so far? (i) To have a noncommutative version of Buchberger’s algorithm for a given (left) admissible system, it is then clear that we have to ﬁrst look for an appropriate analogue of the S-polynomial. For a given admissible system (A, B, ), where A = [ai ]i∈Λ , and for f, g ∈ A with β1 βm αn g 1 LMd (f ) = aα i1 · · · ain = w, LM (g) = aj1 · · · ajm = u ∈ B,

we say that an S-element about f and g, denoted S(f, g), is constructable provided f and g are not quasi-zero elements (see §3 Remark(ii) for the reason) and there are s, r, v, l ∈ B such that LM(sLMd (f )r) = LM(vLMd (g)l) = 0, or LM(sLMd (f )r) = LM(vLMd (g)l) = 0. If it is the case, then we write S(f, g) = sf r − vgl or S(f, g) =

µ sf r − vgl λ

where λ = LC(s(f )r) and µ = LC(v(g)l) in the “or” case. Note that if S(f, g) is constructable, then the quaternion (s, r, v, l) may not be unique, or in other words, taking a unique “l.c.m” (as in the commutative case) is not always possible, that is, all “overlaps” of w and u must be considered in order to construct all S-elements about f, g (e.g., see [Gr], [Mor2]). This may be illustrated by looking at A = kX, the free algebra on X = {Xi }i∈Λ with B its standard k-basis. For u, w ∈ B, there are two possible relations that should be considered: (a) If u and w have proper overlaps, then sur = vwl for some s, r, v, l ∈ B; (b) If (a) is excluded, then trivially 1uw = uw1, wu1 = 1wu. For instance, if w = X1 X2 X3 X2 X3 and u = X3 X2 X3 X4 , we must consider both X1 X2 (X3 X2 X3 ) with (X3 X2 X3 )X4 as one overlap and X1 X2 X3 X2 (X3 ) with (X3 )X2 X3 X4 as another overlap. Now use ≥gr on B. Then, for f = w + X12 + 1 and g = u + 5X2 X3 X1 + 2X3 X2 , we have LMd (f ) = LM(f ) = w, LMd (g) = LM(g) = u, and S1 (f, g) = f X4 − X1 X2 g, S2 (f, g) = f X2 X3 X4 − X1 X2 X3 X2 g. A similar argumentation works for constructing left S-elements in a left admissible system. (ii) We also note that the idea of using S-elements in Buchberger theorem is indeed not restricted to ﬁnite set of generators. More precisely, a version of Buchberger theorem may always be mentioned as follows.

5. Buchberger Theorem

53

5.1. Theorem (noncommutative version of Buchberger theorem) Let (A, B, ) be a (left) admissible system and G = {gi }i∈J a subset of non-quasi-zero elements in A. Suppose that, for each pair (gi , gj ), a (left) S-element S(gi , gj ) is constructable, i, j ∈ J. Then G is a (left) Gr¨obner basis in A if and only if every (left) S-element S(gi , gj ) has a (left) Gr¨ obner representation by G, or in other G

words, if and only if S(gi , gj ) = 0. 2 (iii) Finally, for a given ﬁnite subset F = {f1 , ..., fs } in a (left) admissible system (A, B, ), once the S-elements S(fi , fj ) are well-constructed and form a ﬁnite set, or an “eﬀective” ﬁnite set may be “selected” (as in [Mor2], see CH.III §1), an analogue of the Buchberger’s algorithm on A can be (at least theoretically ﬁrst) reached. And as we have seen from the commutative Buchberger’s algorithm, if A is not (left) Noetherian, the While loop in the noncommutative algorithm may not stop in a ﬁnite steps and may consequently yield an inﬁnite Gr¨ obner basis. For instance, if we work on the free algebra A = kX, Y and set X >grlex Y , then, starting with the element g0 = X 2 − XY yields g1 = XY X − XY 2 = S(g0 , g0 ) and both the programme of [FG] and the algorithm of [Mor2] produce the Gr¨ obner basis G = {gn = XY n X − XY n+1 }n≥0 (see §4). We ﬁnish this section by a couple of examples. Example (i) Set X >grlex Y in the free algebra kX, Y and consider G = {g1 , g2 } ⊂ kX, Y where g1 = X 2 Y − αXY X − βY X 2 − γX g2 = XY 2 − αY XY − βY 2 X − γY,

α, β, γ ∈ k.

Then since LM(g1 ) = X 2 Y , LM(g2 ) = XY 2 , and X · LM(g2 ) · 1 = 1 · LM(g1 )Y , it follows that S(g2 , g1 ) = X · g2 · 1 − 1 · g1 · Y = β(Y X 2 Y − XY 2 X) = 1 · g2 · X − Y · g1 · 1. G

It is easy to see that S(g2 , g1 ) = 0. Hence G, which is the set of deﬁning relations of the down-up algebra A(α, β, γ) (CH.I §5, Example (ix)), is a Gr¨ obner basis i j k in kX, Y . By Theorem 3.1, A(α, β, γ) has a k-basis {u (du) d | i, j, k ≥ 0}. (ii) Let > be a well-ordering on the set Λ. Set in the free algebra kX with X = {Xi }i∈Λ the following Ri = Xi2 , Rk = Xk X + X Xk , k > , G = {Ri , Rk }.

54

II. Gr¨ obner Bases in Algebras

If we consider the admissible system (kX, B, ≥grlex ) as deﬁned in §1 Example (iv), where Xk >grlex X if k > , then since LM(Ri ) = Xi2 , LM(Rk ) = Xk X , it is not hard to see that every S-element obtained by considering diﬀerent overlap has a Gr¨ obner representation by G, for instance for k = i = , k > , Xi2 Xk X − Xi2 (Xk X + X Xk ) = −Xi2 X Xk ; for k > i, Xk Xi2 − (Xk Xi + Xi Xk )Xi = −Xi (Xk Xi + Xi Xk ) + Xi2 Xk . Hence G, which is the set of deﬁning relations of the Grassmann algebra G(x) (CH.I §5, Example (viii)), is a Gr¨ obner basis in kX. By Theorem 3.1, G(x) has a k-basis {xi1 xi2 · · · xim | i1 < i2 < · · · < im , ij ∈ Λ}

6. (Left) Dickson Systems and (Left) G-Noetherian Algebras Let (A, B, ) be a (left) admissible system in the sense of §1. In this section we characterize the G-Noetherian property of A in the sense: A is G-Noetherian if every two-sided ideal of A has a ﬁnite Gr¨ obner basis; and A is left G-Noetherian if every left ideal has a ﬁnite Gr¨ obner basis. As a result, we formulate the (left) Dickson systems (A, B ) in which every (left) ideal has a ﬁnite (left) Gr¨ obner basis. Given a (left) admissible system (A, B, ), the divisibility (Deﬁnition 2.2) yields another order on B as follows: If w, u ∈ B and w = u, then w ≺ u if and only if u = LM(vws) for some v, s ∈ B with v = 1 or s = 1. (For a left admissible system, if and only if u = LM(vw) for some v ∈ B with v = 1). Note that w u actually implies w = u or w ≺ u by Deﬁnition 1.2 (MO2) (but the converse is not necessarily true). Thus, since is a well-ordering, we have the following fact. 6.1. Lemma If S is any nonempty subset of B, then the set of minimal elements in S with respect to , denoted min{S, }, is nonempty. 2 To characterize the existence of ﬁnite Gr¨obner bases (also see the remark given in the end of this section), we assume from now on that

6. Dickson Systems and G-Noetherian Algebras

55

(∗) is a partial ordering on B, i.e., it is reﬂexive, transitive and antisymmetric. Maintaining the notation of §2, if T is a subset of A, we put LM(T ) = LM(f ) f ∈ T ⊂ B, LMd (T ) = LMd (f ) f ∈ T not a quasi-zero element ⊂ B, LMd (T ) = LMd (f ) f ∈ T not a left quasi-zero element ⊂ B. It is clear that if B is closed under multiplication or if A is a domain then LM(T ) = LMd (T ) = LMd (T ). 6.2. Theorem Let F = {f1 , ..., fs } be an s-tuple of nonzero elements in a two-sided ideal I, respectively in a left ideal L of A. With notation and the assumption as above, the following are equivalent. (i) F is a Gr¨ obner basis of I, respectively a left Gr¨obner basis of L; (ii) min{LM(I) ∪ LMd (F), } = min{LM(F) ∪ LMd (F), }, respectively min{LM(L) ∪ LMd (F), } = min{LM(F) ∪ LMd (F), }. Proof We only need to prove the equivalence for I because a similar argumentation works for L. To be convenient, we put min{LM(I) ∪ LMd (F), } = U,

min{LM(F) ∪ LMd (F), } = V.

(i) ⇒ (ii). Suppose that F = {f1 , ..., fs } is a Gr¨ obner basis for I. We may assume that each fi is not a quasi-zero element (see the remark under Proposition 3.2 for the reason). For u ∈ U , if u = LMd (fi ) for some fi ∈ F, then clearly u ∈ V . If u = LM(g) for some g ∈ I, g ∈ F, consider a Gr¨obner representation of g with respect to F: n g= λi ui fi vi with LM(ui fi vi ) LM(g) whenever ui fi vi = 0. i=1

By Lemma 1.3, we have LM(g) = LM(uj fj vj ) for some j(1 ≤ j ≤ n), and it follows from Lemma 2.3 that LM(g) = LM(uj LMd (fj )vj ). Thus, LMd (fj ) LM(g) = u and hence u ∈ V . This proves U ⊂ V . Conversely, let v ∈ V . Suppose that g ∈ I is such that LM(g) ≺ v. Then clearly g ∈ F. Consider a Gr¨ obner representation of g with respect to F: g=

n

λi ui fi vi with LM(ui fi vi ) LM(g) whenever ui fi vi = 0.

i=1

By Lemma 1.3, we have LM(g) = LM(uj fj vj ) for some j(1 ≤ j ≤ n), and it follows from Lemma 2.3 that LM(g) = LM(uj LMd (fj )vj ). Thus, LMd (fj )

56

II. Gr¨ obner Bases in Algebras

LM(g) ≺ v. By the assumption (∗) made on we get LMd (fj ) ≺ v, contradicting the minimality of v. This shows that v ∈ U and hence V ⊂ U . Therefore U =V. (ii) ⇒ (i). Let f ∈ I be nonzero. We prove that f has a Gr¨ obner representation s with respect to F. By the division algorithm (§2) we have f = i=1 vi fi si + r, F

where LM(vi fi si ) LM(f ) whenever vi fi si = 0, and r = f . If r = 0, then since r ∈ I there exists some w ∈ min{LM(I) ∪ LMd (F), } such that w LM(r). But since U = V , there is some fj ∈ F such that w = LM(fj ) or there is some fi ∈ F such that w = LMd (fi ). It is clear that the latter case cannot F happen because r = f . Thus, in the ﬁrst case LM(fj ) = LMd (fj ). Then LM(fj ) is a quasi-zero element and consequently it cannot divide any monomial in B except for itself, i.e., w LM(r) only when w = LM(fj ) = LM(r). But F

LM(fj ) = LM(r) can never happen since r = f . This proves r = 0, as desired. 2 From the above theorem we retain that if F = {f1 , ..., fs } is a left Gr¨obner baobner basis of the ideal sis of the left ideal L = f1 , ..., fs ], respectively a Gr¨ I = f1 , ..., fs , then min{LM(L) ∪ LMd (F), }, respectively min{LM(I) ∪ LMd (F), }, is a ﬁnite set. One may hope that the converse of this consequence is also true. Indeed, we have the following more ﬂexible result. 6.3. Proposition Let I, respectively L, be as above. With notation as in Theorem 6.2, if one of the following conditions is satisﬁed: (i) min{LM(I), }, respectively min{LM(L), }, is ﬁnite; (ii) min{LM(I) ∪ LMd (F), }, respectively min{LM(L) ∪ LMd (F) }, is ﬁnite, then I has a ﬁnite Gr¨ obner basis, respectively L has a ﬁnite left Gr¨ obner basis in A. Proof Suppose that the condition (i) is satisﬁed. (If (ii) is satisﬁed, the proof is similar.) We may assume that the ﬁnitely many minimal elements in min{LM(I), } are w1 , ..., wn . For each wi , we may choose fwi ∈ I with wi = LM(fwi ). Put F = fwi i = 1, ..., n . We now prove that U = min{LM(I) ∪ LMd (F), } = min{LM(F) ∪ LMd (F), } = V. If u ∈ U and u = LM(g) for some g ∈ I, then there is some wi = LM(fwi ) such that wi LM(g) = u. Since fwi ∈ F, it follows that u = wi ∈ V . If u = LMd (fwj ) = wj for some fwj ∈ F, then clearly u ∈ V . This proves U ⊂ V . Conversely, suppose v ∈ V and g ∈ I such that LM(g) ≺ v. Then there exists some wi = LM(fwi ) such that wi LM(g) ≺ v. By the assumption (∗) made

6. Dickson Systems and G-Noetherian Algebras

57

on this is a contradiction. Thus v ∈ U and hence V ⊂ U . Therefore, U = V . It follows from Theorem 6.2 that F is a Gr¨obner basis of I. A similar argumentation for L ﬁnishes the proof. 2 6.4. Deﬁnition Suppose that ≤ is a quasi-ordering on a set M (i.e., ≤ is reﬂexive and transitive). ≤ is called a Dickson quasi-ordering if every subset N of M has a ﬁnite Dickson basis with respect to ≤, that is, there exists a ﬁnite subset S ⊂ N such that for every u ∈ N there is some s ∈ S with s ≤ u. We refer to ([BW] Ch.4) for details concerning Dickson quasi-orderings. The property of a Dickson partial ordering mentioned in the following proposition is fundamental for the applications of the foregoing results. 6.5. Proposition ([BW] Ch.4 Corollary 4.43) Let ≤ be a Dickson partial ordering on M . Then every nonempty subset N of M has a unique minimal ﬁnite basis S, i.e., a ﬁnite basis S such that S ⊆ C for all other bases C of N . S consists of all minimal elements of N . 2 6.6. Theorem Let (A, B, ) be a (left) admissible system and the ordering on B induced by the divisibility in A. If is a Dickson partial ordering on B, then every ideal, respectively every left ideal of A, has a ﬁnite Gr¨obner basis, respectively a ﬁnite left Gr¨ obner basis. Consequently, A is G-Noetherian and left G-Noetherian. Proof This follows from Proposition 6.5 and Proposition 6.3 immediately.

2

6.7. Deﬁnition Let (A, B, ) be a (left) admissible system such that the ordering on B induced by the divisibility in A is a Dickson partial ordering. Then we call (A, B, ) a (left) Dickson system. Remark So far our discussion has been based on Deﬁnition 1.2 in the case that 1 is contained in the k-basis B of A. If one may pay a little more attention to each of the previous sections, however, it is not hard to see that everything we have done may be modiﬁed into the case where 1 ∈ B (i.e., by simply modifying a condition used on B in [Gr]: If w ∈ B and w is not a quasi-zero element, then there are u, v ∈ B such that w = LMd (uwv), so that the division by nonquasi-zero elements is reﬂexive). The reason that we specify this point here is that in practice one does need to construct Gr¨obner bases by using a k-basis B not containing 1, for instance, in the path algebras that are widely used in representation theory. To be convenient we now give the deﬁnition of a path algebra as follows (e.g., see [Gr]). Let Γ be a ﬁnite directed graph with Γ0 = {v1 , ..., vs } = vertices and Γ1 =

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{a1 , ..., am } = arrows. We let B = { ﬁnite directed paths in Γ}. Then Γ0 ⊂ B are viewed as paths of length 0. The path algebra, A = kΓ, is the k-vector space with basis B. Note that usually 1 ∈ B. Multiplication on A is deﬁned as: If p, q ∈ B, then 0 if terminus of p = origin of q, p·q = pq ∈ B if terminus of p = origin of q. That is, multiplication of paths is given by concatenation, when this makes sense. The length-lexicographic order on B, denoted >lglex , is deﬁned as: First ﬁx an arbitrary order on the vertices and arrows, say, vs >lglex · · · >lglex v1 and am >lglex · · · >lglex a1 . If p, q ∈ B, then q >lglex p if either the length of q > the length of p or q = aj1 · · · ajd and p = ai1 · · · aid then there is an , 1 ≤ ≤ d such that aj1 = ai1 , ..., aj−1 = ai−1 but aj >lglex ai . It is not diﬃcult to verify that (A, B, >lglex ) is an admissible system in the sense of §1.

7. Solvable Polynomial Algebras In this section, we introduce the class of solvable polynomial algebras in the sense of [K-RW], which provides all (left) Dickson systems we will mostly work with in later chapters, and from which we also construct Dickson systems of some algebras with divisors of zero. First we need some preliminaries. Let (Mi , ) be quasi-ordered sets, i = 1, ..., n, and M = M1 × · · · × Mn the Cartesian product of Mi . The direct product of (Mi , ) is the quasi-ordered set (M, ), where is deﬁned by (a1 , ..., an ) (b1 , ..., bn ) ⇔ ai bi , i = 1, ..., n. 7.1. Proposition ([BW] CH.4, Corollary 4.47) Let (Mi , ) be Dickson quasiordered sets for i = 1, ..., n, and let (M, ) be the direct product of the (Mi , ). Then (M, ) is a Dickson quasi-ordered set. 2 A special case of the above proposition is the well-known Dickson’s lemma which is mentioned as follows.

7. Solvable Polynomial Algebras

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Dickson’s Lemma Let (IN n , ) be the direct product of n copies of the natural numbers (IN , ≺) with their natural ordering. Then, (IN n , ) is a Dickson partially ordered set. More explicitly, every subset S of IN n has a ﬁnite subset B such that for every (m1 , ..., mn ) ∈ S, there exists (k1 , ..., kn ) ∈ B with ki ≤ mi for i = 1, ..., n. 2 Let A = k[a1 , ..., an ] be a ﬁnitely generated k-algebra with n generators a1 , ..., an , and let n αn 1 B = aα = aα · · · a n α = (α1 , ..., αn ) ∈ IN 1 be the set of all standard monomials in A (see §1). 7.2. Proposition Let A be as above and satisfy (1) B is a k-basis of A (then 1 ∈ B), and (2) aα aβ = λα,β aα+β + f, aα , aβ ∈ B, λα,β ∈ k − {0}, f ∈ A. The following holds. (i) If (A, B, ) is an admissible system such that in the above (2) LM(f ) ≺ aα+β , then (A, B, ) is a Dickson system and a left Dickson system as well. (ii) If (A, B, ) is a left admissible system such that in the above (2) LM(f ) ≺ aα+β , then (A, B, ) is a left Dickson system; If furthermore is either ≥lex or a graded monomial ordering gr (see §1), then (A, B, ) is also an admissible system and hence a Dickson system. Proof Let be the (partial) ordering on B induced by the divisibility in A (§6 (∗)). By the assumption we have, for aβ , aα ∈ B, aβ is divisible by aα , i.e., aα aβ , if and only if aβ = LM(aγ aα aη ) = aγ+α+η if and only if β = α + δ for some δ ∈ IN n . It follows from Dickson’s lemma that is a Dickson partial ordering on B. Hence (A, B, ) is a Dickson system. Since (A, B, ) is also a left admissible system (see §1), a similar argumentation as above shows that (A, B, ) is also a left Dickson system. But then (ii) follows as well. 2 Let A = k[x1 , ..., xn ] be the commutative polynomial k-algebra in n variables, n αn 1 and B = {xα 1 · · · xn | (α1 , ..., αn ) ∈ IN } the standard k-basis for A. Then any admissible system (A, B, ) is a Dickson system. (Note that in this case a left Dickson system is the same as a Dickson system.)

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7.3. Deﬁnition (A. Kandri-Rody and V. Weispfenning, 1990) Let A be as in Proposition 7.2. We call A a solvable polynomial algebra, provided A satisﬁes (i) or (ii) of Proposition 7.2. From Proposition 7.2 we immediately derive the following fact. 7.4. Proposition (i) Solvable polynomial algebras are (left) Noetherian domains. (ii) For any nonzero f and g in a solvable polynomial algebra A with LM(f ) = β1 αn β βn 1 aα = aα 1 · · · an , LM(g) = a = a1 · · · an , putting γi = max{αi , βi }ni=1 , aγ = aγ11 · · · aγnn , λ = LC(aγ−α f ), µ = LC(aγ−β g), then a unique left S-element of f and g is deﬁned as µ S(f, g) = aγ−α f − aγ−β g. λ 2 The class of solvable polynomial algebras includes CH.I §5 Example (i)–(vii) as well as many other skew polynomial algebras (§1 Example (ii), also see CH.III §2). In particular, with respect to any monomial ordering on IN n , a multiplicative analogue On (λji ) of the Weyl algebra (CH.I §5 Example (iii) is a solvable polynomial algebra. If A is a solvable polynomial algebra with respect to some graded monomial ordering gr , then (A, B, gr ) is a Dickson system, and hence A is both GNoetherian and left G-Noetherian. More examples of solvable polynomial algebras are given in next chapter. We now proceed to construct examples of (left) Dickson systems in which the algebras contain divisors of zero. Example (i) Consider the Grassmann algebra G(x) as deﬁned in CH.I §5 with x = {xi }i∈Λ . Then there exists a (left) admissible system (G(x), B, ); if |Λ| = n is ﬁnite, then (G(x), B, ) is a (left) Dickson system. Proof From §5 Example (ii) we know that the set of deﬁning relations of G(x), denoted G = {Ri , Rk }, is a (left) Gr¨ obner basis in kX and that B = {xi1 xi2 · · · xim | ij ∈ Λ, i1 < i2 < · · · < im } forms a k-basis for G(x). It is easy to see that the monomial ordering ≥grlex in kX induces a graded lex ordering on B. Hence (G(x), B, ≥grlex ) forms a (left) admissible system. In the case where |Λ| = n, we see that αn 1 α2 x · · · x B = xα n αi = 0, 1 1 2

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n n n and |B| = 1 + + + ··· + + 1 = 2n = dimk G(x). 1 2 n−1 Observing the multiplication (and hence division) in G(x), (G(x), B, ≥grlex ) is clearly a (left) Dickson system. 2 As we pointed out in CH.I §5 (or from the above proof), in the case where x is a ﬁnite set, G(x) is indeed a homomorphic image of a multiplicative analogue On (λji ) of the Weyl algebra. More generally, let R = k[x1 , ..., xn , xn+1 , ..., xn+m ] be a solvable polynomial k-algebra and (R, B, ) an admissible system associated with R such that xj xi = λij xi xj , 1 ≤ i < j ≤ n + m, λij ∈ k − {0}. obner basis for Then by Proposition 4.1, G = {xpn+1 , xpn+2 , ..., xpn+m } forms a Gr¨ the two-sided ideal I = xpn+1 , ..., xpn+m , where p is a ﬁxed positive integer (also see CH.V §1 Propositon 1.4). Consider the quotient algebra A = R/I and write ai for the coset of each xi in A, 1 ≤ i ≤ n, and write bj for the coset of each βm n β1 xn+j in A, 1 ≤ j ≤ m. Since B = {x1α1 · · · xα n xn+1 · · · xn+m | α = (α1 , ..., αn ) ∈ n m IN , β = (β1 , ..., βm ) ∈ IN } and G is a Gr¨obner basis for I, it follows from Theorem 3.1 and Proposition 3.3 that n αn β1 βm 1 B = aα · · · a b · · · b n 1 m (α1 , ..., αn ) ∈ IN , 0 ≤ βi ≤ p − 1 1 is a k-basis for A. It is clear that A is no longer a solvable polynomial algebra. But by the assumptions on R we easily derive the following result. 7.5. Proposition (A, B, ) is an admissible system and hence a Dickson system if denotes the ordering ≥lex or some graded monomial ordering gr on B.

8. No (Left) Monomial Ordering Existing on ∆(k[x1 , ..., xn ]) with chark > 0 This ﬁnal section is focused on the algebra of linear partial diﬀerential operators with polynomial coeﬃcients because of its importance in practice. Let k be a ﬁeld, k[x1 , ..., xn ] the commutative polynomial algebra in n variables and Endk k[x1 , ..., xn ] the k-algebra of all k-linear operators on k[x1 , ..., xn ]. Recall that the algebra of k-linear diﬀerential operators with polynomial coeﬃcients, denoted ∆(k[x1 , ..., xn ]), is the subalgebra of Endk k[x1 , ..., xn ] generated by k[x1 , ..., xn ] and all partial derivations ∂1 = ∂/∂x1 , ..., ∂n = ∂/∂xn , where each xi acts on k[x1 , ..., xn ] by the left multiplication. More precisely, ∆(k[x1 , ..., xn ]) = k[x1 , ..., xn , ∂1 , ..., ∂n ] ⊂ Endk k[x1 , ..., xn ],

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and the generators of ∆(k[x1 , ..., xn ]) satisfy ∂j xi = xi ∂j + δij (the Kronecker delta), i, j = 1, ..., n, xi xj = xj xi , ∂i ∂j = ∂j ∂i , i, j = 1, ..., n. If chark = 0, then it is well-known (e.g., [Bj], [MR]) that (i) The set of all standard monomials of ∆(k[x1 , ..., xn ]), denoted 2n αn β1 βn 1 B = xα , (α · · · x ∂ · · · ∂ , ..., α , β , ..., β ) ∈ I N 1 n 1 n n 1 1 is a k-basis of ∆(k[x1 , ..., xn ]). (ii) ∆(k[x1 , ..., xn ]) coincides with the ring of diﬀerential operators D(Ank ) of the aﬃne n-space Ank over k. (iii) (See CH.I §5, Example (i)) ∆(k[x1 , ..., xn ]) is isomorphic to the nth Weyl algebra An (k) = k[x1 , ..., xn , y1 , ..., yn ] generated by 2n elements subject to the relations: yj xi = xi yj + δij (the Kronecker delta), xi xj = xj xi , yi yj = yj yi ,

1 ≤ i, j ≤ n, 1 ≤ i < j ≤ n.

(iv) (See CH.I §5, Example (i)) ∆(k[x1 , ..., xn ]) is isomorphic to the iterated skew polynomial k-algebra k[x1 , ..., xn ][y1 ; δ1 ] · · · [yn ; δn ], where δi = ∂/∂xi , i = 1, ..., n. (v) ∆(k[x1 , ..., xn ]) is an inﬁnite dimensional simple k-algebra and is a Noetherian domain. It follows from §7 that (∆(k[x1 , ..., xn ]), B, ) is a (left) admissible system and hence it is a (left) Dickson system (at least when is ≥lex or ≥grlex ), for, in this case, ∆(k[x1 , ..., xn ]) is a solvable polynomial algebra. However, if chark = p > 0 for some prime number p, then ∆(k[x1 , ..., xn ]) no longer has the nice properties 1–4 listed above, in particular, it is not a solvable polynomial k-algebra in the sense of §7. This follows from the following facts. (The following 8.1–8.3, and 8.5 are obtained in [Zh].) It is easy to see that every element f ∈ ∆(k[x1 , ..., xn ]) is of the form αn β1 βn 1 f = λαβ xα 1 · · · xn ∂1 · · · ∂n α,β

where λαβ ∈ k, α = (α1 , ..., αn ), β = (β1 , ..., βn ) ∈ IN n . αn 1 As in §1 we write |α| = α1 + · · · + αn , |β| = β1 + · · · + βn , xα = xα 1 · · · xn , β1 βn β ∂ = ∂1 · · · ∂n . Moreover, we write α! = α1 ! · · · αn !, 0! = 1, and write D ∗ f for the action of an operator D on an element of f ∈ k[x1 , ..., xn ].

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8.1. Lemma With the notation as above, the following holds. (i) ∂ip = 0, i = 1, ..., n. (ii) For any α, β ∈ IN n , if |β| ≥ |α|, then β! (β = α and βi ≤ p − 1, 1 ≤ i ≤ n) ∂ β ∗ xα = 0 otherwise. 2 For the reader’s convenience we include a proof of the next proposition. 8.2. Proposition With the notation as above, B = xα ∂ β α = (α1 , ..., αn ), β = (β1 , ..., βn ) ∈ IN n , 0 ≤ βi ≤ p − 1 forms a k-basis for ∆(k[x1 , ..., xn ]). α β Proof Suppose that D = = 0, where the (α, β) are mu(α,β) λ(α,β) x ∂ tually diﬀerent and λ(α,β) ∈ k. Rewrite D as a polynomial in ∂ with cot (i) α(j) eﬃcients in k[x1 , ..., xn ]: D = fi ∂ γ , where fi = , j=1 λα(j) γ (i) x i=1

(j)

i = 1, ..., t. Note that in the expression of fi the xα are mutually diﬀerent. Using the graded lexicographic ordering on IN n , we may assume that (1) γ (1) >grlex γ (2) grlex · · · >grlex γ (t) . Then 0 = D ∗ xγ = f1 γ (1) !. Since (1) γi ≤ p − 1, i = 1, ..., n, we have f1 = 0. Hence λα(j) γ (i) = 0, j = 1, ..., . Similarly we may get f2 = · · · = ft = 0. Therefore, λα,β = 0 for all (α, β) appearing in the expression of D. This proves the linear independence of elements in B, as desired. 2 It follows from Lemma 8.1, Proposition 8.2 and its proof that we also have the following. 8.3. Corollary (i) The centre of ∆(k[x1 , ..., xn ]) is the subalgebra k[xp1 , ..., xpn ]. (ii) ∆(k[x1 , ..., xn ]) is not a simple algebra. (iii) ∆(k[x1 , ..., xn ]) is a free module of ﬁnite rank over its centre. 2 It is not hard to see that if there was a (left) monomial ordering on B, then (∆(k[x1 , ..., xn ]), B, ) would be a (left) Dickson system. Unfortunately, we have the following bad news (from an online discussion of the author with Zhang Jiangfeng in the beginning of Nov. 2000). 8.4. Conclusion Let ∆(k[x1 , ..., xn ]) be as above, where k is of characeristic p > 0. Then there does not exist a monomial ordering or a left monomial

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ordering on n αn β1 βn 1 (α · · · x ∂ · · · ∂ , ..., α ) ∈ I N , 0 ≤ β ≤ p − 1 . B = xα 1 n i n n 1 1 Proof Consider u = xi ∂ip−1 , w = ∂ip−2 ∈ B. If there was a left monomial ordering on B, then since ∂i xi = xi ∂i + 1 and ∂ip = 0, the formulas ∂i u = ∂ip−1 ,

∂i w = ∂ip−1

would yield a contradiction, i.e., ∂ip−1 ≺ ∂ip−1 . Since any monomial ordering on B is also a left monomial ordering, the above stone has killed two birds. 2 Nevertheless, light still comes, that is, we have two ways to use Gr¨obner bases on ∆(k[x1 , ..., xn ]) where chark = p > 0. To see this, put A = ∆(k[x1 , ..., xn ]), and consider the standard ﬁltration F A on A which is by deﬁnition the ﬁltration: F0 A ⊂ F1 A ⊂ · · · ⊂ Fn A ⊂ · · · with     αn β1 βn 1 Fn A = f = |α| + |β| ≤ n , n ≥ 0. λαβ xα · · · x ∂ · · · ∂ n n 1 1   α,β

The associated graded algebra of A, denoted G(A), which is by deﬁnition the positively graded algebra G(A) = ⊕n∈IN G(A)n with G(A)n = Fn A/Fn−1 A. 8.5. Proposition (i) A is a homomorphic image of the iterated skew polynomial algebra (i.e., the nth Weyl algebra over k) An (k) = k[x1 , ..., xn ][y1 ; δ1 ] · · · [yn ; δn ], where δi = ∂/∂xi , i = 1, ..., n. More precisely, A ∼ = An (k)/I where I = p p δ1 , ..., δn . (ii) G(A) ∼ = k[z1 , ..., z2n ]/J, where k[z1 , ..., z2n ] is the commutative polynomial p p , ..., z2n . k-algebra in 2n variables and J = zn+1 2 Note that An (k) is a solvable polynomial algebra. From Proposition 8.5(i) it is therefore clear that we can apply Gr¨ obner bases to ∆(k[x1 , ..., xn ]) by passing to An (k) (see [Zh] for some applications). On the other hand, from a lifting structure point of view, that is, studying ∆(k[x1 , ..., xn ]) by passing to its associated graded algebra whenever the standard ﬁltration on ∆(k[x1 , ..., xn ]) is considered (see CH.III §3 for a general explanation on this view), Proposition 8.5(ii) now provides us with the possibility to play with the associated graded algebra of ∆(k[x1 , ..., xn ]), namely, it follows from Proposition 7.5 and Proposition 8.5(ii) that we have the following fact.

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8.6. Proposition Let A = ∆(k[x1 , ..., xn ]) be as before, and consider the standard ﬁltration F A on A. If σ(xi ) and σ(∂j ) denote the image of xi and ∂j in G(A)1 = F1 A/F0 A respectively, i, j = 1, ..., n, then (G(A), B, ) is a Dickson system, where B = σ(x1 )α1 · · · σ(xn )αn σ(∂1 )β1 · · · σ(∂n )βn (α1 , ..., αn ) ∈ IN n , 0 ≤ βi ≤ p − 1 and denotes the ordering ≥lex or ≥grlex on B.

CHAPTER III Gr¨ obner Bases and Basic Algebraic-Algorithmic Structures

In view of CH.II §3, we start this chapter with the interaction between the PBW bases of ﬁnitely generated algebras and the algorithmic aspect of very noncommutative Gr¨ obner bases (in the sense of [Mor2]). As a consequence, Theorem 1.5 and Proposition 1.6 enable us to recognize and construct quadric solvable polynomial algebras in an algorithmic way. And then, we determine the homogeneous deﬁning relations of the associated graded structures of a given algebra, based on a combination of noncommutative Gr¨obner basis and the structural trick developed in CH.I §4. The latter result is indeed the noncommutative version of a remarkable application of the commutative Gr¨ obner basis theory to algebraic geometry in determining the homogeneous deﬁning equations of the projective closure of an aﬃne variety. In addition to providing some basic algebraic-algorithmic structures for later study, this chapter also motivates a general ﬁltered-graded transfer of Gr¨ obner bases in associative algebras – the topic of next chapter.

H. Li: LNM 1795, pp. 67–90, 2002. c Springer-Verlag Berlin Heidelberg 2002

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1. PBW Bases of Finitely Generated Algebras Let k be a ﬁeld and A = k[a1 , ..., an ] a ﬁnitely generated k-algebra. We say that A has a PBW k-basis if the set of standard monomials n αn 1 α2 (α B = aα a · · · a , ..., α ) ∈ I N 1 n n 1 2 forms a k-basis for A. Note that we say “a PBW k-basis” instead of “the PBW k-basis” because the presently used “natural order” a1 , a2 , ..., an on generators may be changed by any permutation of generators if it is necessary. We have seen from CH.II that having a k-basis consisting of “monomials” is the basis of having a Gr¨ obner basis theory for a k-algebra A. It is also well known that if a ﬁnitely generated k-algebra A has a PBW k-basis, then the structure theory of A, in particular, the representation theory of A will be more nicer. Inspired by the work of [Mor2] and Berger’s quantum PBW theorem [Ber], in this section, we demonstrate the interaction between the algorithmic aspect of very noncommutative Gr¨ obner bases and PBW bases for ﬁnitely generated algebras with speciﬁc deﬁning relations. To better understand this section, we suggest the reader to go back to CH.II §3. First recall from [Mor2] some generalities of the noncommutative Gr¨ obner bases for two-sided ideals in free algebras. Let kX be the free algebra generated by X = {Xi }i∈Λ over the ﬁeld k, and let S = X be the free semigroup generated by X. With notation as in CH.II, we let (kX, B, ) be an admissible system, where B is the k-basis of kX consisting of words of S and 1. It follows from CH.II Proposition 4.2 that the following deﬁnition is necessarily to be mentioned (but it is better to bear CH.II Deﬁnition 3.2 in mind as well). 1.1. Deﬁnition Let I be an ideal of kX. A set G = {gj }j∈J ⊂ I is called a Gr¨ obner basis of I if LM(G) = LM(I) where LM(G) is the two-sided ideal generated by {LM(gj ) | gj ∈ G}, and a similar interpretation is for LM(I). Given a generating set G = {gi }i∈J of an ideal I ⊂ kX, it is generally diﬃcult to know if G is a Gr¨ obner basis of I. However, if kX is the free k-algebra generated by a ﬁnite set of indeterminates X = {X1 , ..., Xn }, and if G = {g1 , ..., gs } is also ﬁnite, then from [Mor2] we know that the noncommutative version of Buchberger’s algorithm does exist and it can be used to produce a Gr¨obner basis of I starting with G, though the obtained basis is usually no longer ﬁnite (unless I has a ﬁnite Gr¨ obner basis and the procedure halts). The existence of a ﬁnite

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Gr¨ obner basis is based on a technical process in analysing the S-elements (see CH.II §5) that we recall in some detail below. Let X = {X1 , ..., Xn }, (kX, B, ), and G = {g1 , ..., gs } be ﬁxed as above but we assume that all gi are monic, i.e., LC(gi ) = 1, i = 1, ..., s. Adopting the notation as in [Mor2], let S × S be the Cartesian square of the free semigroup S generated by X. If (l, r), (λ, ρ) ∈ S × S are such that lLM(gj )r = λLM(gi )ρ, then an S-element of gj and gi is denoted by S(i, j; l, r; λ, ρ) = lgj r − λgi ρ. We say that S(i, j; l, r; λ, ρ) has a weak Gr¨ obner representation by G if ckµ lkµ gk rkµ in which S(i, j; l, r; λ, ρ) = k,µ

for each k, µ, lku LM(gk )rkµ ≺ lLM(gj )r. The product S × S has a natural S-bimodule structure in the sense that for each t ∈ S, for each (l, r) ∈ S × S, t(l, r) = (tl, r), (l, r)t = (l, rt). To be convenient, we denote this algebraic structure on S × S by S ⊗ S. An ideal of S ⊗ S is a subset J ⊂ S ⊗ S such that if (l, r) ∈ J , t ∈ S, then (tl, r) ∈ J , (l, rt) ∈ J ; a set of generators for J is a (not necessarily ﬁnite) set G ⊂ J such that for each (l, r) ∈ J there are l1 , r1 ∈ S, (wl , wr ) ∈ G such that l = l1 wl and r = wr r1 . For s ≥ j ≥ 1, we write ST (LM(gj )) for the ideal of S ⊗ S generated by the set r = 1 and there is (l, 1) ∈ S ⊗ S SOB(LM(gj )) = (1, r) ∈ S ⊗ S , such that lLM(gj ) = LM(gj )r and we put Tj (G) = (l, r) ∈ S ⊗ S lLM(gj )r ∈ Ij ∪ ST (LM(gj )), where Ij stands for the ideal of S generated by {LM(g1 ), LM(g2 ), ..., LM(gj−1 )}. (One may see that Tj (G) is indeed an ideal of S ⊗ S.) Let U be a minimal generating set of the ideal Tj (G). For each σ = (lσ , rσ ) ∈ U, choose iσ , λσ , ρσ such that lσ LM(gj )rσ

=

λσ LM(giσ )ρσ , iσ < j or iσ = j and there is w ∈ S such that rσ = wρσ ,

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and we let MIN(j) = {(iσ , j; lσ , rσ ; λσ , ρσ )}. An element (i, j; l, r; λ, ρ) ∈ MIN(j) is said to be trivial if there is w ∈ S such that either l = λLM(gi )w (and so ρ = wLM(gj )r) or λ = lLM(gj )w (and so r = wLM(gi )ρ). 1.2. Theorem ([Mor2] Corollary 5.8, Theorem 5.9) Let G = {g1 , ..., gs } be a generating set of the ideal I ⊂ kX where LC(gi ) = 1 for each i. The following holds. (i) The set OBS(j) = {(i, j; l, r; λ, ρ) ∈ MIN(j) and nontrivial} is ﬁnite. (ii) G is a Gr¨ obner basis of I if and only if for each j, for each nontrivial (i, j; l, r; λ, ρ) ∈ MIN(j), the S-element S(i, j; l, r; λ, ρ) has a weak Gr¨ obner representation. 2 1.3. Corollary (compare with the example given in the end of CH.II §4) If A = kX, Y /I with I = R = Y X − λXY − F , where λ ∈ k, F ∈ kX, Y , and if LM(R) = Y X with respect to some monomial ordering on kX, Y , then {R} is a Gr¨ obner basis for I and A has the PBW k-basis {xα y β | (α, β) ∈ IN 2 }, where x and y are the images of X and Y in A respectively. 2 Now let A = k[a1 , ..., an ] be a ﬁnitely generated k-algebra and (kX, B, ) an admissible system as before, where kX = kX1 , ..., Xn . Suppose that A is deﬁned by the following relations Rji = Xj Xi − λji Xi Xj − {Xj , Xi },

1 ≤ i < j ≤ n,

where λji ∈ k, {Xj , Xi } = 0 or {Xj , Xi } ∈ kX − k-span{Xj Xi , Xi Xj }. Writing I for the ideal of kX generated by G = {Rji | 1 ≤ i < j ≤ n}, then A = kX/I. Furthermore, we assume that, with respect to the monomial ordering in the admissible system (kX, B, ), the deﬁning relations satisfy (∗)

LM(Rji ) = Xj Xi ,

1 ≤ i < j ≤ n.

1.4. Lemma With notation as before, if we put G = gk = Rkj , gj = Rji 1 ≤ j, k ≤ n ,

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71

then OBS(k)

(h, k; l, r; λ, ρ) ∈ MIN(k) and nontrivial

=

(j, k; 1, Xi ; Xk , 1) i < j < k .

= Proof Recall that

Tk (G) = (l, r) ∈ S ⊗ S lLM(gk )r ∈ Ik ∪ ST (LM(gk )),

where Ik stands for the ideal of S generated by {LM(g1 ), LM(g2 ), ..., LM(gk−1 )}, and ST (LM(gk )) is the ideal of S ⊗ S generated by the set r = 1 and there is (l, 1) ∈ S ⊗ S SOB(LM(gk )) = (1, r) ∈ S ⊗ S . such that lLM(gk ) = LM(gk )r It is easy to see that SOB(LM(gk )) = {(1, wXk Xj ) | w ∈ S} and SOB(LM(gk )) is a minimal generating set of ST (LM(gk )). It is also not hard to check that, for each (1, wXk Xj ) ∈ SOB(LM(gk )), every (j, k; 1, wXk Xj ; λ, ρ) ∈ MIN(j) is trivial. Furthermore if (l, r) ∈ S ⊗ S is such that lXk Xj r = λXt Xi ρ for some λ, ρ ∈ S ⊗ S, where k > t, then one sees that (t, k; l, r; λ, ρ) is trivial in case j = t; In the case where j = t, one may also easily see that the (1, Xi ) generate all nontrivial elements in MIN(k), or more precisely, OBS(k) = {(h, k; l, r; λ, ρ) ∈ MIN(k) and nontrivial} = {(j, k; 1, Xi ; Xk , 1) | i < j < k}, as desired. 2 1.5. Theorem Let A = k[a1 , ..., an ] = kX/I be the ﬁnitely generated kalgebra as above, where I = Rji | 1 ≤ i < j ≤ n. Suppose that the Rji satisfy the foregoing assumption (∗) with respect to the monomial ordering in (kX, B, ). The following statements are equivalent: (i) The k-algebra A has a PBW k-basis. (ii) G = {Rji | 1 ≤ i < j ≤ n} forms a Gr¨obner basis for the ideal I in kX. (iii) For 1 ≤ i < j < k ≤ n, every Rkj Xi − Xk Rji has a weak Gr¨obner representation by G. Proof By Lemma 1.4, we have OBS(k)

= {(h, k; l, r; λ, ρ) ∈ MIN(k) and nontrivial} (j, k; 1, Xi ; Xk , 1) i < j < k .

=

Also note that for each (j, k; 1, Xi ; Xk , 1) ∈ OBS(k) the corresponding S-element is nothing but Rkj Xi − Xk Rji .

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(i) ⇔ (ii) Since Xj Xi = LM(Rji ) for 1 ≤ i < j ≤ n, this equivalence follows from CH.II Theorem 3.1 and Proposition 3.4. (ii) ⇒ (iii) This follows from Theorem 1.2. (iii) ⇒ (ii) If each Rkj Xi − Xk Rji has a weak Gr¨ obner representation by G, then it follows from Theorem 1.2 that (ii) holds. 2 To realize Theorem 1.5, one may, of course, directly check (iii) by deﬁnition, or verify (ii) by means of the very noncommutative version of Buchberger‘s algorithm given by Mora in [Mor2]. However, to avoid large and tedious noncommutative division procedure, we will see that Berger’s q-Jacobi condition [Ber] is quite helpful in the case where the graded lexicographic ordering ≥grlex is used (indeed ≥grlex is the most commonly used monomial ordering on free algebras in practice, see CH.II Example (iv) for the deﬁnition), though the algebras we are dealing with are not necessarily the q-algebras studied in [Ber] (see Example (vi) in next section). More precisely, Let kX, A = kX/I be as before, where I = Rji | 1 ≤ i < j ≤ n with Rji = Xj Xi − λji Xi Xj − {Xj , Xi }. Then, for 1 ≤ i < j < k ≤ n, the Jacobi sum J(Xk , Xj , Xi ) in the sense of [Ber] is deﬁned as follows. J(Xk , Xj , Xi )

=

{Xk , Xj }Xi − λki λji Xi {Xk , Xj }− −λji {Xk , Xi }Xj + λkj Xj {Xk , Xi }+ +λkj λki {Xj , Xi }Xk − Xk {Xj , Xi }.

1.6. Proposition Let A = kX/I be as above, and let ≥grlex be the graded lexicographic monomial ordering in (kX, B, ≥grlex ) such that Xn >grlex Xn−1 >grlex · · · >grlex X1 , and LM(Rji ) = Xj Xi with respect to ≥grlex , 1 ≤ i < j ≤ n. The following statements are equivalent. (i) G = {Rji | 1 ≤ i < j ≤ n} forms a Gr¨ obner basis for the ideal I = G ⊂ kX with respect to ≥grlex . (ii) For 1 ≤ i < j < k ≤ n, J(Xk , Xj , Xi ) ∈ k-span Rpq 1 ≤ q < p ≤ n +   Xh Rji , Rji Xh , Rji Xk , + k-span X R ,R X ,R X ,  h ki ki h ki k Xh Rkj , Rkj Xm ,

 1 ≤ h < k,  1 ≤ h < k,  1 ≤ h < k, 1 ≤ m < i

Proof From the deﬁning relations we derive {Xk , Xj } = Xk Xj − λkj Xj Xk − Rkj , {Xk , Xi } = Xk Xi − λki Xi Xk − Rki , {Xj , Xi } = Xj Xi − λji Xi Xj − Rji .

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It follows that J(Xk , Xj , Xi )

= Xk Rji − Rkj Xi + + λji Rki Xj − λkj Xj Rki − λkj λki Rji Xk + λki λji Xi Rkj ,

which is obviously contained in I, and consequently Rkj Xi − Xk Rji

=

λji Rki Xj − λkj Xj Rki − λkj λki Rji Xk + λki λji Xi Rkj − −J(Xk , Xj , Xi ).

Noticing the ordering Xn >grlex Xn−1 >grlex · · · >grlex X1 , the equivalence (i) ⇔ (ii) is clear now by Theorem 1.5. 2 Applications of Theorem 1.5 and Proposition 1.6 are given in the next section.

2. Quadric Solvable Polynomial Algebras Based on the results obtained in §1, in this section we formulate and construct quadric solvable polynomial algebras, which form the practical working basis for later chapters. Let A = k[a1 , ..., an ] be a solvable polynomial algebra in the sense of CH.II Deﬁnition 7.3, with the associated (left) admissible system (A, B, gr ), where n αn 1 α2 B = {aα 1 a2 · · · an | (α1 , ..., αn ) ∈ IN }

is the k-basis consisiting of standard monomials and gr is a graded monomial ordering. It follows that the generators of A satisfy only quadric relations, that is, for 1 ≤ i < j ≤ n, λk λh ah + cji , (∗) aj ai = λji ai aj + ji ak a + k≤

where λji , λk ji , λh , cji ∈ k, and λji = 0. This leads to the following speciﬁc class of solvable polynomial algebras. 2.1. Deﬁnition We call the above solvable polynomial algebra A a quadric solvable polynomial algebra. If λh ah + cji = 0 in the formula (∗), we call A a quadratic solvable polyno k mial algebra; if λji ak a = 0 in the formula (∗), we call A a linear solvable k polynomial algebra; if λji ak a + λh ah + cji = 0 in the formula (∗), we call A a homogeneous solvable polynomial algebra. In particular, if k, < j in the formula (∗) whenever λk ji = 0, then we call A a tame quadric solvable polynomial algebra.

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From the deﬁnition it is clear that any linear or homogeneous solvable polynomial algebra is trivially a tame quadric solvable polynomial algebra. More generally, if A = k[x1 ; σ1 , δ1 ][x2 ; σ2 , δ2 ] · · · [xn ; σn , δn ] is an iterated skew polynomial algebra starting with the ground ﬁeld k (see CH.II §1 Example (ii)), where each σi is an algebra automorphism and the δi is a σi -derivation, such that σj (xi ) = λji xi with λji ∈ k−{0} for all j > i, and that δj (xi ) = pji is a linear combination of the α 1 monomials xα 1 · · · x with < j and α1 + · · · + α ≤ 2, for all j > i, then A is a tame quadric solvable polynomial algebra with respect to xn >grlex xn−1 >grlex · · · >grlex x1 . Below we construct quadric solvable polynomial algebras in which some do not seem having a speciﬁc iterated skew polynomial algebra structure starting with the ground ﬁeld k (at least by ﬁrst glance at the given deﬁning relations). And we also show that tame quadric solvable polynomial algebras are completely constructable. Let A = k[a1 , ..., an ] be a quadric solvable polynomial k-algebra with gr . Then, since the generators satisfy the relations as given in previous (∗), and since n αn 1 ∼ B = {aα 1 · · · an | (α1 , ..., αn ) ∈ IN } forms a k-basis for A, we have A = kX/I, where kX = kX1 , ..., Xn is the free associative k-algebra over X = {X1 , ..., Xn } and I is the ideal of kX generated by λk λh Xh − cji , 1 ≤ i < j ≤ n, Rji = Xj Xi − λji Xi Xj − ji Xk X − k≤

or in other words, {Rji | 1 ≤ i < j ≤ n} is a set of deﬁning relations for A. This observation opens to us the door for recognizing and constructing quadric solvable polynomial algebras in an algorithmic way, that is, we have the following consequence of Theorem 1.5. 2.2. Proposition Consider the k-algebra A = kX/I, where I is the ideal of kX generated by the quadric deﬁning relations λk λh Xh − cji , 1 ≤ i < j ≤ n, Rji = Xj Xi − λji Xi Xj − ji Xk X − where λji , λk ji , λh , cji ∈ k. Suppose that (1) λji = 0, 1 ≤ i < j ≤ n, and (2) one of the following conditions is satisﬁed whenever λk ji = 0: (a) k = and k, < j. (b) k = and k, ≤ j, where k = j implies < i and = j implies k < i. Then A = k[a1 , ..., an ] is a quadric solvable polynomial algebra with respect to an >grlex an−1 >grlex · · · >grlex a1 , where each ai is the image of Xi in A, if and only if {Rji | 1 ≤ i < j ≤ n} forms a Gr¨ obner basis in kX with respect to Xn >grlex Xn−1 >grlex · · · >grlex X1 . Proof Suppose that {Rji | 1 ≤ i < j ≤ n} forms a Gr¨obner basis in kX with respect to Xn >grlex Xn−1 >grlex · · · >grlex X1 . Since by the assumption (2) we have LM(Rji ) = Xj Xi , 1 ≤ i < j ≤ n, it follows from Theorem 1.5 that

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75

n αn 1 α2 B = {aα 1 a2 · · · an | (α1 , ..., αn ) ∈ IN } forms a k-basis for A. Now one checks directly that the assumption (1)–(2) and the deﬁning relations together make A into a quadric solvable polynomial algebra with respect to an >grlex an−1 >grlex · · · >grlex a1 . The converse is clear by Theorem 1.5. 2

An immediate application of Proposition 2.2 is to show that tame quadric solvable polynomial algebras are completely constructable. 2.3. Proposition Let A = k[a1 , ..., an ] be a tame quadric solvable polynomial algebra in the sense of Deﬁnition 2.1, that is, the generators of A satisfy λk λh ah + cji , 1 ≤ i < j ≤ n. aj ai = λji ai aj + ji ak a + k≤<j

Setting on the free algebra kX = kX1 , ..., Xn the graded monomial ordering Xn >grlex Xn−1 >grlex · · · >grlex X1 , A is isomorphic to the tame quadric solvable polynomial algebra k[b1 , ..., bn ] = kX/I with respect to bn >grlex bn−1 >grlex · · · >grlex b1 , where each bi is the residue of Xi (modulo I) and I is generated by the relations λk λh Xh − cji , 1 ≤ i < j ≤ n. Rji = Xj Xi − λji Xi Xj − ji Xk X − k≤<j

Proof First note that, with the given data of A, A ∼ = k[b1 , ..., bn ] = kX/I as k-algebras. That k[b1 , ..., bn ] is a tame quadric solvable polynomial algebra with respect to bn >grlex bn−1 >grlex · · · >grlex b1 follows from Theorem 1.5 and Proposition 2.2. 2 We are ready to construct quadric solvable polynomial algebras by using Proposition 2.2 and Proposition 1.6. In the examples given below, notation are maintained as before. Moreover, by abusing language, some examples will be called “deformations” of certain well-known algebras. Example (i) Let X2 X1 −qX1 X2 −aX12 −bX1 −cX2 −d = R21 ∈ kX1 , X2 , where obner basis in kX1 , X2 with q, a, b, c, d ∈ k. By Corollary 1.3, {R21 } is a Gr¨ respect to X2 >grlex X1 . Thus, If q = 0, then the algebra A = kX1 , X2 /R21 is a tame quadric solvable polynomial algebra with respect to ≥grlex (indeed this is a skew polynomial algebra). If q = 0, a = b = c = 0, and d = 1, then A = A1 (q), the additive analogue of the ﬁrst weyl algebra. One sees that here we have got all 2-dimensional quadric solvable polynomial algebras with respect to X2 >grlex X1 . (ii) Deformations of U (sl2 ). Let U (sl2 ) be the enveloping algebra of the 3-dimensional Lie algebra sl2 = kx ⊕ ky ⊕ kz deﬁned by the bracket product: [x, y] = z, [z, x] = 2x, [z, y] = −2y.

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This example provides quadric solvable polynomial algebras which are deformations of U (sl2 ). Let kX1 , X2 , X3 be the free k-algebra over {X1 , X2 , X3 }, and kX1 , X2 , X3 /I = A where I is the two-sided ideal generated by the deﬁning relations R21 = X2 X1 − αX1 X2 − γX2 − F21 , R31 = X3 X1 − α1 X1 X3 + αγ X3 − F31 , R32 = X3 X2 − βX2 X3 − F (X1 ) − F32 , where

α = 0, β, γ ∈ k, F (X1 ) ∈ k-span{X12 , X1 , 1}, F21 , F31 , F32 ∈ kX1 , X2 , X3 .

If α = β = 1, γ = 2, F (X1 ) = X1 , and F21 = F31 = F32 = 0, then A = U (sl2 ). Moreover, in the case where F21 = F31 = F32 = 0, the family of algebras constructed above includes many well-known deformations of U (sl2 ), e.g., Woronowicz’s deformation of U (sl2 ) [Wor], Witten’s deformation of U (sl2 ) [Wit], Le Bruyn’s conformal sl2 enveloping algebra [Le2], Smith’s deformation of U (sl2 ) where the dominant polynomial f (t) has degree ≤ 2 [Sm2], BenkartRoby’s down-up algebra in which β = 0 (cf. [KMP], [CM]). Set on kX1 , X2 , X3 the monomial ordering X3 >grlex X2 >grlex X1 . Then the only Jacobi sum determined by the deﬁning relations of A with respect to the ﬁxed ordering on generators is J(X3 , X2 , X1 )

= {X3 , X2 }X1 − λ31 λ21 X1 {X3 , X2 }− −λ21 {X3 , X1 }X2 + λ32 X2 {X3 , X1 }+ +λ32 λ31 {X2 , X1 }X3 − X3 {X2 , X1 } 1 = (f (X 1 )γ+ F32 )X1 − α · αX1 (f(Xγ1 ) + F32 )− −α − α X3 + F31 X2 + βX2 − α X3 + F31 + +β · α1 (γX2 + F21 )X3 − X3 (γX2 + F21 ) β F21 X3 − X3 F21 . = F32 X1 − X1 F32 − αF31 X2 + βX2 F31 + α

Write F = F32 X1 − X1 F32 − αF31 X2 + βX2 F31 +

β F21 X3 − X3 F21 . α

By Proposition 1.6, if LM(Rji ) = Xj Xi w.r.t. ≥grlex , 1 ≤ i < j ≤ 3, and  R21 , R31 , R32 ,    X1 R21 , R21 X1 , X2 R21 , R21 X2 , R21 X3 , F ∈ k-span X R ,R X ,X R ,R X ,R X ,    1 31 31 1 2 31 31 2 31 3 X1 R32 , X2 R32

      

,

then {R21 , R31 , R32 } forms a Gr¨ obner basis with respect to X3 >grlex X2 >grlex X1 . Below we consider two cases:

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Case I. Input in the deﬁning relations of A the data   α = β = 0, γ, µ, q, ε, ξ, λ, η32 ∈ k,     F (X1 ) ∈ k-span{X12 , X1 , 1},     G(X2 ) ∈ k-span{X22 , X2 , 1},    H(X3 ) ∈ k-span{X3 , 1}, (D1) F21 = µX12 + qX1 + H(X3 ),    2   F31 = ε(X  1 X2 + X2 X1 ) − ξX1 +2 λX1 + G(X2 ),    F32 = µ(X1 X3 + X3 X1 ) − εαX2 + ξα(X1 X2 + X2 X1 )    −λαX2 + qX3 + η32 . Clearly, in this case we have LM(Rji ) = Xj Xi , 1 ≤ i < j ≤ 3, and the conditions of Proposition 2.2 are satisﬁed. Moreover, a direct veriﬁcation shows that F32 X1 − X1 F32 −αF31 X2 + αX2 F31 F21 X3 − X3 F21

µ(X3 X12 − X12 X3 ) + εα(X1 X22 − X22 X1 )+ +ξα(X2 X12 − X12 X2 ) + λα(X1 X2 − X2 X1 )+ +q(X3 X1 − X1 X3 ), = εα(X22 X1 − X1 X22 ) + ξα(X12 X2 − X2 X12 )+ +λα(X2 X1 − X1 X2 ), = µ(X12 X3 − X3 X12 ) + q(X1 X3 − X3 X1 ), =

and concequently, J(X3 , X2 , X1 ) = F = 0. By Proposition 2.2, A is a quadric solvable polynomial algebra. Case II. Input in the deﬁning relations of A the data   α = β = 0, γ, q, ε, ξ, λ, η32 ∈ k,     F (X1 ) ∈ k-span{X12 , X1 , 1},      G(X2 ) ∈ k-span{X22 , X2 , 1}, (D2) H(X3 ) ∈ k-span{X3 , 1},    F21 = qX1 + H(X3 ),     F31 = ε(X1 X2 + X2 X1 ) − ξX12 + λX1 + G(X2 ),    F = −εαX 2 + ξα(X X + X X ) − λαX + qX + η , 32 1 2 2 1 2 3 32 2 As with the data (D1), one checks that in this case we also have J(X3 , X2 , X1 ) = 0. So A is a tame quadric solvable polynomial algebra. By taking the residues (modulo I) in the above cases, one may obtain a set {F21 , F31 , F32 } in which each member Fij is a linear combination of standard monomials. (iii) Non-polynomial central extensions of the deformations of U (sl2 ). These are the 4-dimensional algebras deﬁned by the relations from the free

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algebra kX1 , X2 , X3 , X4 R21 R31 R32 R41 R42 R43

= X2 X1 − αX1 X2 − γX2 − F21 − K21 = X3 X1 − α1 X1 X3 + αγ X3 − F31 − K31 = X3 X2 − αX2 X3 − F (X1 ) − F32 − K32 = X4 X 1 − X 1 X 4 , = X4 X 2 − X 2 X 4 , = X4 X 3 − X 3 X 4 ,

where K21 , K31 , K32 ∈ k-span{X4 , 1} and {α, γ, F (X1 ), F21 , F31 , F32 , F32 } is taken either from (D1) or from (D2) in Example (ii). Since the only possible nonzero Jacobi sums determined by the above relations with respect to X4 >grlex X3 >grlex X2 >grlex X1 are given by J(X3 , X2 , X1 ) = K32 X1 − X1 K32 − αK31 X2 + αX2 K31 + K21 X3 − X3 K21 , J(X4 , X3 , X2 ) = F (X1 )X4 − X4 F (X1 ) + F32 X4 − X4 F32 , J(X4 , X3 , X1 ) = αγ (X4 X3 − X3 X4 ) + F31 X4 − X4 F31 , J(X4 , X2 , X1 ) = γ(X2 X4 − X4 X2 ) + F21 X4 − X4 F21 , it can be further checked that these sums have weak Gr¨ obner representations by {R41 , R42 , R43 }. Thus, Proposition 2.2 and Proposition 1.6 hold. Hence, the algebras deﬁned by the relations given above are quadric solvable polynomial algebras. (iv) Deformations of An (k). Let An (k) be the nth Weyl algebra over k. This example provides quadric solvable polynomial algebras which are deformations of An (k). Set on the free algebra kY, X = kYn , ..., Y1 , Xn , ..., X1 the monomial ordering Yn >grlex Xn >grlex Yn−1 >grlex Xn−1 >grlex · · · >grlex Y1 >grlex X1 . Consider the algebra A with deﬁning relations Hji = Xj Xi − Xi Xj , 1 ≤ i < j ≤ n, ji = Xj Yi − Yi Xj , 1 ≤ i < j ≤ n, H 1 ≤ i < j ≤ n, Gji = Yj Yi − Yi Yj , 1≤i<j≤n Gji = Yj Xi − Xi Yj , Rjj = Yj Xj − qj Xj Yj − Fjj , 1 ≤ j ≤ n, where qj ∈ k, Fjj ∈ kY, X. If in the deﬁning relations Fjj = 1 for 1 ≤ i < j ≤ n, then the additive analogue An (q1 , ..., qn ) of the Weyl algebra (CH.I §5 Example (ii)) is recaptured. A direct veriﬁcation shows that the only possible nonzero Jacobi sums determined by the deﬁning relations and the ordering given on generators are J(Yj , Xj , Xi ) = Fjj Xi − Xi Fjj , 1 ≤ i < j ≤ n, 1 ≤ i < j ≤ n, J(Yj , Xj , Yi ) = Fjj Yi − Yi Fjj , 1 ≤ j < k ≤ n, J(Yk , Yj , Xj ) = Fjj Yk − Yk Fjj , J(Xk , Yj , Xj ) = Fjj Xk − Xk Fjj , 1 ≤ j < k ≤ n.

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79

For 1 ≤ j ≤ n, at least if Fjj ∈ k-span{Xj2 , Xj , Yj , 1}, then all conditions of Proposition 2.2 and Proposition 1.6 are satisﬁed, and one checks that all Jacobi sums have weak Gr¨ obner representations. It follows that A is a tame quadric solvable polynomial algebra with ≥grlex in the case where all qj = 0. (v) Deformations of the Heisenberg enveloping algebra. Let kX ∪ Z ∪ Y be the free k algebra over {Xn , ..., X1 , Zn , ..., Z1 , Yn , ..., Y1 } = X ∪ Z ∪ Y , A = kX ∪ Z ∪ Y /I, where I is the ideal generated by the deﬁning relations x Rji = Xj X i − X i X j , 1 ≤ i < j ≤ n, y = Yj Yi − Yi Yj , 1 ≤ i < j ≤ n, Rji z Rji = Zj Zi − Zi Zj , 1 ≤ i < j ≤ n, δ zy Rji = Zj Yi − λi ji Yi Zj , 1 ≤ i, j ≤ n, δji xz 1 ≤ i, j ≤ n, Rji = Xj Zi − µi Zi Xj , xy = Xj Yi − Yi Xj , i = j, Rji xy Rjj = Xj Yj − qj Yj Xj − Fjj , 1 ≤ i ≤ n, where λi , µi , qj ∈ k, Fjj ∈ kX ∪ Z ∪ Y . If we take qj = 1 Zj = Z and Fjj = Z, 1 ≤ j ≤ n, then the enveloping algebra of 2n + 1-dimensional Heisenberg Lie algebra is recovered. In the case where λi = µi = q = 0, qj = q −1 , and Fjj = zj , we recover the q-Heisenberg algebra (CH.I §5 Example (v)). Set the monomial ordering Xn >grlex · · · >grlex X1 >grlex Zn >grlex · · · >grlex Z1 >grlex >grlex Yn >grlex · · · >grlex Y1 . Then a direct veriﬁcation shows that the only possible nonzero Jacobi sums determined by the deﬁning relations and the ordering given on generators are J(Xk , Xj , Yj ) = Fjj Xk − Xk Fjj , J(Xk , Xj , Yk ) = −Fkk Xj + Xj Fkk , J(Xk , Zj , Yk ) = −Fkk Zj + Zj Fkk , J(Xk , Yk , Yj ) = Fkk Yj − Yj Fkk , J(Xj , Yk , Yj ) = −Fjj Yk + Yk Fjj ,

1 ≤ j < k ≤ n, 1 ≤ j < k ≤ n, 1 ≤ k, j ≤ n, 1 ≤ j < k ≤ n, 1 ≤ j < k ≤ n.

It can be further checked that, for 1 ≤ j ≤ n, at least if Fjj ∈ k-span{Zj2 , Zj , Yj2 , Yj , Xj , 1}, then all conditions of Proposition 2.2 and Proposition 1.6 are satisﬁed, and all Jacobi sums have weak Gr¨ obner representations by the deﬁning relations. It follows that A is a tame quadric solvable polynomial algebra with ≥grlex in the case where all qj = 0.

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(vi) Berger’s q-enveloping algebras. Recall from [Ber] that a q-algebra A = k[a1 , ..., an ] over a commutative ring k is deﬁned by the quadric relations Rji

=

Xj Xi − qji Xi Xj − {Xj , Xi }, 1 ≤ i < j ≤ n, where qji ∈ k, k k αji Xk X + αh Xh + cji , αji , αh , cji ∈ k, and {Xj , Xi } = kl satisfying if αji = 0, then i < k ≤ < j, and k − i = j − .

Deﬁne two k-subspaces of the free algebra kX1 , ..., Xn E1 = k-Span Rji n ≥ j > i ≥ 1 , E2

=

k-Span Xi Rji , Rji Xi , Xj Rji , Rji Xj n ≥ j > i ≥ 1 .

For 1 ≤ i < j < k ≤ n, if every Jacobi sum J(Xk , Xj , Xi ) is contained in E1 + E2 , then A is called a q-enveloping algebra with respect to the natural total ordering an > an−1 > · · · > a1 . A q-enveloping algebra is said to be invertible if in the deﬁning relations all coeﬃcients qji are invertible, 1 ≤ i < j ≤ n. In [Ber] the q-PBW theorem was obtained for q-enveloping algebras, that is, αn 1 the set of standard monomials {aα | (α1 , ..., αn ) ∈ IN n } forms a k1 · · · an basis for a q-enveloping algebra A. Clearly, if we set the monomial ordering Xn >grlex Xn−1 >grlex · · · >grlex X1 , then the deﬁning relations of a q-algebra A satisfy LM(Rji ) = Xj Xi , 1 ≤ i < j ≤ n and k k k, < j in αji Xk X whenever αji = 0. Hence, by Proposition 1.6 and Proposition 2.2, the deﬁning relations of a qenveloping algebra form a Gr¨ obner basis in kX1 , ..., Xn ; if furthermore A is an invertible q-enveloping algebra then A is a tame quadric solvable polynomial algebra (indeed, A is an iterated skew polynomial algebra by the proof of Theorem 2.9.4 in [Ber]). We observe that the conditions i < k ≤ < j and k − i = j − in the deﬁnition of a q-algebra are not necessarily satisﬁed by a quadric solvable polynomial algebra, or more generally, a quadric algebra characterizied by Proposition 1.6 is not necessarily a q-enveloping algebra in the sense of [Ber]. Remark In the end of ﬁrst part of [LWZ], it was pointed out that a q-enveloping algebra over a ﬁeld k is generally not a solvable polynomial algebra with respect to ≥grlex . This is, of course, not true for invertible q-enveloping algebras, as argued in the above example (vi). The author takes this place to correct that incorrect remark.

3. Associated Homogeneous Deﬁning Relations

81

3. Associated Homogeneous Deﬁning Relations of Algebras Let k be any ﬁeld, let kX be the free k-algebra over X = {Xi }i∈Λ , and let S = X be the free semigroup generated by X = {Xi }i∈Λ . From CH.I §4 Example (i) and (iv) we retain that kX has the natural gradation deﬁned by the degree of words in S, i.e., kX = ⊕p∈IN kXp with kXp = { d(w)=p cw w | cw ∈ k, w ∈ S}, and that the standard ﬁltration (or the grading ﬁltration) F kX on kX is given by the k-subspaces: Fp kX = ⊕i≤p kXi ,

p ∈ IN .

If I is a two-sided ideal of kX and A = kX/I, then F kX induces a ﬁltration F A on A: Fp A = (Fp kX + I)/I, p ∈ IN . Indeed, F A coincides with the standard ﬁltration on the k-algebra A = kX/I = k[ai ]i∈Λ where each ai is the image of Xi in kX/I. If we consider the associated graded algebra G(A) = ⊕p∈IN (Fp A/Fp−1 A) of A = ⊕p∈IN Fp A of A, then it follows from CH.I Proposition and the Rees algebra A are determined by I ∗ , where I ∗ is the homogenization ideal 4.4 that G(A) and A of I in the polynomial ring kX[t] in commuting variable t, namely,  ∼ = kX[t]/I ∗ , and  A (∗)  G(A) ∼ = kX[t]/(t + I ∗ ), where t denotes the ideal of kX[t] generated by t. With notation as above, let {fj }j∈J be a set of deﬁning relations of the k-algebra A = kX/I, i.e., the two-sided ideal I is generated by {fj }j∈J . Let F A be the be the associated graded algebra standard ﬁltration on A, and let G(A) and A and Rees algebra of A with respect to F A, respectively. In view of the above (∗), it is natural to ask the following question: in terms of Question Can we determine the deﬁning relations of G(A) and A the deﬁning relations of A? Inspired by a remarkable application of Gr¨ obner basis in algebraic geometry, in this section we give an (even stronger) positive answer to the above question. To be convenient, we recall from (e.g., [CLO ]) the following result. Let k be an algebraically closed ﬁeld of characteristic 0, and k[x1 , ..., xn ] the commutative polynomial k-algebra in n variables. Let I be an ideal of k[x1 , ..., xn ], and let I ∗ denote the homogenization ideal of I in k[x0 , x1 , ..., xn ] with respect

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to x0 . Then the projective algebraic set V (I ∗ ) deﬁned by I ∗ in the projective n-space Pnk gives the projective closure of the aﬃne algebraic set V (I) deﬁned by I in the aﬃne n-space Ank . The following result tells us that, using the Gr¨ obner basis method, the deﬁning equations of the projective closure V (I ∗ ) of the aﬃne algebraic set V (I) can be determined from that of V (I), or equivalently, the deﬁning relations of the graded k-algebra k[x0 , x1 , ..., xn ]/I ∗ can be determined from that of k[x1 , ..., xn ]/I. • If G = {g1 , ..., gs } is a Gr¨obner basis for I with respect to a graded monomial ordering in k[x1 , ..., xn ], then G ∗ = {g1∗ , ..., gs∗ } is a Gr¨ obner basis for I ∗ ⊂ ∗ k[x0 , x1 , ..., xn ], where gi is the homogenization of gi in k[x0 , x1 , ..., xn ]. Before studying the question posed above, let us also recall some well known examples (see CH.I §5). Example (i) Let g = kx1 ⊕ · · · ⊕ kxn be an n-dimensional Lie algebra over k n with [xi , xj ] = h=1 λhij xh , and let A = U (g) be the enveloping algebra of g with the standard ﬁltration F U (g). Then by the PBW theorem we know that G(U (g)) is, as a graded k-algebra, isomorphic to the polynomial k-algebra in n variables. (ii) Let A = An (k) = k[x1 , ..., xn , y1 , ..., yn ] be the nth Weyl algebra over k as stated in CH.I §5. Then it is well known that, with respect to the standard ﬁltration (or Bernstein ﬁltration) on An (k), G(An (k)) is, as a graded k-algebra, isomorphic to the polynomial k-algebra in 2n variables. Note that in both examples (i) and (ii) the proof of the fact about G(A) is nontrivial in the literature (e.g., [Bj], [Dix]). (iii) Let g and A = U (g) be as in (i). Related to the study of quantum groups, (g) which is generated by S.P. Smith proposed in [Sm1] the quadratic algebra U x0 , x1 , ..., xn where x0 is taken to be central and the remaining deﬁning relations n are [xi , xj ] = h=1 cij,h xh x0 . In [LeS] and [LeV], this algebra was called the (g) looks very like the Rees homogenized enveloping algebra. Observe that U (g)/1 − x0 U (g) ∼ algebra of U (g), namely, there is U = U (g). (We will see in the end of this section that U (g) is exactly the Rees algebra of U (g).) A general interpretation on the homogenized algebra of a given algebra is given after Theorem 3.7 below. From the above examples one might expect that for a k-algebra A with standard may be given by ﬁltration F A, the deﬁning relations of G(A), respectively A, simply taking the highest degree homogeneous part of the deﬁning relations of A, respectively by simply taking the homogenizations of the deﬁning relations of A in the sense of CH.I §4. As indicated by the following examples, however, (even in the commutative case) the question we posed above is not so trivial to

3. Associated Homogeneous Deﬁning Relations

83

answer in general. Example (iv) Consider I = f1 , f2 = x2 − x21 , x3 − x31 , the ideal of the aﬃne twisted cubic in IR3 . If we homogenize f1 , f2 , then we get the ideal J = x2 x0 − x21 , x3 x20 − x31 in IR[x0 , x1 , x2 , x3 ]. One may directly check that for f3 = f2 − x1 f1 = x3 − x1 x2 ∈ I, f3∗ = x3 x0 − x1 x2 ∈ J, i.e., J = I ∗ . (v) Let kX be the free k-algebra generated by {X1 , X2 , X3 }, and let f = 2X3 X2 X1 − 3X1 X32 , g12 = X2 X1 − X1 X2 , g13 = X3 X1 − X1 X3 , g23 = X3 X2 − X2 X3 . Considering the two-sided ideal I = f, g12 , g13 , g23 , then it can be directly veriﬁed that h = −3X1 X32 +2X1 X2 X3 = f −2X3 g12 +2g13 x2 +2X1 g23 ∈ ∗ ∗ ∗ I ∗ , but h ∈ f ∗ , g12 , g13 , g23 (note that the latter is equal to f, g12 , g13 , g23 in kX[t]). Remark In the above examples (iv) and (v) there has been nothing about G(A). However, it will be clear from Theorem 3.7 below that generally the deﬁning relations of G(A) cannot be obtained by simply taking the highest degree homogeneous part of the deﬁning relations of A. Nevertheless, based on the foregoing relationships in (∗), the result (•) recalled above still gives us the light, i.e., we may ask Question If the deﬁning relations of A form a Gr¨ obner basis, what will happen to the deﬁning relations of G(A) and A? As a preliminary result, in order to answer the above question, we show that if {fj }j∈J is a standard basis of I in the sense of (e.g., [Gol]), then the deﬁning and G(A) can be completely determined. relations of A To see why the standard basis is the ﬁrst choice in our discussion, we ﬁrst strengthen previous (∗) as follows. For any f ∈ kX we denote by LH(f ) the highest degree homogeneous part of f , i.e., if f = F0 +F1 +· · ·+Fp with Fi ∈ kXi , then LH(f ) = Fp . If I is an ideal of kX, we denote by LH(I) the graded ideal generated by {LH(f ) | f ∈ I} in kX. 3.1. Proposition Let A = kX/I and F A the standard ﬁltration on A. Then G(A) ∼ = kX/LH(I). Proof We know that G(A) ∼ = kX[t]/(t + I ∗ ). To prove the theorem, we ﬁrst recall that if f = F0 + F1 + · · · + Fp ∈ kX, then LH(f ) = Fp and f ∗ = LH(f ) + tFp−1 + · · ·. Hence the inclusion map kX → kX[t] yields the

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inclusion LH(f ) ⊂ t + I ∗ . This, in turn, yields a graded ring homomorphism kX LH(f ) g + LH(f )

ϕ

−→ →

kX[t] (t + I ∗ ) g + (t + I ∗ )

Obviously, ϕ is surjective. On the other hand, each element F ∈ kX[t] has a unique presentation F = F0 + F where F0 ∈ kX, F ∈ t. Moreover, from CH.I Lemma 4.2 we know that each homogeneous element in I ∗ is of the form tr f ∗ for some f ∈ I. If f = Fp + Fp−1 + · · · + F0 with LH(f ) = Fp , then f ∗ = LH(f ) + tFp−1 + · · · + tp F0 . Therefore, each element of t + I ∗ can be written as a sum u + v, where u ∈ LH(f ), v ∈ t. Thus we can deﬁne a ring homomorphism kX kX[t] ψ −→ ∗ (t + I ) LH(I) F + (t + I ∗ )

→

F0 + LH(I)

Since ψ ◦ ϕ = 1, it follows that ϕ is also injective and hence an isomorphism. (Indeed, ψ is the ring homomorphism induced by the canonical homomorphism kX[t] → kX which sends t to 0.) 2 Suppose I = fj j∈J . From the above proposition we certainly expect that LH(I) = LH(fj )j∈J . This leads to the use of standard bases. 3.2. Deﬁnition The set {fj }j∈J is called a standard basis of I if each element f ∈ I with p = d(f ), where f = F0 + F1 + · · · + Fp with Fi ∈ kXi and Fp = 0 (see CH.I §3), has a presentation as a ﬁnite sum f = j gj fj hj , where gj , hj ∈ kX, and d(gj ) + d(fj ) + d(hj ) ≤ p for all j. Let I be a graded ideal of kX, i.e., I = ⊕p∈IN (I ∩ kXp ). If {fj }j∈J is a generating set of I consisting of homogeneous elements, then it is easy to see that {fj }j∈J is a standard basis of I. But generally it is not so easy to check if a generating set of an ideal is a standard basis. We refer to [Gol] for a homological criterion of standard basis. The ﬁrst easy but important property of a standard basis is the following 3.3. Lemma If {fj }j∈J is a standard basis of I, then for any f ∈ I, LH(f ) = LH(gj )LH(fj )LH(hj ) for some gj , hj ∈ kX, fj ∈ {fj }j∈J . Indeed, we have the more stronger result: {fj }j∈J is a standard basis if and only if LH(fj )j∈J = LH(I). Proof If {fj }j∈J is a standard basis of I, then by the deﬁnition it is easy to see that for any f ∈ I, LH(f ) = LH(gj )LH(fj )LH(hj ) for some gj , hj ∈ kX,

3. Associated Homogeneous Deﬁning Relations

85

fj ∈ {fj }j∈J . Hence LH(fj )j∈J = LH(I). Conversely, if LH(fj )j∈J = LH(I), then for any f ∈ I with d(f ) = p, say gj LH(fj )hj f = fp + fp−1 + · · · with fi ∈ kXi , we have LH(f ) = fp = for some gj , hj ∈ kX, fj ∈ {fj }j∈J , and d(gj ) + d(LH(fj )) + d(hj ) = d(gj ) + gj fj hj ∈ I has d(f ) < p, d(fj ) + d(hj ) = p. Now the element f = f − we may repeat the above argumentation and after a ﬁnite number of steps we will reach a presentation f = gj fj hj where gj , hj∈ kX, fj ∈ {fj }j∈J and d(gj ) + d(fj ) + d(hj ) ≤ p for all i. It follows that {fj }j∈J is a standard basis of I. 2 3.4. Proposition With notation as before, if {fj }j∈J is a standard basis of I, then (i) G(A) has deﬁning relations LH(fj ), j ∈ J; and moreover (ii) {LH(fj )}j∈J is a standard basis of LH(I). Proof This follows immediately from Proposition 3.1 and Lemma 3.3.

2

3.5. Proposition With notation as before, if {fj }j∈J is a standard basis of I, then of A, viewed (i) I ∗ is generated by {fi∗ }i∈J , or in other words, the Rees algebra A as a quotient of kX[t] by previous (∗), has deﬁning relations Ri = tXi − Xi t, Rj = fj∗ ,

i∈Λ j ∈ J;

and moreover (ii) {fj∗ }j∈J is a standard basis of I ∗ . Proof By CH.I Lemma 4.2, each homogeneous element in I ∗ is of the form tr f ∗ for some f ∈ I. Suppose f = j hj fj gj . Since {fj }j∈J is a standard basis of I, it follows from Lemma 3.3 and the deﬁnition of homogenization of f that d(h∗j ) + d(fj∗ ) + d(gj∗ ) ≤ d(f ∗ ) and f ∗ − j h∗j fj∗ gj∗ = tr1 m∗1 + tr2 m∗2 + · · · with rj > 0, mj ∈ I, and d(trj m∗j ) ≤ d(f ∗ ) for all mj . Similarly, for each m∗j ∈ I ∗ where mj = i hij fij gij , we have d(h∗ij ) + d(fi∗j ) + d(gi∗j ) ≤ d(m∗j ) and m∗j − i h∗ij fi∗j gi∗j = tr1j m∗1j + tr2j m∗2j + · · · with rkj > 0, mkj ∈ I, and d(trkj m∗kj ) ≤ d(m∗j ) for all mkj . Since d(f ∗ ) is ﬁnite, after a ﬁnite number of steps we will reach f ∗ ∈ fj∗ j∈J , in particular, f ∗ = j h∗j fj∗ gj∗ with d(h∗j ) + d(fj∗ ) + d(gj∗ ) ≤ d(f ∗ ) for all j. (Note that I is a proper ideal, the ﬁnal step of the reduction procedure cannot reach an expression like l tl .) This proves the conclusions of (i) and (ii). 2

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Remark One may also obtain Proposition 3.4 from Proposition 3.5. To see this, suppose I ∗ = fj∗ j∈J . Then since each element F of I ∗ is of the form F = j Hj fj∗ Gj , Hi , Gi ∈ kX[t], we can deﬁne the ring homomorphism ψ and accomplish the argumentation as in the proof of Proposition 3.1. Now we return to use Gr¨ obner bases. Let kX be the free k-algebra with the kbasis B consisting of words in the free semigroup S = X, where X = {Xi }i∈Λ . If (kX, B, gr ) is an admissible system, where gr is some graded monomial ordering on B (CH.II Deﬁnition 1.2), it follows immediately from the deﬁnition that • Any Gr¨ obner basis G for an ideal I in kX with respect to gr is a standard basis of I in the sense of Deﬁnition 3.2. Thus we have reached the following result. 3.6. Theorem With notation as before, let I be an ideal of kX and A = kX/I. Consider the standard ﬁltration F A on A. If G = {fj }j∈J is a Gr¨obner basis for I with respect to gr , then G(A) has the deﬁning relations Rj = LH(fj ), j ∈ J, has the deﬁning relations and A Ri = tXi − Xi t, Rj = fj∗ ,

i ∈ Λ, j ∈ J. 2

Furthermore, let us consider the k-basis B(t) = wtr w ∈ B, r ≥ 0 for kX[t]. Then the ordering gr on B induces a monomial ordering on B(t), again denoted gr , as follows: w1 tr1 gr w2 tr2 if and only if w1 gr w2 , or w1 = w2 and r1 > r2 . With the deﬁnition as abobe, we have Xj gr tr for all j ∈ Λ and all r ≥ 0, and we can discuss Gr¨obner basis in kX[t] exactly as in kX. Before mentioning the next theorem we also note that for any nonzero f ∈ kX, the degree-preserving ordering gr yields the following equalities: LM(f ) = LM(LH(f )) LM(f ∗ ) = LM(f ).

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87

3.7. Theorem With notation as before, let G = {fj }j∈J ⊂ I where I is a two-sided ideal of kX. The following are equivalent: (i) G is a Gr¨ obner basis of I; (ii) {fj∗ }j∈J is a Gr¨ obner basis of I ∗ in kX[t]; obner basis of the two-sided ideal LH(I) in kX. (iii) {LH(fj )}j∈J is a Gr¨ Proof (i) ⇒ (ii). By the above remark, this can be proved exactly as we did in the proof of Proposition 3.5. (ii) ⇒ (i). From CH.I Lemma 4.2 we know that (f ∗ )∗ = f holds for any f ∈ I. So the assertion follows again from the above remark. (i) ⇔ (iii). Using the above remark, this can be directly checked. 2 For a given ﬁnitely generated k-algebra A = k[a1 , ..., an ], except for G(A) and A associated to A with respect to the standard ﬁltration F A on A, another graded structure used by several authors in recent years is the so called homogenized algebra of A. More precisely, suppose that A is deﬁned subject to the relations Fj (a1 , ..., an ) = 0,

j ∈ J,

where the Fj are elements in the free algebra kX1 , ..., Xn . Then the homogenized (graded) algebra of A, usually denoted H(A) in the literature, is deﬁned by the relations: Ri = Xi T − T Xi , 1 ≤ i ≤ n, j ∈ J, Rj = Fj , where each Fi is the homogenization of Fi with respect to T (say from right hand side) in the free k-algebra kX1 , ..., Xn , T (e.g., if F = X1 X22 + X3 , then F = X1 X22 + X3 T 2 ), or in other words, H(A) = kX1 , ..., Xn , T /I, where I is the two-sided ideal generated by {Ri , Fj | 1 ≤ i ≤ n, j ∈ J}. The study of homogenized enveloping algebra was proposed by Smith in [Sm1], and the study of homogenized down-up algebra was proposed by Benkart and Roby in [BR]. The reader is refered to, e.g., [LeS], [LeV] and [Le2] for some representation theory and noncommutative geometry of homogenized enveloping algebras. From the construction of the homogenized algebra of a given algebra A it is clear e.g., H(A)/(1 − T )H(A) ∼ that the behavior of H(A) is similar to A, = A, where and H(A), G(A) and T is the image of T in H(A), and the relations between A H(A)/T H(A) are given by the following exact sequences: →0 H(A) → A H(A)/T H(A) → G(A) → 0 Furthermore, by ([Li1], [LVO4]) there are two basic facts: work can be done through G(A) = A/X A, because X is a. The lifting (to A) may have contained in the graded Jacobson radical of A (be aware that A

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zero Jacobson radical); and the lifting (to A) work can be done through G(A) because A is complete with respect to its ﬁltration topology. because there is b. The dehomogenizing (to A) work can be done through A, a nice relation between the graded module category A-gr and the ﬁltered module category A-ﬁlt. It is easy to see that the similar facts hold for H(A) and H(A)/T H(A). Note that the generators of both G(A) and H(A)/T H(A) satisfy simpler relations given by the highest degree homogeneous part of the deﬁning relations of A. In order to determine the algebraic properties of G(A) or H(A)/T H(A), it remains only to ﬁnd the deﬁning relations of G(A) or H(A)/T H(A). Extending the notion of homogenized algebra to arbitrary k-algebra, below we ∼ apply the results of previous sections to assert that in many cases A = H(A). 3.8. Corollary With notation as above, the equivalent conditions of Theorem 3.7 are equivalent to one of the following two conditions. (i) {Xi T − T Xi , fj | i ∈ Λ, j ∈ J} forms a Gr¨obner basis in kXi , T i∈Λ with respect to ≥grlex such that Xi >grlex T , i ∈ Λ. ∼ (ii) A = H(A), where H(A) is the homogenized (graded) algebra of A = k[xi ]i∈Λ . 2 Certainly, Theorem 3.7 and Corollary 3.8 can be applied to general quadric solvable polynomial algebras deﬁned in previous §2, but we leave this to CH.IV §4 for completeness. Let us ﬁnish this section by returning to the examples given in CH.I §5. 3.9. Corollary If A is one of the k-algebras listed in CH.I §5 Example (i)–(vii), then, after setting a suitable ordering on the generators in order to deﬁne ≥grlex as in the examples of §2, one may make Proposition 2.2 and Proposition 1.6 hold, and therefore, the deﬁning relations of A form a Gr¨ obner basis in the corresponding free algebra with respect to ≥grlex . (Note that in [Mor2] it has been veriﬁed that the deﬁning relations of An (k), respectively the deﬁning relations of U (g), form a Gr¨ obner basis.). Moreover, one may further use Theorem 3.6– Corollary 3.8 to get the deﬁning relations for the associated graded structures are quadratic algebras and most of them are of A, in particular, G(A) and A iterated skew polynomial algebras starting with the ground ﬁeld k. For instance, we mention one of the following consequences: (i) The associated graded algebra G(An (k)) of the nth Weyl algebra An (k) over a ﬁeld k has the deﬁning relations X j Xi − Xi Xj ,

i = j, i, j = 1, ..., n.

Hence G(An (k)) is isomorphic to the commutative polynomial k-algebra in 2n

4. Remark

89

variables. And the Rees algebra A n (k) of An (k) has the deﬁning relations Yj Yi − Yi Yj , Xj Xi − Xi Xj , 1 ≤ i < j ≤ n, 1 ≤ i ≤ n, Yi T − T Yi , Xi T − T Xi , 1 ≤ i, j ≤ n, Yj Xi − Xi Yj − δij T 2 , that make A n (k) into an iterated skew polynomial algebra. (ii) Let g be a ﬁniten- dimensional Lie algebra over a ﬁeld k and U (g) its enveloping algebra. Then the associated graded algebra G(U (g)) of U (g) has the deﬁning relations Xj Xi − Xi Xj ,

i = j, 1 ≤ i < j ≤ n.

Hence G(U (g)) is isomorphic to the commuttaive polynomial k-algebra in n variables. And the Rees algebra U (g) of U (g) has the deﬁning relations Xi T − T Xi , Xj Xi − Xi Xj −

n

1 ≤ i ≤ n, λij X T, 1 ≤ i < j ≤ n.

=1

It follows that the so called homogenized enveloping algebra proposed in [Sm1] and studied in [LeV] and [LeS] is nothing but exactly the Rees algebra of U (g).

4. A Remark on Recognizable Properties of Algebras via Gr¨ obner Bases For a given k-algebra A and a certain algebraic property P of A, in general we hardly know whether or not P can be algorithmically determined (e.g., see [Ufn1]). However, we have seen from CH.II §4 that a stronger Noetherian property, i.e., the G-Noetherian property of an algebra A may be realized by ﬁnding a (left) Dickson system associated with A. Moreover, from previous sections we also have seen that given a k-algebra A = kX/I, where kX is the free kalgebra over X = Xi i∈J and I is an ideal of kX, if I has a Gr¨ obner basis, then the structural properties of A, such as the ideal membership and to have a PBW basis, can be algorithmically established. Not only this, indeed, the earlier work of [Berg] has motivated more discovery of recognizable properties of algebras via very noncommutative Gr¨ obner bases. To be convincible, it is the aim of this remark section to introduce the work of [G-IL] which establishes the existence of algorithms for recognizing certain structural properties of ﬁnitely generated k-algebras. We refer the reader to [G-IL] for detailed proofs of the statements mentioned in this section and for relevant algorithms written in pseudo-codes.

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In what follows, kX = kX1 , ..., Xn , i.e., the algebra A = kX/I is ﬁnitely generated, and we suppose that the two-sided ideal I is generated by G = {g1 , ..., gs }, i.e., A is ﬁnitely presented. 4.1. Theorem ([G-IL] Theorem 1) Suppose that G is a Gr¨obner basis for I. Then the following properties of A are algorithmically recognizable: (i) A has a growth of a given type, in particular, A is ﬁnite dimensional. (ii) A is algebraic over k. (iii) A satisﬁes the polynomial identity [X1 , ..., Xr ] of Lie nilpotency, for a ﬁxed integer r. 2 4.2. Theorem ([G-IL] Theorem 2) Suppose that the generating set G of I consists of monomials (hence G is a Gr¨ obner basis for I by CH.II Proposition 3.5). Then the following properties of A are algorithmically recognizable: (i) A has no nilpotent elements. (ii) A is semisimple (in the sense of Jacobson). (iii) A is prime. (iv) A is semiprime. 2 Note that the algebra A appearing in Theorem 4.2 above has been called a monomial algebra in [G-IL]. We also refer to [G-I1–3] and [Ufn1] for more discussion in algorithmically recognizing some other structural properties of ﬁnitely presented algebras, such as Jacobson radical, Hilbert and Poincar´e series, global (homological) dimension, and noetherianity, etc. In CH.VIII, we will see that the results obtained in §1–§3 of this chapter and CH.IV §4 enable us to show that every quadric solvable polynomial algebra has ﬁnite global dimension and K0 -group ZZ.

CHAPTER IV Filtered-Graded Transfer of Gr¨ obner Bases

Let A = k[ai ]i∈Λ be a k-algebra generated by {ai }i∈Λ over the ﬁeld k. Let F A be the standard ﬁltration on A, G(A) = ⊕p∈IN G(A)p with G(A)p = Fp A/Fp−1 A, p = Fp A, the Rees = ⊕p∈IN A p with A the associated graded algebra of A, and A algebra of A in the sense of CH.I §3. Then by CH.I Proposition 3.16, a. G(A) = k[σ(ai )]i∈Λ where each σ(ai ) is the image of ai in F1 A/F0 A = G(A)1 , and = k[X, b. A ai ]i∈Λ where X is the canonical element of degree 1 represented 1 = F1 A and each ai is the homogeneous element of degree 1 by 1 in A 1 = F1 A. represented by ai in A Suppose that A ∼ = kX/fj j∈J , where kX is the free k-algebra on X = {Xi }i∈Λ and fj j∈J is the two-sided ideal of kX generated by {fj }j∈J . Let B be the k-basis of kX consisting of all words of the free semigroup S = X. If {fj }j∈J forms a Gr¨ obner basis with respect to a graded monomial ordering gr on B, then CH.III Theorem 3.6 entails that, with respect to the standard ﬁltration of F A on A, ∼ c. G(A) ∼ = kX/LH(fj )j∈J and A = kX[t]/fj∗ j∈J , where each LH(fj ) is the highest degree homogeneous part of fj in kX, and each fj∗ is the homogenization of fj in the polynomial ring kX[t] with respect to the commuting variable t. However, if we consider the standard ﬁltration (or the grading ﬁltration) on

H. Li: LNM 1795, pp. 91–105, 2002. c Springer-Verlag Berlin Heidelberg 2002

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IV. Filtered-Graded Transfer of Gr¨ obner Bases

∼ kX, it follows from CH.I Proposition 3.18 that G(kX) ∼ = kX and kX = kX[t]. In other words, an application of CH.III Theorem 3.6 to kX indeed yields the following equivalence: d. {fj }j∈J is a Gr¨ obner basis in kX. obner basis in G(kX). e. {LH(fj )}j∈J is a Gr¨ obner basis in kX. f. {fj∗ }j∈J is a Gr¨ Noticing the the above facts and the interpretation given in CH.III §3, in this chapter we develop a general philosophy of the ﬁltered-graded transfer of Gr¨ obner bases so that, on one hand, certain Gr¨ obner basis theory may be valid for more algebras, and on the other hand, certain computations using noncommutative Gr¨ obner bases may possibly be carried out ﬁrst at a feasible graded level and then be lifted (or dehomogenized) back to ungraded level (as illustrated in later CH.V–CH.VIII). To be convenient, we ﬁx the convention of this chapter once and for all. Let A = k[ai ]i∈Λ be a k-algebra as above, B a ﬁxed k-basis of A consisting of monomials, that is, αn 1 B = w = aα n ≥ 1, α · · · a ∈ I N , i i1 in and F A the standard ﬁltration on A determined by B: Fp A = ci wi ci ∈ k, wi ∈ B, d(wi ) ≤ p ,

p ∈ IN ,

αn 1 where d(w) denote the degree of w (see CH.II §1, e.g., if w = aα i1 · · · ain then d(w) = |α| = α1 + · · · + αn ). We call this ﬁltration the B-standard ﬁltration on A. Moreover, assume that B is a strictly ﬁltered basis in the sense that • B= Bp with Bp = {w ∈ B | d(w) ≤ p}, and if d(w) = p then w ∈ Bp−1 . p≥0

Let A = k[a1 , ..., an ] be a ﬁnitely generated k-algebra. If the set of all standard monomials n αn 1 B = aα 1 · · · an (α1 , ..., αn ) ∈ IN forms a k-basis (i.e., a PBW basis in the sense of CH.III §1), then, the deﬁnition of B-standard ﬁltration F A on A shows clearly that B is strictly ﬁltered. So the assumption (•) is indeed quite mild.

1. Filtered-Graded Transfer of (Left) Admissible Systems Let A, B and F A be as assumed in the beginning of this chapter. Let G(A) = = ⊕p≥0 A p be the associated graded algebra and the Rees ⊕p≥0 G(A)p and A

1. Filtered-Graded Transfer of Admissible Systems

93

algebra of A, respectively. With notation as in CH.I §3, let v be the order function on A determined by F A. If f ∈ A with v(f ) = p (i.e., f ∈ Fp A−Fp−1 A), p is then the homogeneous element represented by f in G(A)p , respectively in A denoted by σ(f ), respectively by f . In this section we discuss the ﬁltered-graded transfer of (left) admissible systems associated with A, G(A), and A. First note some basic facts about a strictly ﬁltered k-basis B deﬁned by the foregoing (•). 1.1. Lemma (i) For each w ∈ B, d(w) = v(w). (ii) For each p ∈ IN , Bp is a k-basis of Fp A. (iii) For w, u ∈ B, σ(u) = σ(w) if and only if u = w.

2

Next we establish the ﬁltered-graded transfer of k-bases between A, G(A), and respectively. A, αn 1 1.2. Lemma For any w ∈ B, if w = aα i1 · · · ain with d(w) = p, then σ(w) = α n = ai11 · · · aα σ(ai1 )α1 · · · σ(ai1 )αn ∈ G(A)p and w in ∈ Ap .

Proof Since w = 0 (element of the k-basis), we conclude by the assumption (•) αn α1 1 · · · σ(ai1 )αn = 0 and on B that w = aα i1 · · · ain ∈ Fp A − Fp−1 A. Hence, σ(ai1 ) it follows from CH.I Lemma 3.11 that σ(w) = σ(ai1 )α1 · · · σ(ai1 )αn ∈ G(A)p and 1 n aα 2 w = aα i1 · · · in ∈ Ap . 1.3. Proposition Let A, B, and F A be as ﬁxed. Then the following are equivalent. (i) B is a strictly ﬁltered k-basis for A. = p} forms a k-basis of G(A)p for each (ii) σ(Bp ) = {σ(w) | w ∈ Bp, d(w) p ∈ IN . Hence σ(B) = σ(w) w ∈ B forms a k-basis for G(A). p for each (iii) Bp = {wX p−s | w ∈ Bp , d(w) = s ≤ p} forms a k-basis of A h where X is p ∈ IN . Hence B = {wX | w ∈ B, h ∈ IN } forms a k-basis for A, (CH.I §3). the canonical element of degree 1 in A Proof (i) ⇔ (ii) Let f = ci wi be an element of A with ci ∈ k, wi ∈ B. If f ∈ Fp A − Fp−1 A, then since F A is the B-standard ﬁltration, we have wi ∈ Bp and σ(f ) = d(wi )=p ci σ(wi ). Thus, G(A)p is, as a k-space, spanned by σ(Bp ). If i λi σ(wi ) = 0 for λi ∈ k and σ(wi ) ∈ σ(Bp ), then i λi wi ∈ Fp−1 A. But this implies λi = 0 for all i as B is a strictly ﬁltered basis by (i). Since F A is separated and d(w) = v(w) for w ∈ B by Lemma 1.1, the implication (ii) ⇒ (i) may be directly veriﬁed. (i) ⇔ (iii) Note that wi ∈Bp λi w i X p−si = 0 implies i λi wi = 0 and λi = 0 by p is spanned by Bp . By CH.I Proposition 3.8. It is suﬃcient to show that A

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p is of the form fX p−s , CH.I Lemma 3.11, every homogeneous element F in A where s = v(f ) ≤ p, i.e., f ∈ Fs A − Fs−1 A. Suppose that f = i ci wi with ci ∈ k and wi ∈ B. Since F A is the B-standard ﬁltration, wi ∈ Fs A ⊆ Fp A and f = i ci w i X s−qi where qi = d(wi ) = v(wi ) = qi ≤ s. Hence, F = fX p−s = p−qi i X with qi ≤ s ≤ p, as desired. Similarly the implication (iii) ⇒ (i) i ci w may be veriﬁed. 2 Now, let (A, B, gr ) be a (left) admissible system with gr a graded monomial ordering on B in the sense of CH.II Deﬁnition 1.2(iii). With notation as in the ordering Proposition 1.3, deﬁne on σ(B), respectively on B,   σ(u) gr σ(w) if and only if u gr w, (∗) respectively  u X h gr wX if and only if u gr w or u = w and h > . From the deﬁnition (∗) above we observe that 1 and all p ≥ 0. a. w gr X p for all w ∈ B with w = b. gr is a well-ordering on σ(B), respectively on B. The next lemma shows that gr is compatible with the B-standard ﬁltration F A on A. 1.4. Lemma Let (A, B, gr ) be as above. Then the following holds. (i) For f ∈ A, f ∈ Fp A if and only if d(LM(f )) = v(f ) ≤ p, and f ∈ Fp A − Fp−1 A if and only if d(LM(f )) = p; for f, g ∈ A, LM(f ) gr LM(g) implies d(LM(f )) = v(f ) ≥ v(g) = d(LM(g)). (ii) For each w ∈ B, σ(w), respectively w, is a monomial of G(A), respectively a and moreover d(w) = d(σ(w)) = d(w). monomial of A, (iii) With the deﬁnition as in (∗) above, for f ∈ A, σ(LM(f )) = LM(σ(f )), ) = LM(f), and LM(X s f) = X s LM(f) for all s ≥ 0. LM(f Proof (i) Since B is strictly ﬁltered and F A is the B-standard ﬁltration, this is clear by Lemma 1.1. (ii) This follows from Lemma 1.2. n (iii) If f ∈ Fp A − Fp−1 A, we may write f = i=1 λi wi , where λi ∈ k − {0}, such that wi ∈ Bp and w1 gr w2 gr · · · gr wn . Then LM(f ) = w1 and d(LM(f )) = v(f ) = d(w1 ) = p by (i). Thus, σ(f ) = d(wi )=p λi σ(wi ) and it follows from the deﬁnition (∗) that LM(σ(f )) = σ(w1 ) = σ(LM(f )). With the n 1 + i=2 λi w i X p−si , where si = d(wi ) = v(wi ) < p, and same f , we have f = λ1 w ), and LM(X s f) = it follows from the deﬁnition (∗) that LM(f) = w 1 = LM(f s 2 X LM(f ) for all s ≥ 0. This ﬁnishes the proof.

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95

We are ready to mention the main result of this section. 1.5. Theorem Let F A be the B-standard ﬁltration on A. The following statements hold. (i) If (A, B, gr ) is a (left) admissible system, where gr is some graded monomial ordering on B, then (G(A), σ(B), gr ) is a (left) admissible system in which gr is deﬁned as in the foregoing (∗). (ii) If (A, B, gr ) is a (left) admissible system, where gr is some graded mono B, gr ) is a (left) admissible system in which gr mial ordering on B, then (A, is deﬁned as in the foregoing (∗). Proof (i) Let (A, B, gr ) be as assumed. By Proposition 1.3 and the observations mentioned below the deﬁnition (∗), we only need to verify CH.II Definition 1.2 (MO1)–(MO2) for the elements of σ(B). using gr as deﬁned in the foregoing (∗), if σ(w), σ(u), σ(v), σ(s) ∈ σ(B) with σ(u) gr σ(w), and if LM(σ(v)σ(w)σ(s)) = 0 and LM(σ(v)σ(u)σ(s)) = 0, then  u gr w,    σ(vws) = σ(v)σ(w)σ(s) = 0, (1) (CH.I Lemma 3.11) σ(vus) = σ(v)σ(u)σ(s) = 0,    LM(vws) = 0, LM(vus) = 0. Hence, LM(vus) gr LM(vws) and this yields σ(LM(vus)) gr σ(LM(vws)). By Lemma 1.4 and (1) above, we have σ(LM(vus)) = LM(σ(vus)) = LM(σ(v)σ(u)σ(s)), σ(LM(vws)) = LM(σ(vws)) = LM(σ(v)σ(w)σ(s)). Therefore, LM(σ(v)σ(u)σ(s)) gr LM(σ(v)σ(w)σ(s)). This veriﬁes (MO1). If σ(u) = LM(σ(v)σ(w)σ(s)) with σ(v) = 1 or σ(s) = 1, then by CH.I Lemma 3.11 and the foregoing Lemma 1.4, we obtain that σ(u) = LM(σ(vws)) = σ(LM(vus)) and consequently u = LM(vws) (Lemma 1.1). But this implies u gr w. Hence, σ(u) gr σ(w) and this veriﬁes (MO2) as well. (ii) Again, by Proposition 1.3 we only need to verify CH.II Deﬁnition 1.2 (MO1)–(MO2). And by Lemma 1.4(iii), we only need to deal with the elements in B which are not of the form X n u with n > 0. Using gr as deﬁned and if LM( v w s) = 0, in the foregoing (∗), if w, u , v, s ∈ B with u gr w, LM( vu s) = 0, then  u gr w,    n = vw s = 0 for some n ≥ 0, X vws (2) (CH.I Lemma 3.11) m X v us = vu s = 0 for some m ≥ 0,    LM(vws) = 0, LM(vus) = 0. Hence LM(vus) gr LM(vws) and consequently (LM(vus)) gr (LM(vws)).

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By Lemma 1.4 and (2) above we obtain us) = LM( vu s), X m (LM(vus)) = LM(X m v n n X (LM(vws)) = LM(X vws) = LM( v w s). Hence, LM( vu s) gr LM( v w s), i.e., (MO1) is veriﬁed. If u = LM( v w s) with v = 1 or s = 1, then by CH.I Lemma 3.11 and the foregoing Lemma 1.4 we have u = LM( v w s) = LM(vwsX n ) = X n (LM(vws)) for some n = 0. Hence u = LM(vws). But this implies u gr w and consequently This veriﬁes (MO2). 2 u gr w. We ﬁnish this section by the following theorem which may be viewed as the converse of Theorem 1.5 but under a stronger condition. 1.6. Theorem Let F A be the B-standard ﬁltration on A. Suppose that G(A) is a domain. The following statements hold. (i) If (G(A), σ(B), gr ) is a (left) admissible system, where gr is some graded monomial ordering on σ(B), then (A, B, gr ) is a (left) admissible system in which gr is deﬁned as u gr w if and only if σ(u) gr σ(w), u, w ∈ B. (Note that by Lemma 1.1, σ(u) = σ(w) implies u = w.) B, gr ) is a (left) admissible system, where gr is some graded mono(ii) If (A, mial ordering on B such that w gr X s for all s ≥ 0, then (A, B, gr ) is a (left) admissible system in which gr is deﬁned as gr w, u, w ∈ B. u gr w if and only if u (Note that by the deﬁnition of u , u =w implies u = w.) Proof (i) By Proposition 1.3 we only need to verify CH.II Deﬁnition 1.2 (MO1)– (MO2) for the elements of B. Let gr be as deﬁned in the ﬁrst part of the theorem. If u, w, v, s ∈ B with u gr w, and if LM(vus) = 0, LM(vws) = 0, then since G(A) is a domain, we have by Lemma 1.4 that  σ(u) gr σ(w),    σ(LM(vus)) = LM(σ(vus)) = LM(σ(v)σ(u)σ(s)) = 0, (3) σ(LM(vws)) = LM(σ(vws)) = LM(σ(v)σ(w)σ(s)) = 0,    LM(σ(v)σ(u)σ(s)) gr LM(σ(v)σ(w)σ(s)). Hence σ(LM(vus)) gr σ(LM(vws)) and consequently LM(vus) gr LM(vws). This veriﬁes (MO1). If u = LM(vws) with v = 1 or s = 1, then since G(A) is a domain, σ(u) = σ(LM(vws)) = LM(σ(vws)) = LM(σ(v)σ(w)σ(s)) by Lemma 1.4. This implies σ(u) gr σ(w) and consequently u gr w, i.e., (MO2) is veriﬁed as well.

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(ii) By Proposition 1.3 we only need to verify CH.II Deﬁnition 1.2 (MO1)– (MO2) for the elements of B. Let gr be as deﬁned in the second part of the theorem. If u, w, v, s ∈ B with u gr w, and if LM(vus) = 0, LM(vws) = 0, then since G(A) is a domain, we have by Lemma 1.4 that  u gr w,     (LM(vus)) = LM( v us) = LM( vu s) = 0, (4)  (LM(vws)) = LM(vws) = LM( v w s) = 0,    LM( vu s) gr LM( v w s) Hence (LM(vus)) gr (LM(vws)) and consequently LM(vus) gr LM(vws). This veriﬁes (MO1). If u = LM(vws) with v = 1 or s = 1, then since G(A) is a domain, u = (LM(vws)) = LM(vws) = LM( v w s) by Lemma 1.4. This implies u gr w and consequently u gr w, i.e., (MO2) is veriﬁed as well. This completes the proof. 2

2. Filtered-Graded Transfer of (Left) Gr¨ obner Bases Let A = k[ai ]i∈Λ , B a strictly ﬁltered k-basis of A, and F A the B-standard ﬁltration on A. If L is a (left) ideal of A with the ﬁltration F L induced by F A ⊂A on L, then there are the associated (left) graded ideals G(L) ⊂ G(A) and L (CH.I §3). In this section we discuss the ﬁltered-graded transfer of Gr¨obner bases between L, G(L), and L. 2.1. Theorem With notation as above, suppose that (A, B, gr ) is a (left) admissible system where gr is some graded monomial ordering on B, and that G(A) is a domain. Let I be a left ideal or a two-sided ideal of A. The following statements hold. (i) If G = {gi }i∈J is a (left) Gr¨ obner basis for I, then σ(G) = {σ(gi ) | gi ∈ G} is a (left) Gr¨ obner bsis for G(I), with respect to the (left) admissible system (G(A), σ(B), gr ) as determined in Theorem 1.5. obner basis of G(I) consisting of homogeneous ele(ii) If {Gi }i∈J is a (left) Gr¨ ments, with respect to the (left) admissible system (G(A), σ(B), gr ) as determined in Theorem 1.5, and if gi is choosen to be such that σ(gi ) = Gi , i ∈ J, then G = {gi }i∈J is a (left) Gr¨ obner basis for I. Proof We prove the theorem for a two-sided ideal I of A (a similar argumentation works for a left ideal).

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(i) We show that every nonzero homogeneous element F ∈ G(I) has a Gr¨ obner presentation by σ(G) in the sense of CH.II Deﬁnition 3.2. Suppose F ∈ G(A)p . Then F = σ(f ) for some f ∈ Fp A ∩ I − Fp−1 A, and f has a s Gr¨ obner presentation by G, say f = i=1 λi ui gi vi , where λi ∈ k−{0}, ui , vi ∈ B, gi ∈ G, with the property that LM(f ) gr LM(ui gi vi ) whenever ui gi vi = 0. Since G(A) is a domain, it follows from Lemma 1.1, the deﬁnition of gr on σ(B) (§1 (∗)), Lemma 1.4 and CH.I Lemma 3.11 that σ(f ) = λi σ(ui gi vi ) = λi σ(ui )σ(gi )σ(vi ) d(LM(ui gi vi ))=p

d(LM(ui gi vi ))=p

and LM(σ(f )) = σ(LM(f ))

gr

σ(LM(ui gi vi ))

=

LM(σ(ui gi vi ))

=

LM(σ(ui )σ(gi )σ(vi )).

This shows that F = σ(f ) has a Gr¨obner presentation by σ(G), as desired. (ii) Suppose that Gi ∈ G(I)hi ⊂ G(A)hi (note that F I is the induced ﬁltration), i ∈ J. For each Gi , chose gi ∈ Fhi A ∩ I − Fhi −1 A such that obner presentation, say Gi = σ(gi ). If f ∈ Fp I − Fp−1 I, then σ(f ) has a Gr¨ s σ(f ) = i=1 λi σ(ui )σ(gi )σ(vi ), where λi ∈ k−{0}, σ(ui ), σ(vi ) ∈ σ(B), with the property that LM(σ(f )) gr LM(σ(ui )σ(gi )σ(vi )) whenever σ(ui )σ(gi )σ(vi ) = 0. Since G(A) is a domain, it follows from Lemma 1.4, the deﬁnition of gr on σ(B) (§1 (∗)) and CH.I Lemma 3.11 that σ(LM(f )) = LM(σ(f ))

gr

LM(σ(ui )σ(gi )σ(vi ))

=

LM(σ(ui gi vi ))

=

σ(LM(ui gi vi ))

LM(f ) gr LM(ui gi vi ). s Note that f − i=1 λi ui gi vi = h ∈ Fp−1 A ∩ I = Fp−1 I. After repeating the above procedure for a ﬁnite number of steps, a Gr¨ obner presentation of f is obtained. In the case where a left ideal L is considered, we only need to observe that σ(f ) s may have a Gr¨obner presentation of the form σ(f ) = i=1 σ(wi )σ(gi ), where σ(wi ) ∈ σ(B), and then use a similar argumentation as in the case of considering a two-sided ideal. 2 2.2. Theorem With notation as above, suppose that (A, B, gr ) is a (left) admissible system where gr is some graded monomial ordering on B, and that

2. Filtered-Graded Transfer of Gr¨ obner Bases

99

G(A) is a domain. Let I be a left ideal or two-sided ideal of A. The following statements hold. obner basis for I, then G = { gi | gi ∈ G} is a (i) If G = {gi }i∈J is a (left) Gr¨ B, gr ) (left) Gr¨ obner bsis for I, with respect to the (left) admissible system (A, as determined in Theorem 1.5. (ii) If {Gi }i∈J is a (left) Gr¨ obner basis of I consisting of homogeneous elements, B, gr ) as determined in Theorem with respect to the (left) admissible system (A, 1.5, and if the gi are choosen to be such that X ri gi = Gi (indeed, each gi is the image of Gi in A, see CH.I Lemma 3.11), i ∈ J, then G = {gi }i∈J is a (left) Gr¨ obner basis for I. Proof We prove the theorem for a two-sided ideal I (a similar argumentation works for a left ideal). (i) We prove that every nonzero homogeneous element F ∈ I has a Gr¨ obner presentation by G in the sense of CH.II Deﬁnition 3.2. Suppose F ∈ Ip . Then since F I is the induced ﬁltration, F = X r f for some r ≥ 0 and some s obner presentaf ∈ I (CH.I Lemma 3.11). Let f = i=1 λi ui gi vi be a Gr¨ tion of f by G, where λi ∈ k − {0}, ui vi ∈ B, which has the property that LM(f ) gr LM(ui gi vi ) whenever ui gi vi = 0. By Lemma 1.1 and Lemma 1.4, ri F = X r f = = λi X r u i gi vi X ri , where 0 ≤ ri ≤ d(f ). We λi X r u i gi vi X by considering two cases. prove that this is a Gr¨ obner presentation of F by G, Case 1. LM(f ) = LM(ui gi vi ). Since G(A) is a domain, it follows from Lemma 1.4 and CH.I Lemma 3.11 that LM(F ) = LM(X r f)

X r LM(f) = X r (LM(f )) = X r (LM(ui gi vi )) = LM(X r ui gi vi )

=

=

LM(X r u i gi vi ).

Case 2. LM(f ) gr LM(ui gi vi ). By the deﬁnition of gr on B (§1 (∗)) we have )X h gr (LM(ui gi vi )) X , h, ∈ IN . LM(f Lemma Then, since G(A) is a domain, it follows from the deﬁnition of gr on B, 1.4 and CH.I Lemma 3.11 that ) LM(F ) = LM(X r f) = X r LM(f

=

X r X ri (LM(ui gi vi )) gi vi ) LM(X r X ri ui

=

LM(X r u i gi vi X ri ).

gr

This ﬁnishes the proof of (i). we have Gi = X ri gi for some (ii) Since the Gi are homogeneous elements in I, gi ∈ I by CH.I Lemma 3.11. Note that since F I is the induced ﬁltration, if s ni i Gi vi X mi , f ∈ I then f has a Gr¨ obner presentation, say f = i=1 λi X u ni mi where λi ∈ k − {0}, X u i , X vi ∈ B, with the property that LM(f) gr

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i Gi vi X mi whenever u i Gi vi = 0. Substituting each Gi by X ri gi and LM(X ni u → A by considering the image of f in A under the ring homomorphism A s sending X to 1, we obtain f = i=1 λi ui gi vi (see CH.I §3). We claim that this is a Gr¨ obner presentation of f by G = {gi }i∈J . Indeed, since G(A) is a domain, it follows from Lemma 1.4 and CH.I Lemma 3.11 that ) = LM(f) LM(f

gr = =

LM(X ni u i Gi vi X mi ) i X ri gi vi X mi ) LM(X ni u X ni +ri +mi (LM(ui gi vi )) .

By the deﬁnition of gr on B we see that LM(f ) gr LM(ui gi vi ), as claimed.2 Remark Let A be a ﬁltered ring with ﬁltration F A and L a left ideal of A generated by {f1 , ..., fs }. If we consider the ﬁltration F L on L induced by F A and identify G(L) with the graded left ideal ⊕n (Fn A ∩ L + Fn−1 A)/Fn−1 A of G(A), then generally {σ(f1 ), ..., σ(fs )} does not generate G(L) in G(A). This had been a problem bothering people who are studying ﬁnitely generated algebras by the ﬁltered-graded method. Now we see that this will be no longer a problem (even much better) when we are working on a (left) admissible system and using (left) Gr¨ obner bases. For example, let us consider the left ideal L in the second Weyl algebra A2 (k) generated by f1 = x1 ∂1 , f2 = x2 ∂12 −∂1 . Then one may check that {∂1 } is a Gr¨ obner basis of L with respect to x1 >grlex x2 >grlex ∂1 >grlex ∂2 . However, with respect to the Bernstein ﬁltration (standard ﬁltration) on A2 (k), it is obvious that σ(f1 ) = σ(x1 )σ(∂1 ) and σ(f2 ) = σ(x2 )σ(∂1 )2 do not generate and L, a similar illustration is omitted. G(L). For A

3. Filtered-Graded Transfer of (Left) Dickson Systems Let A = k[ai ]i∈Λ , B a strictly ﬁltered k-basis of A, and F A the B-standard ﬁltration on A. Let (A, B, gr ) be a (left) admissible system, where gr is some graded monomial ordering on B . Recall from CH.II §6 that the divisibility (CH.II Deﬁnition 2.2) yields another order gr on B as follows: If w, u ∈ B and w = u, then u gr w if and only if u = LM(vws) for some v, s ∈ B with v = 1 or s = 1. (For a left admissible system, if and only if u = LM(vw) for some v ∈ B with v = 1); and (A, B, gr ) is called a (left) Dickson system if gr is a Dickson partial ordering on B (CH.II Deﬁnition 6.4, Deﬁnition 6.7). In this section we discuss

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the ﬁltererd-graded transfer of (left) Dickson systems associated with A, G(A) and A. 3.1. Theorem Let (A, B, gr ) be a (left) admissible system where gr is some graded monomial ordering on B. The following statements hold. (i) Suppose that G(A) is a domain. If (A, B, gr ) is a (left) Dickson system, then so is the system (G(A), σ(B), gr ) determined in Theorem 1.5. (ii) If the system (G(A), σ(B), gr ) determined in Theorem 1.5 is a (left) Dickson system, the so is (A, B, gr ). Proof (i) Let N ⊆ σ(B) be a nonempty subset and put N = w ∈ B σ(w) ∈ N . By the deﬁnition of σ(B) in §1 and Lemma 1.1, each element of N corresponds uniquely to an element in N . Since (A, B, gr ) is a (left) Dickson system, there exists a ﬁnite subset W ⊆ N such that for each u ∈ N there is some w ∈ W with u gr w, i.e., u = LM(vws) for some v, s ∈ B. Since G(A) is a domain, it follows from Lemma 1.4 and CH.I Lemma 3.11 that σ(u)

=

σ(LM(vws))

=

LM(σ(vws))

=

LM(σ(v)σ(w)σ(s)).

Hence σ(u) gr σ(w). Thus, for each element σ(u) ∈ N , there exists some σ(w) ∈ W = σ(W ) = {σ(w) | w ∈ W } ⊆ N such that σ(u) gr σ(w). Note that W is a ﬁnite subset of N , this proves that (G(A), σ(B), gr ) is a (left) Dickson system. (ii) Let N ⊆ B be a nonempty subset and put N = σ(w) w ∈ N . Then N is a nonempty subset of σ(B). Since (G(A), σ(B), gr ) is a (left) Dickson system, there exsists a ﬁnite subset W ⊆ N such that for each σ(u) ∈ N there is some σ(w) ∈ W with σ(u) gr σ(w), i.e., σ(u) = LM(σ(v)σ(w)σ(s)) for some σ(v), σ(s) ∈ σ(B). Note that σ(u) = 0 if and only if σ(v)σ(w)σ(s) = 0. It follows from Lemma 1.4 that σ(u)

=

LM(σ(v)σ(w)σ(s)

=

LM(σ(vws))

= σ(LM(vws)). Hence u = LM(vws) by Lemma 1.1, and consequently u gr w. Thus, for each element u ∈ N , there exists some w ∈ W = {w ∈ N | σ(w) ∈ W } ⊆ N such

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that u gr w. By Lemma 1.1, W is a ﬁnite subset of N . This proves that (A, B, gr ) is a (left) Dickson system. 2 3.2. Theorem Let (A, B, gr ) be a (left) admissible system where gr is some B, gr ) determined in Theograded monomial ordering on B. If the system (A, rem 1.5 is a (left) Dickson system, then so is (A, B, gr ). Proof Let N ⊆ B be a nonempty subset and put w∈N . N = w Since (A, B, gr ) is a (left) Dickson system, Then N is a nonempty subset of B. there exsists a ﬁnite subset W ⊆ N such that for each u ∈ N there is some Note gr w, i.e., u = LM(X n vw sX m ) for some X n v, X m s ∈ B. w ∈ W with u that u = 0 if and only if vw s = 0. It follows from Lemma 1.4 and CH.I Lemma 3.11 that u =

LM(X n vw sX m )

=

X n+m+h LM(vws)

=

X n+m+h (LM(vws)) .

Hence u = LM(vws) by CH.I Proposition 3.8, and consequently u gr w. Thus, for each element u ∈ N , there exists some w ∈ W = {w ∈ N | w ∈ W } ⊆ N such that u gr w. By CH.I Proposition 3.8, W is a ﬁnite subset of N . This 2 proves that (A, B, gr ) is a (left) Dickson system. Remark (i) Let A = k[ai ]i∈Λ be an arbitrary k-algebra with the standard ﬁltration F A. If A is Noetherian, then generally we hardly know whether G(A) is Noetherian or not (on the contrary of CH.I Proposition 3.13). Nevertheless, Theorem 3.1(i) may be viewed as an algorithmic transfer of the (G-)Noetherian property of A to G(A). (Note that the G-Noetherian property is indeed more stronger than the classical Noetherian Property, see CH.II §6). The (ii) It seems that an analogue of Theorem 3.1(i) does not exist for A. problem is that not every homogeneous element of A is of the form f with f ∈ A does (CH.I Lemma 3.11), and consequently the division transfer from A to A not work well. Nevertheless, in the next section we will see that an analogue of Theorem 3.1(i) does exist for a large class of popular algebras (Theorem 4.2).

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4. Filtered-Graded Transfer Applied to Quadric Solvable Polynomial Algebras We ﬁnish this chapter by a closer look at the ﬁltered-graded transfer applied to quadric solvable polynomial algebras. Let A = k[a1 , ..., an ] be a quadric solvable polynomial algebra with the associated (left) admissible system (A, B, gr ), as deﬁned in CH.II §7 and CH.III §2, where n αn 1 B = aα = aα , 1 · · · an α = (α1 , ..., αn ) ∈ IN the set of all standard monomials in A, is a k-basis for A ( and is obviously strictly ﬁltrered), and gr is a graded monomial ordering on B. Moreover, the generators of A satisfy the relations aj ai = λji ai aj + λk λh ah + cji , 1 ≤ i < j ≤ n, ji ak a + k≤

where λji = 0, λk ji , λh , cji ∈ k. 4.1. Theorem Let (A, B, gr ) be as above, and let F A be the B-standard ﬁltration on A. The following holds. (i) G(A) is a domain. (ii) G(A) is a quadratic solvable polynomial algebra with the associated Dickson system (G(A), σ(B), gr ). Moreover, if A is tame then so is G(A). Proof (i) Let F , G be homogeneous elements of degree p, q in G(A). Then there are f ∈ Fp A − Fp−1 A, g ∈ Fq A − Fq−1 A such that σ(f ) = F = 0, σ(g) = G = 0. Supose that LM(f ) = aα , LM(g) = aβ . Since B is strictly ﬁltered, by Lemma 1.1 we have d(LM(f )) = |α| = p, d(LM(g)) = |β| = q. By CH.II Proposition 7.2, we obtain LM(f g) = LM(LM(f )LM(g)) = aα+β and and d(aα+β ) = |α| + |β| = p + q. Again, since B is strictly ﬁltered, we have aα+β ∈ Fp+q−1 A and hencce f g ∈ Fp+q−1 A. This shows that σ(f )σ(g) = F G = 0. Therefore, G(A) is a domain. (ii) This follows from the construction of G(A), Theorem 1.5, part (i) of the present theorem , and Theorem 3.1(i). 2 4.2. Theorem (a complement of Theorem 3.2) Let (A, B, gr ) be as in Theorem is a quadratic solvable polynomial algebra with the associated 4.1. Then A B, gr ). Dickson system (A, satisﬁes the conditions of CH.II Proposition 7.2. By Proof We prove that A = k[ CH.I Proposition 3.16, A a1 , ..., an , X], where X is the canonical homoge1 = F1 A. By Lemma 1.2 and neous element of degree 1 represented by 1 in A α1 αn β an X | (α1 , ..., αn , β) ∈ IN n+1 } forms a k-basis Proposition 1.3, B = { a1 · · ·

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Since G(A) is a domain, it follows consisting of all standard monomials in A. from CH.I Lemma 3.11 that aj ai = a = λji ai aj + λk a a + λ a + c j ai k h h ji ji k≤

= λji ai aj +

λk ak a + ji

λh ah X + cji X 2 .

k≤

aj X, j = 1, ..., n, and that ai aj gr LM(Fij ) by the deﬁnition Note that X aj = k of gr on B, where Fij = k≤ λji ak a + λh ah X + cji X 2 . So we are done.2 Furthermore, applying CH.III Proposition 1.5, Theorem 3.6, and Theorem 3.7 to a quadric solvable polynomial algebra, we fulﬁl the promise made after Corollary 3.8 in CH.III §3. Let A = k[a1 , ..., an ] be a quadric solvable polynomial algebra with the associated (left) admissible system (A, B, gr ). By the discussion of CH.III §2, A has the deﬁning relations Rji = Xj Xi − λji Xi Xj − λk λh Xh − cji , 1 ≤ i < j ≤ n. ji Xk X − k≤

4.3. Proposition With notation as above, if, for 1 ≤ i < j ≤ n, LM(Rji ) = Xj Xi with respect to Xn >grlex · · · >grlex X1 . Then, with respect to the standard ﬁltration F A on A, the following holds. (i) The associated graded algebra G(A) of A has the quadratic deﬁning relations σ(Rji ) = Xj Xi − λji Xi Xj − λk ji Xk X , 1 ≤ i < j ≤ n, which form a Gr¨ obner basis in the free algebra kX1 , ..., Xn with respect to Xn >grlex · · · >grlex X1 . of A has the quadratic deﬁning relations (ii) The Rees algebra A T Xi − Xi T, 1 ≤ i ≤ n, ji = Xj Xi − λji Xi Xj − λk Xk X − λh Xh T − cji T 2 , 1 ≤ i < j ≤ n, R ji which form a Gr¨ obner basis in the free algebra kX1 , ..., Xn , T with respect to Xn >grlex · · · >grlex X1 >grlex T . 2 At this stage, we mention the following application of the foregoing results to those examples which have been considered before (CH.I §5, CH.II §7, CH.III §3).

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4.4. Corollary Let A be one of the following k-algebras: (i) The nth Weyl algebra An (k); (ii) The additive analogue An (q1 , ..., qn ) of An (k); (iii) The multiplicative analogue On (λji ) of An (k); (iv) The enveloping algebra U (g) of an n-dimensional k-Lie algebra g; (v) The nth q-heisenberg algebra hn (q); (vi) Any quadric solvable polynomial algebra constructed in CH.III §2. Let (A, B, ≥grlex ) be the (left) Dickson system associated with A, where B is the standard k-basis of A consisting of all standard monomials and ≥grlex is the graded lexicographic ordering that is suitably deﬁned on B. Consider the B B, ≥lexgr ) are (left) standard ﬁltration on A. Then (G(A), σ(B), ≥grlex ) and (A, are all (left) G-Noetherian. Dickson systems, and hence A, G(A) and A

CHAPTER V GK-dimension of Modules over Quadric Solvable Polynomial Algebras and Elimination of Variables

A well-known theme in commutative computational algebra is to ﬁnd the symbolic solutions of polynomial equations, or in other words, is the elimination of variables, and on this aspect, the Wu’s method and Gr¨ obner basis method have been quite powerful (cf. [BW], [Cho], [CLO ], [Wan]). In the elimination theory using Gr¨ obner bases, a notable fact is that there is a nice relation between the number of eliminated variables and the dimension of the algebraic variety determined by the system of polynomial equations. More precisely, let A = k[x1 , ..., xn ] be the commutative polynomial k-algebra in n variables and I a proper ideal of A such that the dimension of the aﬃne algebraic variety V (I) is d. Then it follows from ([LVO5] CH.V §7) that • for every subset U = {xi1 , ..., xid+1 } ⊂ {x1 , ..., xn }, k[xi1 , ..., xid+1 ] ∩ I = {0}, i.e., there is a nonzero member of I that only depends on the generators in U . In particular, k[x1 , ..., xd , xd+i ] ∩ I = {0}, i = 1, ..., n − d. At this stage, the wonderful thing is that, once a Gr¨ obner basis is produced,

H. Li: LNM 1795, pp. 107–132, 2002. c Springer-Verlag Berlin Heidelberg 2002

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V. Dimension and Elimination of Variables

the computation of the dimension of V (I) and the elimination process can be algorithmically realized simultaneously (e.g., see [BW], [CLO ]). Since the natural noncommutative algebraic translation of the dimension of an algebraic variety is the Gelfand-Kirillov dimension (GK-dimension) of a module over a k-algebra (e.g., see [KL]), it is natural to ask Question Can we use Gr¨ obner bases to compute GK-dimension for modules over a certain kind of algebra and have an analogue of the above (•)? Let A = k[a1 , ..., an ] be a quadric solvable polynomial algebra as deﬁned in in CH.II §7 and CH.III §2, and let L be a left ideal of A with the left Gr¨ obner basis G = {g1 , ..., gs }. For each gj with the leading monomial α

α

jn LM(gj ) = a1 j1 a2 j2 · · · aα n ,

j = 1, ..., s,

we associate gj to the n-tuple of exponents Rj = (αj1 , ..., αjn ) ∈ IN n ,

j = 1, ..., s.

In this chapter, using the ﬁltered-graded transfer of Gr¨ obner bases developed in CH.IV, we ﬁrst demonstrate how to compute the GK-dimension of the cyclic left module M = A/L (and hence of any ﬁnitely generated left module) in the case where A is linear (Ch.III Deﬁnition 2.1), by manipulating only the data (R1 , R2 , ..., Rs ) actually as in the commutative case; and furthermore, we use a double ﬁltered-graded transfer of Gr¨ obner bases linked by the gr -ﬁltration (see §6 for the deﬁnition) to obtain the same result for general quadric solvable polynomial algebras. This, in turn, yields an elimination (of variables) lemma for quadric solvable polynomial algebras. In other words, we have a complete answer to the above question for the class of quadric solvable polynomial algebras.

1. Gr¨ obner Bases in Homogeneous Solvable Polynomial Algebras We start with homogeneous solvable polynomial algebras. Let A = k[a1 , ..., an ] be a homogeneous solvable polynomial algebra in the sense of Ch.III Deﬁnition 2.1, and let (A, B, gr ) be the associated (left) admissible system of A, where n αn 1 B = aα = aα 1 · · · an α = (α1 , ..., αn ) ∈ IN is the standard k-basis of A and gr is a graded monomial ordering on B. Then A is a Noetherian domain (CH.II §7). In this section, we record brieﬂy some easily veriﬁed properties of Gr¨obner bases for left ideals in A (so Gr¨ obner basis is refered to left Gr¨ obner basis). Since

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109

our discussion will not only deal with the monomials in B but will also deal with the coeﬃcient of the product of two monomials in computation, due to the noncommutativity, we write T(A) for the set of all terms in A, i.e., T(A) = caα c ∈ k, aα ∈ B . Moreover, for a nonzero element f ∈ A with f= cα aα , cα ∈ k − {0}, aα ∈ B, we write LT(f ) = LC(f )LM(f ) for the leading term of f (see CH.II Deﬁnition 1.1). 1.1. Lemma (i) A is a positively graded k-algebra, i.e., A = ⊕m∈IN Am with αn 1 Am = { |α|=m cα aα | aα = aα 1 · · · an , |α| = α1 + · · · + αn }, in particular, n A0 = k, A1 = i=1 kai . (ii) If cα aα , cβ aβ ∈ T(A), then cβ aβ is divisible by cα aα (from left hand side, see CH.II Deﬁnition 2.2) if and only if there is some cγ aγ ∈ T(A) such that cβ aβ = cα cγ aγ aα . 2 Next, we deal with monomial left ideals. A left ideal L of A is said to be a monomial left ideal if it has a generating set consisting of standard monomials in B, i.e., L = α∈Λ Aaα , Λ ⊂ IN n , aα ∈ B. By Lemma 1.1, the following two properties of a monomial left ideal are derived as for a commutative monomial ideal in the commutative polynomial algebra k[x1 , ..., xn ]. α 1.2. Lemma Let L = α∈Λ Aa be a monomial left ideal of A. Then a β standard monomial a ∈ L if and only if aβ is divisible by aα for some α ∈ Λ. 2 1.3. Lemma Let L be a monomial left ideal of A, and let f ∈ A. Then the following are equivalent: (i) f ∈ L; (ii) Every term of f lies in L; (iii) f is a k-linear combination of the standard monomials in L ∩ B. 2 Let L be a left ideal of A, and LT(L) the set of leading terms of elements of L. We denote by LT(L)] the monomial left ideal generated by LT(L) in A. If furthermore G = {g1 , ..., gs } is a left Gr¨obner basis of L, then we denote by LT(G)] the monomial left ideal generated by LT(G) = {LT(g1 ), ..., LT(gs )} in

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A. Generally, in a solvable polynomial algebra we do not have LT(L)] = LT(G)] because of CH.II Proposition 4.2. However, for a homogeneous solvable polynomial algebra A, it follows from Lemma 1.2 and Lemma 1.3 that the following result holds as in the commutative case. 1.4. Proposition Let L be a left ideal of A and G = {g1 , ..., gs } ⊂ L. (i) (compare with CH.II Proposition 4.1) If L = aα ∈ B | α ∈ Λ ⊂ IN n ] is a obner basis for monomial left ideal, then L = LT(L)] and {aα | α ∈ Λ} is a Gr¨ L. (ii) (compare with CH.II Proposition 4.2) G = {g1 , ..., gs } is a Gr¨ obner basis of L if and only if LT(L)] = LT(G)]. (iii) (compare with CH.II Theorem 3.1) Let G be a Gr¨ obner basis of L, then A/L has a k-basis {[aα ] | aα ∈ B − LT(G)]}, where [aα ] is the class of aα in A/L. 2

2. The Hilbert Function of A/L Let A = k[a1 , ..., an ] be a homogeneous solvable polynomial algebra with the associated (left) admissible system (A, B, gr ), and let L be a left ideal of A. The aim of this section is to introduce the Hilbert function HFL of the A-module A/L, and reduce the study of HFL to the study of HFLT(L)] . Consider the standard ﬁltration F A on A: Fm A = cα aα ∈ A aα ∈ B, |α| ≤ m ,

m ∈ IN ,

in which F0 A = k, and each Fm A is a ﬁnite dimensional k-space. Then F A induces the ﬁltration F L on L: Fm L = L ∩ Fm A, m ∈ IN . The Hilbert function of the A-module A/L, denoted HFL , is deﬁned by putting

Fm A , m ∈ IN . HFL (m) = dimk Fm L The key idea making the Hilbert function computable comes from the following noncommutative version of a celebrated result obtained from (F.S. Macaulay, Some properties of enumeration in the theory of modular systems, Proc. London math. Soc., 26(1927), 531–555). The next proposition will be recaptured in a more general setting in later §7, and the proof given below also shows why a graded monomial ordering is necessary.

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111

2.1. Proposition Let (A, B, gr ), L be as given above, and let LT(L)] be the left ideal generated by LT(L), the set of leading terms of all elements in L. Then HFL (m) = HFLT(L)] (m),

m ∈ IN .

Proof Taking a left Gr¨ obner basis G of L and putting B≤m = B ∩ Fm A, we claim that there are k-space isomorphisms: ∼ Fm A Fm A ∼ = = −→ k-Span aα aα ∈ B≤m − LT(L)] ←− Fm L Fm LT(L)] Since LT(L)] is a monomial left ideal, the second isomorphism is easily obtained by Lemma 1.3 and Proposition 1.4. We will obtain the ﬁrst isomorphism by showing that the k-spaces Fm L and Fm LT(L)] have the same dimension. First note that (1) LM(f ) f ∈ Fm L = {LM(f1 ), ..., LM(fs )} for a ﬁnite number of f1 , ..., fs ∈ Fm L. By rearranging and deleting duplicates, we may assume that (2)

LM(f1 )gr LM(f2 )gr · · · gr LM(fs ).

We claim that {f1 , ..., fs } forms a basis of Fm L as a k-space. To see this, consider a nontrivial linear combination c1 f1 + · · · + cs fs and choose the smallest i such that ci = 0. It follows from (2) above that no ci LT(fi ) can be canceled, and hence the linear combination is nonzero. Thus, f1 , ..., fs are linearly independent. Next, let W = k-Span{f1 , ..., fs } ⊂ Fm L. If W = Fm L, pick f ∈ Fm L − W with LM(f ) minimal (note that gr is a well-ordering). By (1), LM(f ) = LM(fi ) for some i, and hence, LT(f ) = λLT(fi ) for some λ ∈ k. Then f − λfi ∈ Fm L has a smaller leading monomial, so that f − λfi ∈ W by the minimality of LM(f ). This implies f ∈ W , which is a contradiction. It follows that W = Fm L, and we conclude that {f1 , ..., fs } forms a k-basis. A similar argumentation as above shows that LM(f1 ), ..., LM(fs ) are k-linearly independent. Note that we are using gr . This means that for any nonzero f ∈ A, if LM(f ) ∈ Fm A, then so does f (CH.IV Lemma 1.4). It follows immediately from Lemma 1.3 and the above (1) that Fm LT(L)] = k-Span{LM(f1 ), ..., LM(fs )}. So we are done.

2

Now, recall from Lemma 1.1(i) that A = k[a1 , ..., an ] is positively graded where deg(ai ) = 1, i = 1, ..., n, i.e., A = ⊕m∈IN Am with Am = { cα aα ∈ A | aα ∈ B, |α| = m}, and each Am is a ﬁnite dimensional k-space. Let L be a graded

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left ideal of A. Then L = ⊕m∈IN Lm with Lm = Am ∩ L. The Hilbert function of the graded A-module A/L, denoted HFLg , is deﬁned by putting

Am HFLg (m) = dimk , m ∈ IN . Lm 2.2. Proposition With notation as above, HFLg (m) = HFL (m) − HFL (m − 1), Proof Since in this case we have Fm A = ⊕p≤m Ap , isomorphism of k-spaces:

m ∈ IN . m ∈ IN , there is a natural

Fm A + L Am ∼ = −→ Lm Fm−1 A + L It follows that dimk (Am /Lm ) = dimk (Fm A/Fm L)− dimk (Fm−1 A/Fm−1 L), as desired. 2

3. The Hilbert Polynomial of A/L Let (A, B, gr ) be the (left) admissible system associated with a homogeneous solvable polynomial algebra A, and let L be a left ideal of A with the Hilbert function HFL as deﬁned in §2. Based on Proposition 2.1 of last section, in this section we prove that there exists a polynomial hL (x) ∈ Q [x] with positive leading coeﬃcient such that, for m 0, HFL (m) = hL (m). We approach this by mimicking the commutative case [BW], where the main idea is to • classify S = B≤m − LT(L)], where B≤m = B ∩ Fm A, by a suitably deﬁned equivalence relation on S, and then • compute the number of elements of each equivalence class. To this end, we need a few more technicalities. First note that LM(ts) = aα+β = LM(st) for t = aα , s = aβ ∈ B. Next, αn 1 for t = aα 1 · · · an ∈ B, M ∈ IN , we write topM (t) for the set consisting of all indices ij ∈ {1, ..., n} with αij ≥ M . Suppose topM (t) = {i1 , ..., is } with i1 < · · · < is . We deﬁne αis+1 αin M shM (t) = aM , a · · · a · · · a i1 is is+1 in where is+1 < · · · < in . Since A is homogeneous, we have shM (t) ∈ T(A) (see §1). Then it is clear that (∗)

shM (t) = shM (LM(shM (t))).

3. The Hilbert Polynomial

113

3.1. Lemma With notation as above, let S ⊂ B be a subset and M ∈ IN . The following hold. (i) The relation ∼M deﬁned by s ∼M t

if and only if shM (s) = shM (t)

is an equivalence relation on S. (ii) Let [s]∼M denote the equivalence class of s ∈ S. If LM(shM (s)) ∈ S for all s ∈ S, then the set RM = t ∈ S LM(shM (t)) = t is a system of unique representatives for the partition of S into equivalence classes with respect to ∼M . The set RM can also be described as αn 1 RM = t ∈ S t = aα · · · a , α ≤ M, 1 ≤ i ≤ n . i n 1 (iii) Let RM be as in (ii). For t ∈ RM , we have [t]∼M = LM(st) LM(st) ∈ S, s = aγ11 · · · aγnn with γi = 0 for i ∈ topM (t) Proof (i) This is clear. (ii) By the assumption we have LM(shM (s)) ∈ S for all s ∈ S. Hence s ∼M LM(shM (s)) by the above (∗), i.e., [s]∼M = [LM(shM (s))]∼M with LM(shM (s)) ∈ RM . If s ∼M t, then shM (s) = shM (t). This means that LM(shM (s)) = LM(shM (t)), or in other words, the representative of [s]∼M αn 1 in RM is unique. Furthermore, if t ∈ RM , t = aα and shM (t) = 1 · · · an α α i s+1 M M n (ai1 · · · ais )(ais+1 · · · ain ), then clearly αi ≤ M for 1 ≤ i ≤ n.

β1 αn βn 1 (iii) Let t ∈ RM , t = aα 1 · · · an . Suppose t = a1 ·α· · an ∈ S and t ∼M t. i α i s+1 M n Then shM (t ) = shM (t), i.e., shM (t) = (aM i1 · · · ais )(ais+1 · · · ain ) = shM (t ). Since LM(shM (t)) = t, for ik ∈ topM (t) we have βik = αik < M ; for ik ∈ topM (t) we have βik ≥ M = αik by (ii). Hence there exists s = aγ11 · · · aγnn with γi = 0 for i ∈ topM (t), such that t = LM(st). Conversely, for any s = aγ11 · · · aγnn with γi = 0, i ∈ topM (t), if LM(st) ∈ S then 2 it is clear that t ∼M LM(st).

3.2. Lemma Let L be a left ideal of A and G a Gr¨obner basis of L. If we put M = max βi LT(g) = cβ aβ1 1 · · · aβnn , g ∈ G, 1 ≤ i ≤ n , then for every t ∈ B, whenever i ∈ topM (t) and ν ∈ IN , t ∈ B − LT(L)] if and only if LM(aνi t) ∈ B − LT(L)]. αn 1 Proof Writing t = aα 1 · · · an , topM (t) = {i1 , ..., is }, then αis+1 α M ais+1 · · · ainin . shM (t) = aM i1 · · · ais

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If LM(aνi t) ∈ B−LT(L)], then t ∈ B−LT(L)]. Conversely, let t ∈ B−LT(L)]. Assume the contrary that LM(aνi t) ∈ LT(L)]. Then aνi t is divisible by some LT(g) = cβ aβ1 1 · · · aβnn . Since i ∈ topM (t), we have αi ≥ M ≥ βi . It follows that t is divisible by LT(g), i.e., t ∈ LT(L)] (see Lemma 1.1(ii)), a contradiction. Hence LM(aνi t) ∈ B − LT(L)]. 2 Before proving the main result, let us recall a fact from Proposition 2.1 and its proof: ()

HFL (m)

=

HFLT(L)] (m)

=

|B≤m − LT(L)]|,

m ∈ IN ,

where B≤m = B ∩ Fm A. obner 3.3. Theorem Let (A, B, gr ) and L be as before. Let G be a left Gr¨ basis of L (note that A is solvable), and put M = max βi LT(g) = cβ aβ1 1 · · · aβnn , g ∈ G, 1 ≤ i ≤ n . Then there exists a unique polynomial hL (x) ∈ Q [x] with positive leading coefﬁcient such that HFL (m) = hL (m),

for all m ≥ n · M.

If the ground ﬁeld is computable, then hL (x) and the number n · M can be computed from any given generating set of L. Proof By the previous formula (), the desired polynomial hL (x) must satisfy (1)

hL (m) = |B≤m − LT(L)]|

for all m ≥ n · M . We will arrive at such a polynomial by counting the elements of B≤m − LT(L)]. To this end, we let m ∈ IN with m ≥ n · M . It follows from Lemma 3.2 that S = B≤m − LT(L)] satisﬁes LM(shM (s)) ∈ S for all s ∈ S. Hence S is the disjoint union of the equivalence classes with respect to the equivalence relation ∼M of Lemma 3.1(i). Using the set RM of Lemma 3.1(ii) as a system of unique representatives, we have (2) |B≤m − LT(L)]| = |[t]∼M |, t∈RM

where [t]∼M is the equivalence class of t ∈ RM . Now from Lemma 3.1(iii) and Lemma 3.2 it is clear that for every t ∈ RM , if αn 1 t = aα 1 · · · an , then [t]∼M = LM(st) s = aγ11 · · · aγnn , γi = 0, for i ∈ topM (t), |γ| ≤ m − |α| .

4. GK.dim(A/L) = d = n − min {|J| | J ∈ M} .

It follows that |[t]∼M | =

m − |α| + |topM (t)| |topM (t)|

115

which is a polynomial of degree |topM (t)| in m − |α|. Combining this with (2), we have obtained a polynomial hL (x) ∈ Q [x] with positive leading coeﬃcient that satisﬁes (1) and has deghL (x) = max |topM (t)| t ∈ RM =

max |topM (s)| s ∈ B≤m − LT(L)] .

Since by Proposition 1.4 we have LT(L)] = LT(G)], the existence proof of the polynomial hL (x) that we have just given shows that the last statement of the theorem concerning computability is true. It is also clear that there can be only one hL (x) ∈ Q [x] satisfying HFL (m) = hL (m) for inﬁnitely many m ∈ IN . 2 3.4. Deﬁnition The polynomial hL (x) obtained in the above theorem is called the Hilbert polynomial of A/L. Finally, since A = k[a1 , ..., an ] is positively graded (see Lemma 1.1) with deg(ai ) = 1, i = 1, ..., n, if L be a graded left ideal of A, then Proposition 2.2 enables us to mention the following result. 3.5. Theorem With notation as in §2, there exists a unique polynomial in Q [x] with positive leading coeﬃcient, which is called the Hilbert polynomial of the graded A-module A/L and is denoted by hgL (x), such that for m 0

Am g = hgL (m). HFL (m) = dimk Lm Moreover, if hL (x) has degree d then hgL (x) has degree d − 1.

2

4. GK-dimension Computation and Elimination of Variables (Homogeneous Case) Let A = k[a1 , ..., an ], (A, B, gr ) and L be as in §3. The argumentation of previous sections has indeed shown that the degree of the Hilbert polynomial hL (x) of the A-module A/L is nothing but the Gelfand-Kirillov dimension of

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V. Dimension and Elimination of Variables

A/L (e.g., see [KL]) which is usually denoted by GK.dim(A/L), i.e., deghL (x) = GK.dim(A/L). In this section, we show that there is an algorithm for computing GK.dim(A/L) directly without computing hL (x); and we also show that a noncommutative analogue of the result (•) (as mentioned in the beginning of this chapter) exists for homogeneous solvable polynomial algebras. As in the commutative case (e.g., [BW], [CLO ]), we ﬁrst show that deghL (x) is closely related to the “independence” of generators of A (modulo L). If U = {ai1 , ..., air } ⊂ {a1 , ..., an } with i1 > · · · > ir , then the subalgebra k[U ] = k[ai1 , ..., air ] of A generated by U and k is also a homogeneous solvable polynomial k-algebra. Let T(U ) be the set of of all terms in k[U ]. We say that U is independent (modulo L), if T(U ) ∩ LT(L)] = {0}. 4.1. Proposition With notation as above, put d = max |U | U ⊂ {a1 , ..., an } independent (modulo L) . Then deghL (x) = d. Proof Let G be a Gr¨ obner basis of L. First recall from §3 that for M = max βi LT(g) = cβ aβ1 1 · · · aβnn , g ∈ G, 1 ≤ i ≤ n , we have

deghL (x) = max |topM (t)| t ∈ B≤m − LT(L)], m ≥ n · M .

To prove the inequality “≤”, assume for a contradiction that there exists t ∈ αn 1 B≤m − LT(L)], t = aα 1 · · · an with αi ≥ M for more than d many indices. Then there exists a subset U = {ai1 , ..., air } ⊂ {a1 , ..., an } with i1 > · · · > ir , r > d, and a decomposition t = t1 · t2 with t2 ∈ T(U ) and t1 ∈ T(U c ), where U c = {a1 , ..., an } − U , such that α

α

t2 = ai1i1 · · · airir ,

αij ≥ M, 1 ≤ j ≤ r.

We must have t2 ∈ B≤m − LT(L)] because t2 is a factor of t. On the other hand, since r > d, we have T(U ) ∩ LT(L)] = {0}, and so there exists g ∈ G with LT(g) ∈ T(U ). Note that since for LT(g) = cβ aβ1 1 · · · aβnn we have βi ≤ M , 1 ≤ i ≤ n, it follows that t2 is divisible by LT(g) and thus t ∈ LT(L)], a contradiction. For the inequality “≥”, let U = {ai1 , ..., aid } ⊂ {a1 , ..., an } be such that i1 > · · · > id and T(U ) ∩ LT(L)] = {0}. Then for m ≥ n · M , it is easy to see that M 2 t = aM i1 · · · aid ∈ B≤m − LT(L)] and |topM (t)| = d.

4. GK.dim(A/L) = d = n − min {|J| | J ∈ M} .

117

s Furthermore, let J = j=1 Amj be a left ideal of A generated by standard monomials (i.e., it is a monomial left ideal), where α

jn mj = a1 j1 · · · aα n ,

j = 1, ..., s.

Then we may associate each mj to the n-tuple of exponents integers Rj = (αj1 , αj2 , ..., αjn ) ∈ IN n , and put

j = 1, ..., s,

Mj = i ∈ {1, ..., n} αji = 0 in Rj , j = 1, ..., s. M = J ⊂ {1, ..., n} J ∩ Mj = ∅, 1 ≤ j ≤ s .

From Proposition 1.4 and Proposition 2.1 we know that if L is a left ideal of A and G is a Gr¨ obner basis of L, then hL (x) = hLT(L)] (x) = hLT(G)] (x). This fact enables us to derive the following algorithmic method for computing GK.dim(A/L). 4.2. Theorem With notation as before, let L be a left ideal of A and let obner basis of L. Put G = {g1 , ..., gs } be a left Gr¨ α

jn mj = LM(gj ) = a1 j1 · · · aα n , j = 1, ..., s.

Then

GK.dim(A/L) = d = n − min |J| J ∈ M .

Consequently, (i) d = n if and only if L = {0}; and (ii) if G is a reduced Gr¨ obner basis (CH.II Deﬁnition 3.2(iii)), then d = 0 (i.e., A/L is a ﬁnite dimensional k-space) if and only if s = n and (after reordering r mj ’s if necessary) mj = aj j , rj > 0, j = 1, ..., n. (Also see Theorem 8.1.) Proof First note that for a subset U ⊂ {a1 , ..., an }, T (U ) ∩ LT(L)] = {0} if and only if T (U ) ∩ LT(G) = ∅ where LT(G) = {LT(g1 ), ..., LT(gs )}. Now suppose U = {ai1 , ..., air } ⊂ {a1 , ..., an } with i1 > · · · > ir and T (U ) ∩ LT(G) = ∅. Then the set JU = {1, ..., n} − {i1 , ..., ir } satisﬁes JU ∩ Mj = ∅, 1 ≤ j ≤ s. Hence JU ∈ M, and consequently r = n − |JU | ≤ n − min |J| J ∈ M . Conversely, for any {i1 , ..., it } = J ⊂ M, if we put U = {aj1 , ..., ajn−t }, where {j1 , ..., jn−t } = {1, ..., n} − J, then it is easy to see that T (U ) ∩ LT(G) = ∅. Hence, |U | = n − t ≤ d. Thus we have proved the requred equality. 2 Summing up, over a computable ground ﬁeld k, one can proceed to compute GK.dim(A/L) as follows.

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V. Dimension and Elimination of Variables

• Using a graded monomial ordering on B to compute a left Gr¨obner basis for L. • Compute the number d = GK.dim(A/L) by using Theorem 4.2. Note that our discussion has indeed given an algorithm for doing this. We now go on to show that the independence condition for subsets of the generating set of A given above may be replaced by a weaker independence condition which will enable us to obtain an elimination lemma for homogeneous solvable polynomial algebras. With notation as before, we say that a subset U = {ai1 , ..., air } ⊂ {a1 , ..., an } with i1 < i2 < · · · < ir is weakly independent (modulo L) if k[U ] ∩ L = {0}. It is clear that if U is independent (modulo L), then U is weakly independent (modulo L). Hence, if we put d = max |U | U ⊂ {a1 , ..., an } weakly independent (modulo L) , it follows from Proposition 4.1 that (∗)

d ≥ d = deghL (x).

4.3. Proposition With notation as before, we have d = d = deghL (x). Proof In view of the above (∗), we only have to show d ≥ d . If d = 0,then since L is a proper nonzero left ideal we have hL (m) ≥ 1 =

m+0 . So we suppose d > 0, and without loss of generality we let U = 0 {a1 , ..., ad } ⊂ {a1 , ..., an } be weakly independent (modulo L). Consider the standard ﬁltration F k[U ] on k[U ] as we deﬁned for A. Then Fm k[U ] ⊂ Fm A for all m ∈ IN . Since all standard monomials in variables a1 , ..., ad form a k-basis for k[U ], it follows from the weak independence condition k[U ] ∩ L = {0} that

Fm k[U ] + L m + d = dimk Fm k[U ] = dimk d L Fm A + L ≤ dimk L = HFL (m). Thus for m 0 we obtain

m + d = f (m), hL (m) ≥ d

x + d . Hence d = deghL (x) ≥ where f (x) denote the polynomial d 2 degf (x) = d , as desired.

5. Linear Solvable Polynomial Algebras

119

It follows immediately from Proposition 4.3 that the following lemma holds. 4.4. Lemma (elimination Lemma for homogeneous solvable polynomial algebras) Let A = k[a1 , ..., an ] be a homogeneous solvable polynomial algebra, and let L be a proper left ideal of A such that the A-module A/L has Gelfand-Kirillov dimension d, i.e., deghL (x) = d. Then, for every subset U = {ai1 , ..., aid+1 } ⊂ {a1 , ..., an }, L ∩ k[U ] = {0}, where k[U ] = k[ai1 , ..., aid+1 ] ⊂ A.

5. GK-dimension Computation and Elimination of Variables (Linear Case) In this section we consider the GK-dimension computation of modules and the elimination of variables by restricting to the (left) admissible system (A, B, gr ), where A = k[a1 , ..., an ] is a linear solvable polynomial algebra in the sense of CH.II Deﬁnition 2.1. 5.1. Proposition Let A = k[a1 , ..., an ] be a ﬁnitely generated k-algebra with n αn 1 α2 the k-basis B = {aα = aα 1 a2 · · · an | α = (α1 , α2 , ..., αn ) ∈ IN }. Consider the standard ﬁltration F A. Then A is a linear solvable polynomial algebra with respect to some graded monomial ordering gr on B if and only if the associated graded k-algebra G(A) = k[σ(a1 ), ..., σ(an )] is a homogeneous solvable polynomial algebra with respect to gr on the k-basis σ(B) = {σ(aα ) | aα ∈ B}. Proof This easily follows from CH.IV Theorem 1.6 and Theorem 4.1. Or we refer to [LW] for a direct proof. 2 5.2. Theorem Let (A, B, gr ) be the (left) admissible system associated with a linear solvable polynomial algebra A, and let F A be the standard ﬁltration on A. If L is a nonzero left ideal of A, then there exists a unique polynomial hL (x) ∈ Q [x] with positive leading coeﬃcient such that for m 0,

Fm A = hL (m), dimk Fm L where Fm L = Fm A ∩ L. Moreover, if the ground ﬁeld is computable, then the polynomial hL (x) can be computed from any given generating set of L. Proof Note that Fm A/Fm L ∼ = (Fm A + L)/L, and the latter is the ﬁltration on A/L induced by F A. It follows from CH.I §3 that we have G(A/L) ∼ =

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V. Dimension and Elimination of Variables

G(A)/G(L), and hence

Fm A ⊕p≤m G(A)p + G(L) = dimk , dimk Fm L G(L)

m ∈ IN .

Note that ⊕p≤m G(A)p = Fm G(A), the mth part of the standard ﬁltration F G(A) on G(A). On the other hand, Proposition 5.1 entails that G(A) is a homogeneous solvable polynomial k-algebra with gr . Now the existence of the required polynomial hL (x) follows from Theorem 3.3. s If furthermore k is computable and L = i=1 Aξi , ξi ∈ L, then we can compute hL (x) as follows. • Compute a Gr¨ obner basis G = {g1 , ..., gs } of L starting with {ξ1 , ..., ξs }. • Take σ(G) = {σ(g1 ), ..., σ(gs )} as a Gr¨obner basis of G(L) by CH.IV Theorem 2.1. • Proceed from here as in the case of Theorem 3.3. 2 5.3. Deﬁnition The polynomial obtained in Theorem 5.2 is called the Hilbert polynomial of the A-module A/L and is denoted by hL (x). The above argumentation has indeed shown that the degree of the Hilbert polynomial of the A-module A/L is nothing but the Gelfand-Kirillov dimension of A/L, i.e., deghL (x) = GK.dim(A/L) (∇)

= GK.dim(G(A/L)) = GK.dim(G(A)/G(L)) = deghG(L) (x).

Hence, by §4, we immediately have the following 5.4. Corollary Let A and L be as in Theorem 5.2. If the ground ﬁeld k is computable, then there is an algorithm for computing GK.dim(A/L) from any given generating set of L. 2 Example (i) Let A = A1 (q) be the additive analogue of the ﬁrst Weyl algebra over a ﬁeld k (CH.I §5), i.e., A = k[x, y] subject to the relation yx − qxy = 1, where q is a nonzero element of k. Then A is a linear solvable polynomial algebra with respect to x >grlex y. Let L = Af + Ag, where f = qx, g = xy + q −1 . By a direct calculation, the S-element S(f, g) of f and g is 0. Hence, G = {f, g} is a left Gr¨ obner basis of L. Since LM(f ) = x, LM(g) = xy, it follows from the procedure of §3 that the Hilbert polynomial of A/L is hL (x) = x ∈ Q [x]. Thus, GK.dim(A/L) = 1 and A/L is an inﬁnite dimensional A-module.

5. Linear Solvable Polynomial Algebras

121

We now proceed to determine the existence of an elimination lemma for a linear solvable polynomial algebra by computing GK.dim(A/L) directly without computing hL (x). Let A = k[a1 , ..., an ] be a linear solvable polynomial algebra with the standard ﬁltration F A. Note that the relations satisﬁed by the generators of A (CH.III §2) entails that, for a nonempty subset U = {ai1 , ..., air } of {a1 , ..., an }, the subalgebra k[U ] of A generated by U over k may contain elements which are not the linear sum of the standard monomials in variables in U . Hence, the method we used for the homogeneous case in §4 cannot be carried over to the linear solvable case. However, since G(A) is a homogeneous solvable polynomial algebra (Proposition 5.1), by using the ﬁltered-graded transfer trick again, we may still arrive at an elimination lemma for linear solvable polynomial algebras. Let L be a left ideal of A with the ﬁltration F L induced by the standard ﬁltration F A on A, and let G = {g1 , ..., gs } be a left Gr¨obner basis of L. It follows from CH.IV Theorem 2.1 that σ(G) = {σ(g1 ), ..., σ(gs )} is a left Gr¨ obner basis of G(L) in G(A). Writing α α mj = LM(gj ) = a1 j1 · · · anjn , j = 1, ..., s, Rj = (αj1 , ..., αjn ), then CH.IV Lemma 1.2 and Theorem 4.1 yield (∆)

LM (σ(gj ))

= σ (LM(gj )) = σ(a1 )αj1 · · · σ(an )αjn .

Using notation as above and as in §4: Mj = i ∈ {1, ..., n} αji = 0 in Rj , j = 1, ..., s, M = J ⊂ {1, ..., n} J ∩ Mj = ∅, 1 ≤ j ≤ s , we may also compute GK.dim(A/L) as follows. 5.5. Proposition GK.dim(A/L) = n − min |J| J ∈ M . Consequently, (i) d = n if and only if L = {0}; and (ii) if G is a reduced Gr¨ obner basis (CH.II Deﬁnition 3.2(iii)), then d = 0 (i.e., A/L is a ﬁnite dimensional k-space) if and only if s = n and (after reordering r mj ’s if necessary) mj = aj j , rj > 0, j = 1, ..., n. (Also see Theorem 8.1 in this chapter.) Proof This follows immediately from the foregoing (∇), (∆), and Theorem 4.2. 2

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V. Dimension and Elimination of Variables

Now, for a nonempty subset U = {ai1 , ..., air } ⊂ {a1 , ..., an } with i1 < i2 < · · · < ir , we write αr r 1 , T(U ) = λaα i1 · · · air λ ∈ k, (α1 , ..., αr ) ∈ IN and we say that U is independent (modulo L) if T(U ) ∩ LT(G) = ∅, where LT(G) = {LT(g1 ), ..., LT(gs )}. 5.6. Proposition With notation as above, and putting d = max |U | U ⊂ {a1 , ..., an } independent (modulo L) , then GK.dim(A/L) = d . Proof let J ∈ M with J = {i1 , ..., ir } ⊂ {1, ..., n}. Then J ∩Mj = ∅, j = 1, ..., s. Write U = {aj1 , ..., ajn−r } with {j1 , ..., jn−r } = {1, ..., n} − J. It is easy to see that T(U ) ∩ LT(G) = ∅, i.e., n − r = |U | ≤ d . Since J is arbitrary in M it follows from Proposition 5.5 that GK.dim(A/L) ≤ d . To obtain the opposite inequality, let U = {ai1 , ..., aid } ⊂ {a1 , ..., an } be independent (modulo L), and put J = {1, ..., n} − {i1 , ..., id }. Then we see that J ∈ M. Thus, d = |U | = n − |J| ≤ GK.dim(A/L) by Proposition 5.5. This proves the proposition. 2 Furthermore, let U = {ai1 , ..., air } ⊂ {a1 , ..., an } with i1 < i2 < · · · < ir , and let αr r 1 . (α · · · a , ..., α ) ∈ I N B(U ) = aα 1 r i1 ir Write V(U ) for the k-vector space spanned by B(U ) in A. We say that U is weakly independent (modulo L), if V(U ) ∩ L = {0}. 5.7. Theorem With notation as above, if we put d = max |U | U ⊂ {a1 , ..., an } weakly independent (modulo L) , then GK.dim(A/L) = d . Proof We ﬁrst prove that if U = {ai1 , ..., air } ⊂ {a1 , ..., an } is independent (modulo L) with respect to some ﬁxed Gr¨ obner basis G of L, then U is weakly independent (modulo L), and hence GK.dim(A/L) ≤ d by Proposition 5.6. To see this, let G = {g1 , ..., gs }, and suppose that f ∈ V(U ) ∩ L. If f = 0, then f has a Gr¨ obner presentation by G: f=

s i=1

hi gi ,

LM(f ) gr LM(hi gi ) whenever hi gi = 0.

5. Linear Solvable Polynomial Algebras

123

Hence, LM(f ) appears as one of the LM(LT(hi )LT(gi )). But LM(f ) ∈ B(U ), this implies that some LT(gi ) must be contained in T(U ) ∩ LT(G), contradicting U being independent (modulo L). To prove d ≤ GK.dim(A/L), let hL (x) be the Hilbert polynomial of the A module A/L.

If d = 0, then since L is a proper left ideal we have hL (m) ≥ 1 = m+0 . So we may suppose d > 0, and without loss of generality we let 0 U = {a1 , ..., ad } ⊂ {a1 , ..., an } be weakly independent (modulo L). Consider the ﬁltration F V(U ) on the k-vector space V(U ) induced by F A: Fm V(U ) = Fm A ∩ V(U ), m ∈ IN , and put αd 1 ≤ m B(U )≤m = aα α · · · a + · · · + α , m ∈ IN . 1 d 1 d Then B(U )≤m ⊂ Fm V(U ), and it follows from the weak independence condition V(U ) ∩ L = {0} that for m 0

Fm V(U ) + L m + d = |B(U )≤m | = dimk d L Fm A + L ≤ dimk L = hL (m). Thus we obtain

hL (m) ≥

m + d d

where f (x) denote the polynomial

= f (m), m 0, x + d d

. Hence GK.dim(A/L) =

deghL (x) ≥ degf (x) = d , as desired.

2

We arrive at the following lemma. 5.8. Lemma (elimination Lemma for linear solvable polynomial algebras) Let A = k[a1 , ..., an ] be a linear solvable polynomial algebra with the associated (left) admissible system (A, B, gr ), and let L be a proper left ideal of A such that the A-module A/L has Gelfand-Kirillov dimension d, i.e., the Hilbert polynomial of the A-module A/L has degree d. Then for every subset U = {ai1 , ..., aid+1 } ⊂ {a1 , ..., an } with i1 < i2 < · · · < id+1 , V(U ) ∩ L = {0}, αd+1 d+1 1 where V(U ) = k-Span aα . (α · · · a , ..., α ) ∈ I N 1 d+1 i1 id+1

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V. Dimension and Elimination of Variables

6. The gr -ﬁltration on a Quadric Solvable Polynomial Algebra Let A = k[a1 , ..., an ] be a quadric solvable polynomial algebra with the associated (left) admissible system (A, B, gr ), where n αn 1 α = (α · · · a , ..., α ) ∈ I N B = aα = aα 1 n n 1 is the standard k basis and gr is a graded monomial ordering on B. Recall from CH.III Deﬁnition 2.1 that the generators of A satisfy λk λh ah + cji , 1 ≤ i < j ≤ n. aj ai = λji ai aj + ji ak a + k≤

Suppose that A is neither homogeneous nor linear. Then it is clear that, with respect to the standard ﬁltration F A on A, G(A) is no longer a homogeneous solvable polynomial algebra (though G(A) is a solvable polynomial algebra by CH.IV Theorem 4.1). It follows that the method we used in computing GKdimension of modules over linear solvable polynomial algebras in §5 cannot be directly carried over to arbitrary quadric solvable polynomial algebras. To remedy this problem, we ﬁrst introduce the gr -ﬁltration FA on A in this section. With notation as above, for each α ∈ IN n , construct the k-subspace Fα A = k-span aβ ∈ B α gr β . Clearly, if α gr γ, then Fγ A ⊂ Fα A. Thus, since gr is a monomial ordering, we have a IN n -ﬁltration on A satisfying (1) 1 ∈ F0 A, (2) every Fα A is a ﬁnite dimensional k-space, A = ∪α∈IN n Fα A, and (3) (Fα A)(Fβ A) ⊂ Fα+β A. To emphasise the role of gr in our discussion, we prefer calling this ﬁltration FA the gr -ﬁltration instead of simply a IN n -ﬁltration as it looks like. Since gr is a monomial ordering, it is known that α gr 0 = (0, ..., 0) for all α ∈ IN n ; and since gr is a graded monomial ordering, for each α ∈ IN n , there exists α∗ = max γ ∈ IN n α gr γ . Then we have a well-deﬁned IN n -graded algebra Fα A GF (A) = , GF (A)α with GF (A)α = Fα∗ A n α∈IN

in which the addition is given by the componentwise addition and the multiplication is given by GF (A)α × GF (A)β (f , g)

−→ GF (A)α+β →

fg

6. gr -ﬁltration

125

where, if f ∈ Fα A, then f stands for the image of f in GF (A)α = Fα A/Fα∗ A. GF (A) is called the associated graded algebra of A with respect to FA. As dealing with a ZZ-ﬁltration in CH.I §3, for an element f ∈ Fα A − Fα∗ A, we write σ(f ) for the image of f in GF (A)α . For each i = 1, ..., n, write ei for the ith unit vector (0, ..., 0, 1, 0, ..., 0) in IN n . Then 0 = σ(ai ) ∈ GF (A)ei , and it is not hard to see that, for α = (α1 , ..., αn ) ∈ IN n and aα ∈ B, αn α 1 σ(a1 )α1 · · · σ(an )αn = σ(aα 1 · · · an ) = σ(a ).

Hence, for α = (α1 , ..., αn ) ∈ IN n , GF (A)α = k-span{σ(a1 )α1 · · · σ(an )αn } (i.e., a 1-dimensional space), and consequently, we have indeed proved the following result. 6.1. Proposition With notation as above, the following holds. (i) GF (A) = k[σ(a1 ), ..., σ(an )] is a IN n -graded k-algebra generated by σ(a1 ) ,..., σ(an ). The generators of GF (A) satisfy σ(aj )σ(ai ) = λij σ(ai )σ(aj ), j > i, λij = 0. (ii) The set of monomials σ(B) = σ(a1 )α1 · · · σ(an )αn (α1 , ..., αn ) ∈ IN n forms a k-basis for GF (A). It follows that GF (A) forms a homogeneous solvable polynomial algebra (in the sense of CH.III Deﬁnition 2.1) with respect to the monomial ordering gr . And it follows that GF (A) is a Noetherian domain. 2 For each integer p ∈ IN , we now deﬁne the p-block of IN n as Γp = α ∈ IN n |α| = p , where if α = (α1 , ..., αn ), then |α| = α1 + · · · + αn . Writing s(p) = |Γp |, then

n+p−1 s(p) = . p Hence, we can write Γp = {α(1), α(2), ..., α(s(p) − 1), α(s(p))} , and without loss of generality, we may assume that α(s(p)) gr α(s(p) − 1) gr · · · gr α(2) gr α(1).

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Returning to the standard ﬁltration F A on A, the next lemma makes the key link between FA and F A. 6.2. Lemma With notation as above, we have Fα(s(p)) A = Fp A, for all p ∈ IN . Proof Since gr is a graded monomial ordering, using the p-block deﬁned above, this follows from a careful comparison of Fα(s(p)) A and Fp A.

7. GK-dimension Computation and Elimination of Variables (General Quadric Case) let A = k[a1 , ..., an ] be an arbitrary quadric solvable polynomial algebra with the associated (left) admissible system (A, B, gr ), and let FA be the gr -ﬁltration on A (as deﬁned in §6). For a given left ideal L ⊂ A, we now start to compute the Gelfand-Kirillov dimension GK.dimM of the left A-module M = A/L by using a (left) Gr¨ obner basis of L, and then derive an elimination lemma for A. All notation are retained from previous sections. 7.1. Lemma (i) Let f, g ∈ A. Then σ(f )σ(g) = σ(f g) in GF (A). s (ii) If f = i=1 ci aα(i) , where aα(i) ∈ B and 0 = ci ∈ k, is such that αn α(1) 1 LM(f ) = aα gr aα(2) gr · · · gr aα(s) , 1 · · · an = a

Then LM(σ(f )) = σ(LM(f )) = σ(a1 )α1 · · · σ(an )αn = σ(f ) ∈ GF (A)α(1) . Proof Since GF (A) is a homogeneous solvable polynomial algebra with respect to gr (Proposition 6.1), (i) and (ii) are then clear by the deﬁnition of FA, the deﬁnition of LM(f ), and the deﬁnition of σ(f ). 2 obner 7.2. Proposition Let L be a left ideal of A and G = {g1 , ..., gs } a Gr¨ basis for L. Then σ(G) = {σ(g1 ) , ..., σ(gs )} = {σ(LM(g1 )) , ..., σ(LM(gs ))} and σ(G) forms a Gr¨ obner basis for GF (L) ⊂ GF (A), where GF (L) is the associated graded left ideal of L with respect to the induced ﬁltration: Fα L = L ∩ Fα A, α ∈ IN n . Proof The fact that σ(G) = {σ(g1 ) = σ(LM(g1 )) , ..., σ(gs ) = σ(LM(gs ))} follows from Lemma 7.1. Since GF (A) is a homogeneous solvable polynomial algebra by Proposition 6.1, it follows from CH.V §1 that σ(G) forms a left Gr¨obner

7. Quadric Solvable polynomial algebra

127

basis in GF (A). To see that σ(G) is a Gr¨obner basis for GF (L) in GF (A), we need to prove that every homogeneous element of GF (L) has a Gr¨ obner presentation by σ(G). Let f, g, h ∈ A be such that LM(f ) gr LM(hg). Then it follows from Proposition 6.1 and Lemma 7.1 that LM(σ(f ))

= σ(LM(f )) gr σ(LM(hg)) = LM(σ(hg)) = LM(σ(h)σ(g)). s Now, if f ∈ L and f = obner presentation of f by G, i=1 hi gi is a Gr¨ σ(hi gi ) = then LM(f ) gr LM(hi gi ) whenever hi gi = 0. Thus, σ(f ) = σ(hi )σ(gi ) with the property that LM(σ(f )) gr LM(σ(hi )σ(gi )) whenever σ(hi gi ) = 0. This shows that every homogeneous element of GF (L) has a Gr¨ obner presentation by σ(G), and hence, σ(G) is a Gr¨obner basis for GF (L) F 2 in G (A). 7.3. Proposition Let L be as in Proposition 7.2 and M = A/L. Then GK.dimM = GK.dim

GF (A) GF (L)

Proof In order to compute GK.dimM , our idea is to use the relation between the gr -ﬁltration FA and the standard ﬁltration F A (see Lemma 6.2):   Fα(s(p)) A = Fp A, and hence (1) n+p F A = dim F A = , p ∈ IN . dim  k α(s(p)) k p p To better understand the proof given below, we refer the reader to CH.VIII §2. Let FM be the gr -ﬁltration on M induced by FA: Fα M = (Fα A + L)/L, α ∈ IN n . Then M forms a gr -ﬁltered A-module in the sense that Fα A·Fβ M ⊂ Fα+β M for all α, β ∈ IN n . Let F M be the ﬁltration on M induced by F A: Fp M = (Fp A + L)/L, p ∈ IN . Then (1) yields (2)

Fα(s(p)) M = Fp M, p ∈ IN .

Consider the associated graded GF (A)-module GF (M ) = ⊕α∈IN n GF (M )α , where GF (M )α = Fα M/Fα∗ M and the module action of GF (A) on GF (M ) is the action induced by the action of A on M . If we use the ﬁltration FL on L induced by FA, then there is the isomorphism of IN n -graded GF (A)-modules  F GF (A)α + GF (L)  F F  ∼ G (A)  with  G (M ) = G (A/L) = GF (L) = GF (L) n α∈I N (3)  GF (A)α + GF (L)   , α ∈ IN n =  GF (M )α ∼ GF (L)

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After a ﬁnitely many seccessive use of the exact sequence of k-vector spaces 0 → Fα∗ M −→ Fα M −→ GF (M )α → 0 we have

(4)

  dimk Fα(s(p)) M         

If we deﬁne (5)

dimk GF (M )α

α(s(p)) gr α

=

dimk 

 GF (M )α  .

α(s(p)) gr α

Fp

=

GF (A) GF (L)

=

α(s(p)) gr α

GF (A)α + GF (L) , GF (L)

p ∈ IN ,

then this is a IN -ﬁltration on GF (A)/GF (L) consisting of ﬁnite dimensional kspaces. With this ﬁltration, GF (A)/GF (L) forms a ﬁltered GF (A)-module with respect to the standard ﬁltration F GF (A) on GF (A), i.e., by deﬁnition,  GF (A)β , q ∈ IN , and Fq GF (A) =      α(s(q)) gr β (6)

F

F    G (A) G (A)   Fq GF (A) · Fp ⊂ F , p, q ∈ IN . q+p GF (L) GF (L) (Indeed, one may see that the ﬁltration we deﬁned on GF (A)/GF (L) above is nothing but the ﬁltration induced by the standard ﬁltration F GF (A): Fp (GF (A)/GF (L)) = (Fp GF (A) + GF (L))/GF (L), p ∈ IN .) Consequently, the above (2)+(3)+(4)+(5)+(6) yields

F G (A) dimk Fp M = dimk Fα(s(p)) M = dimk Fp , p ∈ IN , GF (L) and therefore, as ﬁltered modules over the ﬁltered algebra A with the standard ﬁltration F A and the ﬁltered algebra GF (A) with the standard ﬁltration F GF (A), respectively, M and GF (A)/GF (L) have the same growth measured by the same dimension function. Consequently, GK.dimM = GK.dim(GF (A)/GF (L)). 2 Thus, the computation of GK.dimM has been translated into the computation F F F of GK.dim G (A)/G (L) while G (A)/GF (L) is a left graded module over the homogeneous solvable polynomial algebra GF (A). However, note that if G = {g1 , ..., gs } is the Gr¨obner basis for L as given in Proposition 7.2, then by α α Lemma 7.1, for LM(G) = {LM(gj ) = a1 j1 · · · anjn | j = 1, ..., s}, σ(G) = σ(LM(gj )) = σ(a1 )αj1 · · · σ(an )αjn j = 1, ..., s

7. Quadric Solvable polynomial algebra

129

is a Gr¨obner basis for GF (L) in GF (A). It follows from the results of §§3–4 for homogeneous solvable polynomial algebras that the computation of GK.dimM can be realized via LM(G) as follows. 7.4. Theorem With notation as above, the following holds. (i) There exists a unique polynomial hL (x) of rational coeﬃcients with positive leading coeﬃcient such that for m 0, dimk (Fp M ) = hL (m). Hence, GK.dimM = deghL (x). Moreover, if the ground ﬁeld is computable, then the polynomial hL (x) can be computed as in §3. α α (ii) For LM(G) = {LM(gj ) = a1 j1 · · · anjn | j = 1, ..., s}, put Rj = (αj1 , ..., αjn ), j = 1, ..., s, Mj = i ∈ {1, ..., n} αji = 0 in Rj , j = 1, ..., s, M = J ⊂ {1, ..., n} J ∩ Mj = ∅, 1 ≤ j ≤ s . Then we have

GK.dimM = n − min |J| J ∈ M .

Consequently, (a) d = n if and only if L = {0}; and (b) if G is a reduced Gr¨ obner basis (CH.II Deﬁnition 3.2(iii)), then d = 0 (i.e., A/L is a ﬁnite dimensional k-space) if and only if s = n and (after reordering r mj ’s if necessary) mj = aj j , rj > 0, j = 1, ..., n. (Also see Theorem 8.1 in next section.) 2 As before, we call the polynomial hL (x) obtained in Theorem 7.4(i) the Hilbert polynomial of the module M = A/L. Eventually, the above theorem enables us to use exactly the same argumentation as in the proof of Proposition 5.6 and Theorem 5.7 for obtaining the following lemma. 7.5. Lemma (elimination lemma for quadric solvable polynomial algebras) Let A = k[a1 , ..., an ] be a quadric solvable polynomial algebra with the associated (left) admissible system (A, B, gr ), and let L be a proper left ideal of A such that the A-module A/L has Gelfand-Kirillov dimension d, i.e., the Hilbert polynomial of the A-module A/L has degree d. Then for every subset U = {ai1 , ..., aid+1 } ⊂ {a1 , ..., an } with i1 < i2 < · · · < id+1 , V(U ) ∩ L = {0},

130

V. Dimension and Elimination of Variables α

d+1 d+1 1 }. where V(U ) = k-span{aα i1 · · · aid+1 | (α1 , ..., αd+1 ) ∈ IN

2

Remark (i) In the commutative case, the notion of (weak) independence (modobner in ([Gr¨ o], ulo a polynomial ideal I ⊂ k[x1 , ..., xn ]) was introduced by W. Gr¨ 1968, 1970). Its usefulness in algebraic geometry was realized after combination with the algorithmic techniques of Gr¨ obner bases in order to compute the dimension dimV(I) of the aﬃne algebraic set V(I) determined by the ideal I, or the degree of the Hilbert polynomial of the k[x1 , ..., xn ]-module k[x1 , ..., xn ]/I. The notion of (strong) independence (modulo a polynomial ideal I) was introduced in ([KW], 1989) as a key link between the (weak) independence (modulo I) and a Gr¨ obner basis of I. (ii) The reason that we call Lemma 4.4, Lemma 5.8, and Lemma 7.5 the “elimination lemma” is in the light of [Zei1] and [WZ], where an elimination lemma for the Weyl algebra was given in [Zei1] and an existence proof was due to I.N. Bernstein on the basis of holonomic module theory, and another more “eﬀective” proof was given in [WZ] and the lemma was therefore called the “fundamental lemma” in the automatic proving of hypergeometric (ordinary and “q”) multisum/integral identities (see CH.VII §2 for an introductory interpretation of the latter topic). Note that we have established the elimination lemma in a quite general extent without using any “holonomicity”. As the reader will see soon in CH.VII, however, a more general holonomic module theory in connection with the function identity theory does exist, comparing with the holonomic module theory in the sense of Bernstein. (iii) Let I be an ideal in the commutative polynomial algebra A = k[x1 , ..., xn ] and V (I) the aﬃne algebraic variety deﬁned by I. We see that the use of gr ﬁltration in computing the GK-dimension of M = A/I indeed recaptures the fact that dimV (I) = dimV (LM(I)). And moreover, the “passing to GF (A)” trick can also be used to simplify the computation of syzygies, but we do not pursue this topic in this book.

8. Finite Dimensional Cyclic Modules Let k be an algebraically closed ﬁeld of characteristic zero, and I an ideal in the commutative polynomial algebra k[x1 , ..., xn ]. A well-known result in algebraic geometry states that the algebraic set V (I) in the aﬃne n-space Ank is a ﬁnite set if and only if the k-algebra k[x1 , ..., xn ]/I is ﬁnite dimensional over k. Furthermore, this property of V (I) has been algorithmically recognized via Gr¨ obner bases, as recalled from (e.g., [CLO ], [BW]) below.

8. Finite Dimensional Cyclic Modules

131

Theorem Let V (I) be as above and ﬁx a monomial ordering in k[x1 , ..., xn ]. Then the following statements are equivalent. (i) V (I) is a ﬁnite set. i (ii) For each i, 1 ≤ i ≤ n, there is some αi ≥ 0 such that xα ∈ LM(I), i where the latter is the ideal of k[x1 , ..., xn ] generated by all leading monomials of elements in I. (iii) Let G be a Gr¨ obner basis for I. Then for each i, 1 ≤ i ≤ n, there is some αi αi ≥ 0 such that xi = LM(gi ) for some gi ∈ G. (iv) The k-vector space S = k-Span{xβ | xβ ∈ LM(I)} is ﬁnite dimensional, where xβ = xβ1 1 · · · xβnn . (v) The k-vector space k[x1 , ..., xn ]/I is ﬁnite dimensional. 2 An immediate consequence of the above theorem is the quantitative estimate of the number of solutions of a system of equations when the number is ﬁnite. Corollary Let I ⊂ k[x1 , ..., xn ] be an ideal such that for each i, 1 ≤ i ≤ n, i some power xα ∈ LM(I). Then the number of points of V (I) is at most i α1 · α2 · · · αn . 2 Concerning modules over solvable polynomial algebra with GK-dimension 0, for the sake of making the text complete, in this section we introduce the noncommutative verison of the above result obtained from [K-RW], i.e., all ﬁnite dimensional cyclic modules over an arbitrary solvable polynomial algebra may be recognized via left Gr¨ obner bases. For the reader’s convenience, we also include a proof here. 8.1. Theorem [K-RW] Let A = k[a1 , ..., an ] be an arbitrary solvable polynomial algebra and (A, B, ) the left admissible system associated with A. Let L = g1 , ..., gs ] be a left ideal of A generated by a left Gr¨obner basis G = {g1 , ..., gs }. Put M = A/L. The following statements are equivalent. (i) GK.dimM = 0; (ii) The left A-module M is a ﬁnite dimensional k-space; (iii) For each i, 1 ≤ i ≤ n, there exists gi ∈ G with that LM(gi ) = aβi i for some βi ≥ 0. Proof (i) ⇔ (ii) This is well-known in general. (ii) ⇒ (iii) Note that B − LM(L) yields a k-basis for M = A/L (CH.II Theorem 3.1). For each variable ai of A, 1 ≤ i ≤ n, considering the sequence {a0i , ai , a2i , ..., am obner representations of elements in L by G, i , ...} and the Gr¨ the assumption (ii) and the division algorithm in a solvable polynomial algebra (CH.II Proposition 7.2, Corollary 5.3) entail that there exists gi ∈ G with

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LM(gi ) = aβi i for some βi ≥ 0. (iii) ⇒ (ii) Let f ∈ A and consider the division by G. Since G contains g1 , ..., gn G i with LM(gi ) = aα i , it is then clear that the remainder r = f upon division by G β1 β βn is a linear combination of the monomials a = a1 · · · an ∈ B with 0 ≤ βi ≤ αi , i = 1, ..., n. This shows that M = A/L is a ﬁnite dimensional k-space. 2 8.2. Corollary Let L be a left ideal of the solvable polynomial algebra i k[a1 , ..., an ] such that for each i, 1 ≤ i ≤ n, some power aα i ∈ LM(L). Then dimk (A/L) ≤ α1 · α2 · · · αn .

CHAPTER VI Multiplicity Computation of Modules over Quadric Solvable Polynomial Algebras

Let A = k[a1 , ..., an ] be a quadric solvable polynomial algebra in the sense of CH.III Deﬁnition 2.1, L a left ideal of A, and M = A/L the left A-module with its Hilbert polynomial hL (x) = cd xd + cd−1 xd−1 + · · · + c1 x + c0 . Furthermore, let G = {g1 , ..., gs } be a left Gr¨obner basis for L with leading monomials α α jn LM(gj ) = a1 j1 a2 j2 · · · aα j = 1, ..., s. n , Set, as in CH.V, Rj = (αj1 , αj2 , ..., αjn ) ,

j = 1, ..., s.

The main result of this chapter is to show that • the leading coeﬃcient cd of hL (x), or the multiplicity e(M ) = d!cd of the module M = A/L, can be computed by manipulating only the data (R1 , R2 , ..., Rn ) given above, without computing hL (x).

H. Li: LNM 1795, pp. 133–151, 2002. c Springer-Verlag Berlin Heidelberg 2002

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Combining with CH.V, it follows that the leading term cd xd of hL (x) can be determined completely in terms of the data (R1 , R2 , ..., Rn ).

1. The Multiplicity e(M ) of a Module M Let A = k[a1 , ..., an ] be a quadric solvable polynomial k-algebra as before. If L is a proper left ideal of A and M = A/L, then it follows from CH.V §7 that GK.dim(M ) = deghL (x), where hL (x) is the Hilbert polynomial of the A-module M . Moreover, write hL (x) = cd xd + cd−1 xd−1 + · · · + c1 x + c0 , ci ∈ Q . Then since L is a proper left ideal, cd , the leading coeﬃcient of hL (x), is a positive rational number (CH.V Theorem 7.4). We call d!cd the multiplicity of the left A-module M and denote it by e(M ). The aim of this section is to interpret the name of e(M ) from a viewpoint of representation theory of A. (The reader may compare e(M ) deﬁned here with the multiplicity of a module over a commutative Noetherian local ring in commutative algebra (algebraic geometry) by refering to, e.g., [Eis] ch.12.) Note that for any ﬁnitely generated A-module s N = i=1 Aξi with ξi ∈ N , GK.dimN = max{GK.dim(Aξi ) | i = 1, ..., s} (cf. [KL]). The interpretation given in this section is valid for all ﬁnitely generated A-modules. The ﬁrst obvious fact is that if GK.dimM = 0, then M is a ﬁnite dimensional k-space of dimk M = cd = e(M ), and hence the length of any composition series of M is ≤ e(M ). If GK.dimM = d > 0, then M is inﬁnite dimensional over k. Suppose that M is d-pure in the sense that for every sequence of nonzero A-submodules V ⊂ W ⊂ M , the euqality GK.dim(W/V ) = GK.dimW = d holds. If we consider the ﬁltrations on W and W/V induced by the ﬁltration F M , which is induced by the standard ﬁltration F A on A, the following result is easily proved. 1.1. Proposition If M = A/L is d-pure, then M has ﬁnite length and any composition series of M has length ≤ e(M ). Therefore, if e(M ) = 1, then M is irreducible. 2 Let S be an A-module. Recall from homological algebra that the number jA (S) = min i ∈ IN ExtiA (S, A) = 0

2. Computation of e(M )

135

is called the grade number of S. (For a right module, there is a similar deﬁnition.) A is said to be an Auslander regular algebra if A is left and right Noetherian with ﬁnite global homological dimension gl.dimA = ω, and for each nonzero ﬁnitely generated A-module M , each i ≥ 0 and each nonzero submodule N ⊂ ExtiA (A, M ), the inequality jA (N ) > i holds. If furthermore the equality (∗)

jA (M ) + GK.dimM = n = GK.dimA

holds for every nonzero ﬁnitely generated module M , then A is said to have the Cohen-Macaulay property. If jA (M ) = ω = gl.dimA, then M is called a holonomic A-module (see CH.VII §2 for a background introduction to holonomic modules). Any holonomic module over an Auslander regular algebra is of ﬁnite length (e.g., see [Li1], [LVO4]). If the above (∗) holds for A, then any holonomic A-module M is d-pure where d = n − ω. Example Let A be one of the following k-algebras or their associated Graded and Rees algebras with respect to the standard ﬁltration: (i) The nth Weyl algebra An (k); (ii) The enveloping algebra U (g) of a ﬁnite dimensional Lie algebra g; (iii) The additive analogue An (q1 , ..., qn ) of the Weyl algebra; (iv) Any homogeneous solvable polynomial algebra; (v) The q-Heisenberg algebra hn (q). Then A is a quadric sovable polynomial algebra satisfying the above (∗). (We refer to CH.I §5, [Lev], [Li1], [LVO1], [LVO2] and [LVO4] for the reason why (∗) holds for these algebras.)

2. Computation of e(M ) Let A = k[a1 , ..., an ] be an arbitrary quadric solvable polynomial k-algebra with the associated (left) admissible system (A, B, gr ), where n αn 1 B = aα = aα α = (α · · · a , ..., α ) ∈ I N 1 n n 1 and gr is some graded monomial ordering on B, and let L be a proper left ideal of A with a ﬁxed Gr¨ obner basis G = {g1 , ..., gs }. Suppose that the Hilbert polynomial of the left A-module M = A/L is given by hL (x) = cd xd + cd−1 xd−1 + · · · + c1 x + c0 . This section is devoted to the computation of the multiplicity e(M ) = d!cd for M , without computing hL (x).

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We ﬁrst deal with homogeneous solvable polynomial algebras, i.e., we assume that A is a homogeneous solvable polynomial algebra. All notation are retained from previous chapters. Since L is a proper left ideal of A and G = {g1 , ..., gs } is a Gr¨obner basis of L, then it follows from CH.V §2 that A/L, A/LM(L)], and A/LM(G)] have the same Hilbert polynomial hL (x), where LM(L)] is the monomial left ideal of A generated by the leading monomials of all elements in L and LM(G)] is the monomial left ideal of A generated by LM(G) = {LM(g1 ), ..., LM(gs )} (note that LT(L)] = LT(G)] by CH.V §1). For the left A-module M = A/L, in order to compute e(M ) = d!cd without computing hL (x), it is therefore suﬃcient to assume that L is a monomial left ideal, say L = m1 , ..., ms ] where m1 , ..., ms ∈ B. Thus G = {m1 , ..., ms } is a left Gr¨obner basis for L by CH.V §1. With the ﬁxed notation and assumption as above, the computation of e(M ) will stem from another constructive proof of the existence of the Hilbert polynomial hL (x) of M by using G. (Compare with CH.V §3, and see [CLO ] CH.9 for the constructive proof of the existence of the Hilbert polynomial for a polynomial ¨ ideal in the commutative case, which is based on Hilbert’s famous article Uber die Theorie der Algebraischen Formen (1890)). Adopting similar notation as in [CLO ] CH.9, we start by writing C(L) = α = (α1 , ..., αn ) ∈ IN n aα ∈ B, aα ∈ L for the complement of L in B, i.e., C(L) = B − L, and writing e1 e2 .. .

= =

(1, 0, ..., 0) (0, 1, ..., 0)

en

=

(0, 0, ..., 1)

where each ei is called a unit vector. 2.1. Deﬁnition (i) For {ei1 , ..., eir } ⊂ {e1 , ..., en } with i1 < · · · < ir , r ≤ n, the subset   r   [ei1 , ..., eir ] = qj eij qj ∈ IN for 1 ≤ j ≤ r ⊂ IN n   j=1 is called an r-dimensional coordinate subspace of IN n determined by ei1 , ..., eir . (ii) For β = j ∈{i1 ,...,ir } qj ej ∈ IN n with qj ∈ IN , the subset β + [ei1 , ..., eir ] = β + γ γ ∈ [ei1 , ..., eir ] is called a translate of the r-dimensional coordinate subspace [ei1 , ..., eir ].

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137

2.2. Lemma Let β + [ei1 , ..., eir ] be a translate of the coordinate subspace [ei1 , ..., eir ] ⊂ IN n with β = j ∈{i1 ,...,ir } qj ej . (i) If s > |β|, the number of points β + γ in β + [ei1 , ..., eir ] with |β| + |γ| ≤ s is equal to

r + s − |β| . s − |β|, (ii) For s > |β|, the number of points obtained in (i) above is a polynomial 1 function of s of degree r, and the coeﬃcient of sr is . r! 2 The next proposition is a noncommutative analogue of ([CLO ] Ch.9 Theorem 3). Though its proof is similar to that given in the commutative case, to see how the Noetherian property is essential in the proof (as that for Buchberger’s Algorithm) and why a similar argumentation works for a noncommutative homogeneous solvable polynomial algebra, it is worthwhile to write down the detailed proof here. 2.3. Proposition With notation as above, if L is a proper monomial left ideal, then the set C(L) ⊂ IN n can be written as a disjoint union C(L) = C0 ∪ C1 ∪ · · · ∪ Cn−1 ∪ Cn , where each Ci is a ﬁnite (not necessarily disjoint) union of translates of idimensional coordinate subspaces of IN n . And Cn = ∅ if and only if L = {0}. Proof Suppose L = {0}. We do induction on the number of variables n. If n = 1, then A = k[a1 ] is the polynomial k-algebra in one variable and L = a1 for some integer > 0. The only monomials not contained in L are 1, a1 , ..., a−1 1 . Hence, C(L) = {0} ∪ {1} ∪ · · · ∪ { − 1}, where each {j} is a 0-dimensional coordinate subspace of IN (a translate of the origin {0}). Assume that the result holds for homogeneous solvable polynomial algebras of n − 1 generators. Then the result holds for the subalgebra k[a1 , ..., an−1 ] of A. Let L be a monomial left ideal of A. For each j ≥ 0, let Lj be the left ideal αn−1 1 in k[a1 , ..., an−1 ] generated by monomials aα = aα 1 · · · an−1 with the property that aα · ajn ∈ L. Then it follows from CH.V §1 that C(Lj ) = β ∈ IN n−1 aβ · ajn ∈ L . If j < j = j + s and if aα ∈ Lj , then since aα · (asn ajn ) = λα,s asn aα · ajn ∈ L for some λα,s ∈ k −{0}, we have Lj ⊂ Lj . Thus, there exists an integer j0 such that Lj = Lj0 for all j ≥ j0 (note that solvable polynomial algebras are Noetherian domains). By the induction hypothesis, for each of the Lj , 0 ≤ j ≤ j0 , there is a decomposition j C(Lj ) = C0j ∪ C1j ∪ · · · ∪ Cn−1 ,

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where Ckj is a ﬁnite (not necessarily disjoint) union of translates of k-dimensional coordinate subspaces in IN n−1 . For each j ∈ IN , write Ckj × {j} = (β, j) ∈ IN n β ∈ Ckj ⊂ IN n−1 . We now claim that (∗) where

C(L) = C0 ∪ C1 ∪ · · · ∪ Cn−1 ∪ Cn , 0 −1 j j Ck+1 = Ckj0 × IN ∪ Ck+1 × {j}

j=0

for 0 ≤ k ≤ n − 1, and

" C0 = C(L) −

n

# Ck

.

k=1

To see this, ﬁrst note that Ckj × {j} ⊂ C(L) by the deﬁnition of C(Lj ). To show that Ckj0 × IN ⊂ C(L), note that Lj = Lj0 for j ≥ j0 . Hence it is clear that Ckj0 × {j} ⊂ C(L) for j ≥ j0 ’s. For j < j0 , we have aβ · ajn ∈ L whenever aβ · ajn0 ∈ L since A is a homogeneous solvable polynomial algebra and L is a left ideal of A. This shows that Ckj0 × {j} ⊂ C(L) for j < j0 . Thus, Ck+1 ⊂ C(L) for all k ≥ 0, and it follows from the deﬁnition of C0 that the above (∗) holds. Finally, we show that each Ck is a ﬁnite union of translates of k-dimensional coordinate subspaces in IN n . But this is clear for k > 0. For k = 0, suppose that there were a point α = (α1 , ..., αn ) ∈ C0 with αn ≥ j0 . Then α ∈ C(Lj0 ) × {αn } and consequently α ∈ Ckj0 × IN ⊂ Ck+1 for some k. This contradicts the deﬁnition of C0 . Hence we must have αn < j0 for all points of C0 . It turns out that if C0 were inﬁnite, there would be some j < j0 such that inﬁnitely many points of C0 would lie in C(Lj ) × {j} because A is a homogeneous solvable polynomial algebra. Note that C0j is ﬁnite by our inductive hypothesis. It follows that some of these points would have to be in Ckj × {j} for some k > 0. But the latter set is contained in Ck for k > 0 and this again contradicts the deﬁnition of C0 . This shows that C0 must be ﬁnite, ﬁnishing the proof. 2 To go further, let us write the {m1 , ..., ms } of L as  m1     m2 (1) ..  .    ms

members of the given left Gr¨ obner basis G = = = =

11 α12 1n aα · · · aα n , 1 a2 α21 α22 2n a1 a2 · · · aα n , .. . s1 αs2 sn aα · · · aα n . 1 a2

2. Computation of e(M )

139

Recall from CH.V Theorem 4.2 that if we put Rk

=

Mk

=

M

=

(αk1 , ..., αkn ) , k = 1, ..., s, i ∈ {1, ..., n} αki = 0 in Rk , k = 1, ..., s,

J ⊂ {1, ..., n} J ∩ Mk = ∅, k = 1, ..., s ,

then GK.dimM = n − min {|J| | J ∈ M}, where M = A/L. Since A is a homogeneous solvable polynomial algebra, by the deﬁnition of C(L), CH.V §1, and Proposition 2.3, an easy but very useful observation is recorded as follows. 2.4. Observation With notation as before, let β + [ei1 , ..., eir ] be a translate contained in Cr with β = j ∈{i1 ,...,ir } qj ej . Putting J = {1, ..., n} − {i1 , ..., ir }, then J ∈ M. 2 2.5. Theorem With notation as above, for M = A/L, if GK.dimM = d, then the following hold. (i) C(L) = C0 ∪ C1 ∪ · · · ∪ Cd , with Cd = ∅, where each Ci is a ﬁnite (not necessarily disjoint) union of translates Tij of i-dimensional coordinate subspaces in IN n , i.e., (2).

Ci = Ti1 ∪ Ti2 ∪ · · · ∪ Tim .

(ii) For each s ≥ 0 and each Ci , Tij in the above (2), if we put Cis = aα ∈ Ci |α| ≤ s , Tisj = aα ∈ Tij |α| ≤ s , then |Cis | is a polynomial of degree i in s, in particular, |Cds | is a polynomial of degree d when s is large enough, and the leading term of the Hilbert polynomial hL (x) is given by the leading term of the polynomial |Cds | which is of the form N d d! t , where N is the number of distinct Tdj appearing in the above decomposition (2). Proof (i) If GK.dimM = d, then there is some J ∈ M with |J| = n − d such that J ∩ Mk = ∅, k = 1, ..., s. Put {i1 , ..., id } = {1, ..., n} − J. By CH.V §1, it is clear that [ei1 , ..., eid ] ⊂ Cd , i.e., Cd = ∅. If d < n, then note that since n − d is the smallest number of |J| with J ∈ M, Observation 2.4 yields Cd+i = ∅ for i > 0.

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VI. Multiplicity Computation of Modules

(ii) First note that for s 0, dimk (Fs A + L/L) = |C(L)≤s | = hL (s), where hL (x) is the Hilbert polynomial of M , F A is the standard ﬁltration of A, and C(L)≤s = C(L) ∩ Fs A. When Lemma 2.2 applies to the decomposition (2) in part (i), the counting principle (inclusion and exclusion for ﬁnite sets from elementary combinatorics) yields the desired result. 2 From the last theorem it is then clear that e(M ) = N . We now proceed to compute the number N without computing hL (x). For each J ∈ M with J = {1 , ..., n−d } where d = GK.dimM , put {i1 , ..., id } = {1, ..., n} − J. Assume that 1 < 2 < · · · < n−d and i1 < i2 < · · · < id . Since A is homogeneous solvable, by CH.V §1, every mk in the foregoing (1) may be written as αk αki α α α α (1 ) mk = µ · a1k1 a2k2 · · · an−dn−d · ai1ki1 ai2ki2 · · · aid d , k = 1, ..., s, for some µ ∈ k − {0}. To be convenient, let us use the above (1’) to associate G with an s × n matrix   α11 · · · α1n−d α1i1 · · · α1id  .. .. ..  M (G, J) =  ... . . .  αs1 · · · αsn−d αsi1 · · · αsid and call M (G, J) the matrix of G determined by J. For each p = 1, 2, ..., n − d, put Lp = J − {p },

p = 1, ..., n − d.

Since GK.dimM = d and J ∩ Mj = ∅, j = 1, ..., s, there exists some Mk such that Lp ∩ Mk = ∅, and consequently, the kth row of M (G, J), denoted Rk , has the form Rk = (0, ..., 0, αkp , 0, ..., 0, γk ) with αkp = 0 and γk = (αki1 , ..., αkid ) ∈ [ei1 , ..., eid ]. Putting (∗∗) αp = min αkp Lp ∩ Mk = ∅ , p = 1, ..., n − d, and interchanging the rows of M (G, J) (if necessary), the matrix M (G, J) now has the conﬁguration   α1 0 ··· 0 γ1   0 α2 ··· 0 γ2     .. .. .. .. ..   . . . . .     CM (G, J) =  0 0 ··· αn−d γn−d     αn−d+1,1 αn−d+1,2 · · · αn−d+1,n−d γn−d+1    .. .. .. .. ..     . . . . . αs1 αs2 ··· αsn−d γs

2. Computation of e(M )

141

2.6. Proposition With notation as above, the following holds. (i) Let {αp | p = 1, ..., n − d} be deﬁned by some J ∈ M as in (∗∗) above. If β + [ei1 , ..., eid ] is a translate of some d-dimensional coordinate subspace [ei1 , ..., eid ] contained in Cd with β = p ∈{i1 ,...,id } qp ep , then qp < αp . It follows that, if we write E for the number of distinct J ∈ M with |J| = n − d, then e(M ) ≤

E n−d & αp , p=1

where each product

n−d &

αp is determined by some J ∈ M with |J| = n − d.

p=1

(ii) e(M ) can be computed by using only the entries of the associated s × n matrix M (G, J) of G determined by some chosen J ∈ M with |J| = n − d, without computing the Hilbert polynomial hL (x) of M . More precisely, let {αp | p = 1, ..., n − d} be deﬁned by J as in (∗∗) above. Considering the last s − n + d rows of CM (G, J) as resulted above, if we set, for i = 1, ..., s − n + d, α(n − d + i) = αn−d+i,1 , αn−d+i,2 , ..., αn−d+i,n−d , 0, ..., 0 , then ' n−d e(M ) = β = qp ep p=1

( qp < αp , p = 1, ..., n − d, α(n − d + i) β, i = 1, ..., s − n + d ,

where is the Dickson partial order on IN n induced by gr or by the divisibility in A (see CH.II §6 and the proof of CH.II Proposition 7.2). Proof (i) Since β + [ei1 , ..., eid ] is contained in Cd , the ﬁrst part of the statement follows from CH.V §1 by looking at the conﬁguration CM (G, J) of M (G, J) obtained above, and the inequality follows from Observation 2.4. and the rule of product. (ii) For the chosen J ∈ M with |J| = n − d and J = {1 , ..., n−d } = {1, ..., n} − {i1 , ..., id }, consider β = p ∈J qp ep with qp < αp . By the foregoing discussion and CH.V §1, we easily see that e(M ) is nothing but the number of all aβ ’s which cannot be divided by any one of the monomials: α

α

αn−d+1,n−d

α

α

αn−d+2,n−d

mn−d+1

=

a1n−d+1,1 a2n−d+1,2 · · · an−d

mn−d+2 .. . ms

= .. . =

a1n−d+2,1 a2n−d+2,2 · · · an−d .. . αsn−d αs1 αs2 a1 a2 · · · an−d

2

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VI. Multiplicity Computation of Modules

Furthermore, let J ∈ M with J = {1 , ..., n−d } = {1, ..., n} − {i1 , ..., id }. For p = 1, ..., n − d, put (∗ ∗ ∗) α p = min αkp αkp = 0 in the above CM (G, J), k = 1, ..., s . Again by an easy combinitorial computation and combining Proposition 2.6, we can mention the following estimation formula for multiplicity. 2.7. Proposition With notation as above, we have E n−d E n−d & & αp ≥ e(M ) ≥ α p , p=1

where each product

p=1

n−d &

n−d &

p=1

p=1

αp , respectively

α p , is determined by some J ∈ M

with |J| = n − d, and each αp , respectively each α p , is as in the foregoing (∗∗), respectively in the foregoing (∗ ∗ ∗). The equalities hold in the above inequalities if αp = α p , in particular, if the given Gr¨ obner basis G of L consists of s = n − d monomials. 2 The results established above have two immediate consequences in certain special cases. 2.8. Corollary Let L, G, and M = A/L be as before. With notation as above, e(M ) = 1 if and only if there is only one J ∈ M with |J| = n − d, say J = {1 , ..., n−d } ⊂ {1, ..., n}, such that αp = 1, where αp is as in the foregoing (∗∗), p = 1, ..., n − d. 2 2.9. Corollary Let L, G, and M = A/L be as before. Suppose GK.dimM = n − 1. If Ci = {α1i , ..., αsi } denotes the ith column of M (G, J) for some ﬁxed J ∈ M with |J| = 1, then e(M ) = αi , where αi = min {α1i , ..., αsi } ,

i = 1, ..., n. 2

Summing up, let K be an arbitrary left ideal of the homogeneous solvable polynomial algebra A, S = A/K. If G = {g1 , ..., gs } is a left Gr¨obner basis of K, then we write LM(G)] for the monomial left ideal generated by the leading monomials m1 = LM(g1 ), ..., ms = LM(gs ). Put L = LM(G)] in Proposition 2.6. Then it follows from previous 2.6–2.7 that the following theorem holds.

2. Computation of e(M )

143

2.10. Theorem Let K and S be as above, and suppose GK.dimS = d. With notation as before, the following hold. (i) E n−d E n−d & & αp ≥ e(S) ≥ α p , p=1

where each product

p=1

n−d &

n−d &

p=1

p=1

αp , respectively

α p , is determined by some J ∈ M

with |J| = n − d, and each αp , respectively each α p , is as in the foregoing (∗∗), respectively as in the foregoing (∗ ∗ ∗). The equalities hold in the above inequalities if αp = α p , in particular, if the given left Gr¨ obner basis G of K consists of n − d elements. (ii) e(S) can be computed by manipulating only the entries of the associated matrix M (G, J) of G determined by some J ∈ M with |J| = n − d, as in the proof of Proposition 2.6(ii). (iii) e(S) = 1 if and only if there is only one J ∈ M with J = {1 , ..., n−d } such that αp = 1, where αp is as in the foregoing (∗∗), p = 1, ..., n − d. (iv) Suppose GK.dimS = n − 1. If Ci = {α1i , ..., αsi } denotes the ith column of M (G, J) for some ﬁxed J ∈ M with |J| = 1, then e(S) = αi , where for i = 1, ..., n αi = min {α1i , ..., αsi } , i = 1, ..., n. 2 Now we return to an arbitrary quadric solvable polynomial algebra A = k[a1 , ..., an ]. As dealing with GK-dimension of modules in CH.V §7, we establish the computational result of e(M ) for the module M = A/L, where L is a proper left ideal of A, by passing to the homogeneous solvable polynomial algebra GF (A) with respect to the gr -ﬁltration FA on A. Fix a left Gr¨ obner basis G = {g1 , ..., gs } for L. Consider the ﬁltrations on L and M = A/L induced by the ﬁltration FA on A as in CH.V §7. It follows from CH.V Proposition 6.1, Lemma 7.1, and Proposition 7.2 that GF (A) is a homogeneous solvable polynomial algebra and that σ(G) = {σ(LM(g1 )), ..., σ(LM(gs ))} is a left Gr¨ obner basis for GF (L) consisting of monomials, say σ(LM(gi )) = LM(σ(gi )) = σ(a1 )αi1 σ(a2 )αi2 · · · σ(an )αin , i = 1, ..., s. Moreover, the argumentation of CH.V §7 entails that M and GF (M ) = GF (A)/GF (L) have the same Hilbert polynomial hL (x). Thus, putting in Theorem 2.10 K m1

= GF (L), S = GF (A)/GF (L), = LM(σ(g1 )), m2 = LM(σ(g2 )), ..., ms = LM(σ(gs )),

we are now able to state the main result of this section, as follows.

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VI. Multiplicity Computation of Modules

2.11. Theorem With notation as before, suppose GK.dimM = d. Then the following holds. (i) e(M ) can be computed by manipulating only the entries of the associated matrix M (G, J) of G determined by some J ∈ M with |J| = n − d, as in the proof of Proposition 2.6(ii). (ii) e(M ) = 1 if and only if there is only one J ∈ M with J = {1 , ..., n−d } such that αp = 1, where αp is as in the foregoing (∗∗), p = 1, ..., n − d. (iii) Suppose GK.dimS = n − 1. If Ci = {α1i , ..., αsi } denotes the ith column of M (G, J) for some ﬁxed J ∈ M with |J| = 1, then e(M ) = αi , where for i = 1, ..., n αi = min {α1i , ..., αsi } , i = 1, ..., n.

3. Computation of GK.dim(M ⊗k N ) and e(M ⊗k N ) Let A = k[a1 , ..., an ] and B = k[b1 , ..., bm ] be two quadric solvable polynomial k-algebras with the associated (left) admissible systems (A, B(A), gr ) and (B, B(B), gr ), respectively. In this section we compute GK.dim(M ⊗k N ) and e(M ⊗k N ) for the left A⊗k B-module M ⊗k N , where M = A/L for some left ideal L of A and N = B/K for some left ideal K of B, and (a ⊗ b)(x ⊗ y) = (ax ⊗ by) for a ⊗ b ∈ A ⊗k B, x ⊗ y ∈ M ⊗k N . First note that since k is a ﬁeld, under the canonical k-algebra homomorphisms ϕA : A → A ⊗k B, ϕA (a) = a ⊗ 1, and ϕB : B → A ⊗k B, ϕB (b) = 1 ⊗ b, we have A ∼ = ϕB (B) (e.g., see [Pie] CH.9). Hence, A and B may = ϕA (A) and B ∼ be viewed as subalgebras of A ⊗k B, and each a ∈ A, respectively, each b ∈ B, may be identiﬁed with a ⊗ 1, respectively, with 1 ⊗ b. Moreover, there is the left A ⊗k B-module isomorphism M ⊗k N =

A B ∼ A ⊗k B ⊗k , = L K L, K]

where L, K] is the left ideal of A ⊗k B generated by L and K, identiﬁed with ϕA (L) and ϕB (K) respectively. Secondly, since A and B are quadric solvable algebras, i.e., aj ai = λji ai aj + λk λh ah + λ, 1 ≤ i < j ≤ n, ji ak a + , λh , λ ∈ k; where λji = 0, λk ji µt at + µ, 1 ≤ q < p ≤ m, bp bq = µpq bq bp + µuv pq au av + , µ , µ ∈ k, where µpq = 0, µuv t pq

3. GK.dim(M ⊗k N ) and e(M ⊗k N )

145

putting T = A ⊗k B and Xi = ai ⊗ 1, i = 1, ..., n,

Yj = 1 ⊗ bj , j = 1, ..., m,

then T = k[X1 , ..., Xn , Y1 , ..., Ym ] satisﬁes Xj Xi = λji Xi Xj + λk Xk X + λh Xh + λ, 1 ≤ i < j ≤ n, ji µt Yt + µ, m ≥ p > q ≥ 1, Yp Yq = µpq Yq Yp + µuv pq Yu Yv + Yj Xi = Xi Yj , 1 ≤ i ≤ n, 1 ≤ j ≤ m, and the set of standard monomials B(T ) = X α Y β = X1α1 · · · Xnαn Y1β1 · · · Ymβm

α = (α1 , ..., αn ) ∈ IN n β = (β1 , ..., βm ) ∈ IN m

forms a k-basis of T . Thus, T forms a quadric solvable polynomial k-algebra with respect to the graded monomial ordering gr on B(T ), where gr is deﬁned by extending the ordering gr on B(A) and B(B) to B(T ) in a natural way: If in B(A), a1 gr a2 gr · · · gr an , and in B(B), b1 gr b2 gr · · · gr bm , then set in B(T ) X1 gr X2 gr · · · gr Xn gr Y1 gr Y2 gr · · · gr Ym , and for α, γ ∈ IN n , β, η ∈ IN m , we deﬁne X α Y β gr X γ Y η , if either |α| + |β| > |γ| + |η| or |α| + |β| = |γ| + |η| and α gr γ, or |α| + |β| = |γ| + |η|, α = γ and β gr η. 3.1. Lemma With notation as above, if G1 = {f1 , ..., fs } is a left Gr¨obner basis obner basis of the left of the left ideal L in A, and G2 = {g1 , ..., gh } is a left Gr¨ ideal K in B, then G = {f1 , ..., fs , g1 , ..., gh } is a left Gr¨obner basis for the left ideal L = L, K] in T with respect to gr . Proof Since the gr on both B(A) and B(B) extends to B(T ), one may directly check that every nonzero element in L has a left Gr¨obner presentation by G. 2 Note that IN n and IN m may be viewed as subsets of IN n+m under the natural embeddings −→ IN n+m IN n (α1 , ..., αn ) → (α1 , ..., αn , 0, ..., 0) and

IN m −→ IN n+m (β1 , ..., βm ) → (0, ..., 0, β1 , ..., βm )

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VI. Multiplicity Computation of Modules

If we consider the gr -ﬁltrations FA, FB, and FT respectively (as deﬁned in CH.V §6), it is easy to see that Fγ T = (Fα A ⊗k Fβ B) , γ ∈ IN n+m , α ∈ IN n , β ∈ IN m , γ gr α+β

Fα A ) Fβ B ∼ Fγ T , γ ∈ IN n+m , α ∈ IN n , β ∈ IN m , α + β = γ. = Fα∗ A Fβ ∗ B Fγ ∗ T k

Thus, since GF (A) and GF (B) are homogeneous solvable polynomial algebras by CH.V Proposition 6.1, we have GF (A) ⊗k GF (B) ∼ = GF (T ) as homogeneous solvable polynomial algebras. Therefore, by the discussion in previous §2 and Lemma 3.1, the following theorem may be obtained by assuming that (∗) A and B are homogeneous solvable polynomial algebras, and consequently T = A ⊗k B is a homogeneous solvable polynomial algebra. 3.2. Theorem Let A and B be quadric solvable polynomial algebras and M = A/L, N = B/K be as before. We have GK.dim(M ⊗k N ) = GK.dimM + GK.dimN, e(M ⊗k N ) = e(M )e(N ). Proof We use the above assumption (∗). By Lemma 3.1, we may further assume as in §2 that L and K are monomial left ideals, and so L is a left monomial ideal of T . Suppose GK.dimM = p, e(M ) = ; GK.dimN = q, e(N ) = v. Adopting the notation of §2, by Theorem 2.5 we have = C0 ∪ C1 ∪ · · · ∪ Cp with Cp = ∅, p x + lower terms in x; hL (x) = p! C(K) = C0 ∪ C1 ∪ · · · ∪ Cq with Cq = ∅, v q x + lower terms in x, hK (x) = q!

C(L)

where hL (x), respectively, hK (x), is the Hilbert polynomial of M , respectively, of N . We claim that C(L) = C0 ∪ C1 ∪ · · · ∪ Cp+q with Cp+q = ∅. To see this, let us write e1 , ..., en , en+1 , ..., en+m for the unit vectors in IN n+m , as deﬁned in §2, and identify e1 , ..., en with the unit vectors in IN n , en+1 , ..., en+m

3. GK.dim(M ⊗k N ) and e(M ⊗k N )

147

with the unit vectors in IN m , where both IN n and IN m are viewed as subsets of IN n+m . Note that if β +[ei1 , ..., eir ] is a translate of the r-dimensional coordinate subspace [ei1 , ..., eir ] contained in Cr , then [ei1 , ..., eir ] is also contained in Cr . Hence, if [ei1 , ..., eip ] is a p-dimensional coordinate subspace contained in Cp and [ej1 , ..., ejq ] is a q-dimensional coordinate subspace contained in Cq , then by the construction of L it is easy to see that [ei1 , ..., eip , ej1 , ..., ejq ] is a p+q-dimensional coordinate subspace contained in Cp+q . This shows that Cp+q = ∅. Moreover, if C(L) could contain a p + q + z-dimensional coordinate subspace with z ≥ 1, then again from the construction of L it is easy to see that C(L) would contain a p + z1 -dimensional coordinate subspace with z1 ≥ 1, or C(K) would contain a q + z2 -dimensional coordinate subspace with z2 ≥ 1, a contradiction. Therefore, the largest coordinate subspace contained in C(L) is of dimension p + q, and the desired equality GK.dim(M ⊗k N ) = p + q = GK.dimM + GK.dimN follows from Observation 2.4, the proof of Theorem 2.5(i), and the computation of GKdimension of modules over homogeneous solvable polynomial algebras in CH.V. In order to prove the equality for the multiplicity, let α + [ei1 , ..., eip ] be a translate in Cp ⊂ C(L), and β + [ej1 , ..., ejq ] a translate in Cq ⊂ C(K). Then by the construction of L and the above argumentation we easily see that (α + β) + [ei1 , ..., eip , ej1 , ..., ejq ] is a translate in Cp+q ⊂ C(L). Conversely, let γ + [ek1 , ..., ekp+q ] ∈ Cp+q be a translate of the p + q-dimensional coor dinate subspace [ek1 , ..., ekp+q ], where γ = j ∈{k1 ,...,kp+q } sj ej . Again by the above argumentation [ek1 , ..., ekp+q ] must be of the form [ei1 , ..., eip , ej1 , ..., ejq ] where [ei1 , ..., eip ] is a p-dimensional coordinate subspace contained in Cp and [ej1 , ..., ejq ] is a q-dimensional coordinate subspace contained in Cq . If we put JX = j ∈ {k1 , ..., kp+q } ej ∈ {e1 , ..., en } − {ei1 , ..., eip } α = j∈JX sj ej , JY = j ∈ {k1 , ..., kp+q } ej ∈ {en+1 , ..., en+m } − {ej1 , ..., ejq } , β = j∈JY hj ej , it is easily veriﬁed that α + [ei1 , ..., eip ] is a translate in Cp , β + [eji , ..., ejq ] is a translate in Cq . Furthermore, it follows from Observation 2.4 and CH.V Theorem 4.2 that |JX | = n − p, |JY | = m − q. Hence γ + [ek1 , ..., ekp+q ] = (α + β) + [ei1 , ..., eip , ej1 , ..., ejq ]. Thus, we have shown that Cp+q contains exactly v = e(M )e(N ) distinct translates of the p + q-dimensional coordinate subspaces. Therefore, by Theorem 2.5,

A B T e = ab = e e , L L K as desired.

2

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Example (i) Let A = A1 (q) = k[a, b] be the additive analogue of the ﬁrst Weyl algebra subject to the relation ba = qab + 1 (CH.I §5 Example (ii)), and let B = A 1 (k) the Rees algebra of the ﬁrst Weyl algebra A1 (k) with respect to the standard ﬁltration on A1 (k) (CH.I §3). Then by CH.III §3, B = k[˜ a, ˜b, t] is the algebra generated by a ˜, ˜b, t subject to the relations a ˜t = t˜ a, ˜bt = t˜b, ˜b˜ a=a ˜˜b + t2 . Since A is a solvable polynomial algebra with respect to a >grlex b, by CH.I §3 and CH.IV §4, B is a quadratic solvable polynomial algebra with respect to a ˜ >grlex ˜b >grlex t. Let L be the left ideal of A generated by f = a2 , g = ab + (q + 1)q −2 . A direct calculation yields ba2 = q 2 a2 b + (q + 1)a. It turns out that the S-element S(f, g) is zero. Hence {f, g} is a left Gr¨ obner basis for L, and GK.dim(A/L) = 1, e(A/L) = 1. Let K be the left ideal of B generated by F = a ˜2 , G = a ˜˜b + t2 . Then since S(F, G) = H = ˜bF − a ˜G = a ˜t2 , S(F, H) = ˜H = 0, S(G, H) = t2 G − ˜bH = 0, it follows that {F, G, H} forms a left t2 F − a Gr¨ obner basis for K. By the foregoing results, we read that GK.dim(B/K) = 2 and e(B/K) = 2. Thus, for the left ideal L of T = A ⊗k B generated by L and K, we have GK.dim(T /L) = 3 and e(T /L) = 2.

4. An Application to An (q1 , ..., qn ) In view of the ﬁrst section, the advantage of §3 Theorem 3.2 in the the constructive study of the module representations of a quadric solvable polynomial algebra is obvious, in case the algebra considered is a tensor product of some smaller and nicer quadric solvable polynomial algebras. We illustrate this by an application to the additive analogue A = An (q1 , ..., qn ) of the Weyl algebra (CH.I §5 Example (ii)). Example (i) Recall that A = k[a1 , ..., an , b1 , ..., bn ] is deﬁned by the relations aj ai = ai aj , bj bi = bi bj , 1 ≤ i < j ≤ n, bi ai = qi ai bi + 1, i = 1, ..., n, i= j, bj ai = ai bj , where qi ∈ k − {0}. Using CH.I Theorem 1.1, one may verify that A is the tensor product of its subalgebras A1 (qi ) = k[ai , bi ], i = 1, ..., n, i.e., A∼ = A1 (q1 ) ⊗k A1 (q2 ) ⊗k · · · ⊗k A1 (qn ). Put A(i) = A1 (qi ), i = 1, ..., n. If L(i) denotes the left ideal of A(i) generated by {fi = a2i , gi = ai bi + (qi + 1)qi−2 }, and L denotes the left ideal of A generated

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149

by {a2i , ai bi + (qi + 1)qi−2 }ni=1 , then by §3 Theorem 3.2 and Example (i), the A-module A ∼ A(1) A(2) A(n) M= ⊗k ⊗k · · · ⊗k = L L(1) L(2) L(n) has GK.dimM = n and e(M ) = 1. Focusing on the (classical) nth Weyl algebra An (k) = k[x1 , ..., xn , y1 , ..., yn ], we characterize and construct the “smallest” modules over An (k), algorithmically. For any nonzero An (k)-module M , it is well-known that GK.dim(M ) ≥ n (Bernstein inequality), or in other words, An (k) does not have ﬁnite dimensional module representation. A nonzero ﬁnitely generated An (k)-module is said to be holonomic if GK.dim(M ) = n (= gl.dimAn (k)). A holonomic An (k)-module is cyclic, and is n-pure in the sense of §1 (e.g., see [Bor]) and hence is of ﬁnite length ≤ e(M ). If M is a holonomic module with e(M ) = 1, then M is an irreducible An (k)-module by Proposition 1.1. So we may say that such modules are the “smallest” modules over An (k). Since An (k) is a linear solvable polynomial algebra, Theorem 2.11 and CH.V Theorem 7.4 immediately yield the following algorithmic characterization of the smallest An (k)-modules. 4.1. Theorem Let L be a left ideal of An (k) and G = {g1 , ..., gs } a left Gr¨obner basis of L with respect to >grlex on An (k). Consider the standard ﬁltration on An (k) and put M K m1

= An (k)/L, = G(L), = LM(σ(g1 )), m2 = LM(σ(g2 )), ..., ms = LM(σ(gs )).

With notation as in §2, the following statements hold. (i) GK.dim(M ) = n and e(M ) = 1 if and only if (a) n = min{|J| | J ∈ M}; and (b) there is only one J ∈ M with J = {1 , ..., n } such that α1 = α2 = · · · = αn = 1, where αp is as in §2 (∗∗), p = 1, ..., n. (ii) If GK.dim(M ) = 2n − 1, then e(M ) = αi , where αi = min αki αki is as in §2 (1), k = 1, ..., s , i = 1, ..., 2n. 2 Here let us point out that in case n = 1, the above (i) actually gives more about the generating set of a left ideal in A1 (k). obner 4.2. Proposition Let L be a left ideal of A1 (k) and G = {g1 , ..., gs } a Gr¨ basis of L with respect to >grlex on A1 (k). Furthermore let α1 , α2 be as in

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Theorem 4.1(ii). If we put α1 = α1 , β1 = α2 , and suppose LM(g1 ) = xα1 y β ,

LM(g2 ) = xα y β1 ,

then L is generated by {g1 , g2 }. Proof Uising the division algorithm in A1 (k), for f ∈ L, if we consider the remainder of f on division by {g1 , g2 }, it follows from the deﬁnition of α1 and β1 that L/L is a ﬁnite dimensional k-space, where L is the left ideal of A1 (k) 2 generated by {g1 , g2 }. Hence L = L follows from Bernstein inequality. It is well known (e.g., [Bj]) that every nonzero left ideal of An (k) is generated by two elements. Proposition 4.2 may be regarded as an algorithmic realization of this fact in the special n = 1 case. We also refer to [Gal] for another algorithmic realization of this special case. In the foregoing Example (i), if we take all qi = 1, then M becomes a smallest An (k)-module. Let us ﬁnish this chapter by constructing a family of the “smallest” modules over Weyl algebras. Example (ii) Consider the ﬁrst Weyl algebra A1 (k) with generators x, y. In [Dix] it is proved that the module M = A1 (k)/A1 (k)(xy − β) with β ∈ k is simple if and only if β ∈ ZZ. By Theorem 4.1 we see immediately that e(M ) = 2 because {xy − β} is a Gr¨obner basis of the left ideal A1 (k)(xy − β) with respect to x >grlex y. However, we claim that • for every integer n ≥ 1, the module M = A1 (k)/L with L being generated by {xy + n, xn } is a smallest A1 (k)-module. Proof By checking the S-element of xy + n and xn (CH.II §7) it is easy to see that {xy + n, xn } is a Gr¨ obner basis of L. Hence L is a proper left ideal of A1 (k) and GK.dim(M ) = 1. From Theorem 4.1 it is also clear that e(M ) = 1. This shows that M is a smallest A1 (k)-module. 2 If we further consider the k-algebra automorphism σ: A1 (k) −→ A1 (k) with σ(y) = x, σ(x) = −y, and the module M in Example (i), then it is easy to see that the twisted A1 (k)-module, denoted M σ , where M σ ∼ = M as additive groups and the module operation is deﬁned by f mσ = (f m)σ for f ∈ A1 (k) and mσ ∈ M σ , is of the form A1 (k)/L where L is the left ideal of A1 (k) generated by {xy − (n − 1), y n }. It is not hard to verify that M σ is a simple A1 (k)-module, and by a similar veriﬁcation as in Example (i) it is also a simple module of multiplicity 1. Example (iii) For n ≥ 1, regarding the subalgebra A(j) of An (k) generated by xj , yj as the ﬁrst Weyl algebra, let L(j) be the left ideal of A(j) generated by {xj yj + j, xjj }, j = 1, ..., n, and K the left ideal of An (k) generated by {xj yj + i,

4. Application to An (q1 , ..., qn )

151

xij }ni=j=1 . Then by Example (i)–(ii) and Theorem 3.2, An (k) ∼ A(1) A(2) A(n) ⊗k ⊗k · · · ⊗k = K L(1) L(2) L(n) is a simple An (k)-module with Gelfand-Kirillov dimension n and multiplicity 1.

CHAPTER VII (∂-)Holonomic Modules and Functions over Quadric Solvable Polynomial Algebras

In this chapter, we show how the noncommutative structure theory is possible to interact with the function identity theory “eﬀectively” via Gr¨obner bases, in order to support the idea of generalizing Zeilberger’s algebraic-algorithmic approach to the automatic proving of holonomic function identities over Weyl algebras [Zei1]. The main point is to clarify how to use the elimination lemma obtained in CH.V §7 in formulating a ∂-holonomicity for both “functions” and modules over certain quadric solvable polynomial algebras (including Weyl algebras), so that automatic proving of multivariate identities over more operator algebras may be carried out by using Gr¨ obner bases. This eﬀectiveness may be illustrated by going back to the work of [Tka3] and [CS] where Gr¨ obner bases were used for eliminating variables and for constructing algorithms such as Creative Telescoping in the automatic proving of multivariate identities. (In the Weyl algebra case the Creative Telescoping is an algorithm written for computing equations satisﬁed by deﬁnite sums or integrals of holonomic functions, e.g., [Zei2]. In [CS] this algorithm has been extended to a relatively wide context.)

H. Li: LNM 1795, pp. 153–173, 2002. c Springer-Verlag Berlin Heidelberg 2002

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To better understand the title of this chapter, we recall in §§2–3 certain relevant notions and background results from the literature. For a remarkable introduction to automatic proving of function identities, we refer to the book: A = B, written by M. Petkovˇsek, H. Wilf and D. Zeilberger, published by A.K. Peters, Ltd. 1996. Throughout this chapter k = C , the ﬁeld of complex numbers. Notation for operators adopt those from the literature for the reader’s convenience of referring to relevant references.

1. Some Operator Algebras In this section we list some well-known important operator algebras (e.g., see [CS], [WZ]) that will be used in later sections. (i) Algebra of linear partial diﬀerential operators As we have pointed out in CH.I §5 and have seen in CH.II §8, the nth Weyl algebra An (k) over k coincides with the k-algebra of linear partial diﬀerential operators with polynomial coeﬃcients (note that here k = C ), i.e., An (k) = ∆(k[x1 , ..., xn ]), the subalgebra of the k-algebra Endk k[x1 , ..., xn ] of all k-linear operators on the commutative polynomial algebra k[x1 , ..., xn ] generated by the operators {x1 , ..., xn , ∂1 , ..., ∂n ], where ∂i = ∂/∂xi , i = 1, ..., n, and each xi acts by the left multiplication. The generators of An (k) satisfy the following relations. xi xj = xj xi , ∂i ∂j = ∂j ∂i , 1 ≤ i < j ≤ n, 1 ≤ i, j ≤ n. ∂j xi = xi ∂j + δij , Similarly, let k(x1 , ..., xn ) be the ﬁeld of rational functions in n variables. Then the k-algebra of linear partial diﬀerential operators with rational function coefﬁcients is the algebra Bn (k) = k(x1 , ..., xn )[∂1 , ..., ∂n ] ⊂ Endk k(x1 , ..., xn ) where the generators satisfy the same relations as that for An (k). (ii) Algebra of linear partial shift operators The k-algebra of linear partial shift (recurrence) operators with polynomial coeﬃcients, respectively with rational coeﬃcients, is: k[x1 , ..., xn ][E1 , ..., Em ], respectively k(x1 , ..., xn )[E1 , ..., Em ], n ≥ m, subject to the relations: xj xi = xi xj 1 ≤ i < j ≤ n, Ei xi = (xi + 1)Ei = xi Ei + Ei , 1 ≤ i ≤ m, Ej xi = xi Ej , i = j, Ej Ei = Ei Ej , 1 ≤ i < j ≤ m. For any given sequence u = (uα )α∈IN n with α = (α1 , ..., αi , ..., αn ), changing xi

1. Some Operator Algebras

155

into integer variable ni , the actions of ni and Ei on u are deﬁned as: ni u = (uα )α ∈IN n with α = (α1 , ..., αi−1 , ni αi , αi+1 , ..., αn ); Ei u = (uα )α ∈IN n with α = (α1 , ..., αi−1 , αi + 1, αi+1 , ..., αn ). (iii) Algebra of linear partial diﬀerence operators The k-algebra of linear partial diﬀerence operators with polynomial coeﬃcients, respectively with rational coeﬃcients, is: k[x1 , ..., xn ][∆1 , ..., ∆m ], respectively k(x1 , ..., xn )[∆1 , ..., ∆m ], n ≥ m, subject to the relations: xj xi = xi xj 1 ≤ i < j ≤ n, ∆i xi = (xi + 1)∆i + 1 = xi ∆i + ∆i + 1, 1 ≤ i ≤ m, ∆j xi = xi ∆j , i = j, ∆j ∆i = ∆i ∆j , 1 ≤ i < j ≤ m. For any given function f = f (x1 , ..., xn ), the action of each ∆i on f is deﬁned as: ∆i f = f (x1 , ..., xi−1 , xi + 1, xi+1 , ..., xn ) − f. (iv) Algebra of linear partial q-dilation operators For a ﬁxed q ∈ k − {0}, the k-algebra of linear partial q-dilation operators with polynomial coeﬃcients, respectively with rational coeﬃcients, is: (q)

(q)

(q)

(q)

k[x1 , ..., xn ][H1 , ..., Hm ], respectively k(x1 , ..., xn )[H1 , ..., Hm ], n ≥ m, subject to the relations: xj xi = xi xj , 1 ≤ i < j ≤ n, (q) (q) Hi xi = qxi Hi , 1 ≤ i ≤ m, (q) (q) Hj xi = xi Hj , i = j, (q) (q) (q) (q) Hj Hi = Hi Hj , 1 ≤ i < j ≤ m. (q)

For any given function f = f (x1 , ..., xn ), the action of each Hi

is deﬁned as:

(q)

Hi f = f (x1 , ..., xi−1 , qxi , xi+1 , ..., xn ). (v) Algebra of linear partial q-diﬀerential operators For a ﬁxed q ∈ k − {0}, the k-algebra of linear partial q-diﬀerential operators with polynomial coeﬃcients, respectively with rational coeﬃcients is: (q)

(q)

(q)

(q)

k[x1 , ..., xn ][D1 , ..., Dm ], respectively k(x1 , ..., xn )[D1 , ..., Dm ] n ≥ m, subject to the relations: xj xi = xi xj , 1 ≤ i < j ≤ n, (q) (q) Di xi = qxi Di + 1, 1 ≤ i ≤ m, (q) (q) Dj xi = xi Dj , i = j, (q) (q) (q) (q) Dj Di = Di Dj , 1 ≤ i < j ≤ m.

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For any given function f = f (x1 , ..., xn ), the action of each Di (q)

Di f =

is deﬁned as:

f (x1 , ..., xi−1 , qxi , xi+1 , ..., xn ) − f . (q − 1)xi

Clearly, if n = m, then this operator algebra coincides with the additive analogue of An (k) (CH.I §5). It is not hard to see that all algebras (i) – (v) above are iterated skew polynomial algebras, the algebras of (i), (ii), (iii) and (v) with polynomial coeﬃcients are quadric solvable polynomial algebras, and the algebra of (iv) with polynomial coeﬃcients is a homogeneous solvable polynomial algebra.

2. Holonomic Functions From both a theoretic and a practical point of view, so far the most successfully machine-proved (found) function identities are those formed by the so called holonomic functions. The class of holonomic functions is a subclass of the so called special functions which may go back to the earlier book of E.D. Rainville (Special Functions, Macmillan, New York, 1960; reprinted: Chelsea, New York, 1971), and may be described as follows. (From the references given below the reader may also trace a short history of automatic proving of function identities.) • Certain functions appear so often that it is convenient to give them names. These functions are collectively called special functions. There are many examples and no single way of looking at them can illuminate all examples or even all the important properties of a single example of a special function. (— Special Funtions, Group Theoretical Aspects and Applications, R.A. Askey, T.H. Koornwinder and W. Schempp, Eds., Reidel, Dordrecht, 1984.) • Spectial functions have been created and explored to describe scientiﬁc and mathematical phenomena. Thrigonometric functions give the relation of angle to length. Riemann’s zeta function was invented in order to describe the prime number distribution. Legendre’s spherical functions and Bessel’s functions were born in connection with the eigenvalue problems for partial diﬀerential equations. On the other hand, progress in maths was often made through attempts to understand special functions more deeply. The role of hypergeometric functions in the theory of generalized functions are notable in this respect. Sometimes, developments in maths shed light on previously known special functions from a completely diﬀerent angle. The rediscover of Painlev´e transcendents as correlation functions in statistical mechanical models is one of the best examples of this sort. Several approaches were pursued to handle various special functions in a uniﬁed manner, for example, by

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157

means of diﬀerential equations, or by group representations. However, the nature of special functions is not yet fully explored. To make further progress, entirely new standpoints are to be sought. The theory of quantum groups applied to functions in q-analogue and the Hodge theory as an abstract theory of algebraic integrals are successful examples of recent innovations. ... (— Special Functions, Proc. of the Hayashibara Forum, 1990, Eds. M. Kashiwara and T. Miwa, Springer-Verlag, 1991.) More about special functions may be found in, e.g., [AW], [Foa], [GR], [GZ]. The study of holonomic function is motivated by a nice fusion of the “continuous” and “discrete” in mathematics, that is the story recalled from (e.g., [CS], [Lip2], [Stan], [Zei1]) as follows. In view of §1, the nth Weyl algebra An (k) is viewed as the diﬀerential operator algebra of k[x1 , ..., xn ], i.e, An (k) = k[x1 , ..., xn ][∂1 , ..., ∂n ] ⊂ Endk k[x1 , ..., xn ]. αn β1 βn 1 λαβ xα Since every element of An (k) is of the form D = 1 · · · xn ∂1 · · · ∂n , D may be written as a polynomial in ∂1 , ..., ∂n with coeﬃcients in k[x1 , ..., xn ], i.e., D = D(∂1 , ..., ∂n ). Let f be a nonzero member of a family F on which the Weyl algebra An (k) acts naturally. Put If = D ∈ An (k) Df = 0 . Then If is a left ideal of An (k) and is called the annihilator ideal of f . (I) C-ﬁnite functions In the case n = 1, A1 (k) = k[x][∂]. Recall that the solutions of (homogeneous) linear ordinary diﬀerential equations with constant coeﬃcients (1)

D(∂)f ≡ 0

exactly consist of ﬁnite linear combinations of exponential polynomial solutions, i.e., R f= Pr (x)eλr x , r=1

where {λr } are the roots of the characteristic equation D(z) = 0, and the degree of Pr (x), r = 1, ..., R, is one less than the multiplicity of the root λr . A solution of (1) is called a C-ﬁnite function. 2.1. Proposition [Zei1] With notation as above, the following are equivalent. (i) f is C-ﬁnite. (ii) The vector space k[∂]f = k-Span ∂ i f i ≥ 0

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is ﬁnite dimensional. (iii) k[∂]/(If ∩ k[∂]) is a ﬁnite dimensional k-space. (iv) The algebraic set Vf = z ∈ k D(z) = 0 for D(∂) ∈ If ∩ k[∂] is zero-dimensional (i.e., Vf is ﬁnite). 2 In the case where n > 1, by the celebrated Ehrenpreis-Palamadov theorem, the solutions of the so called overdetermined systems Di (∂1 , ..., ∂n )f ≡ 0,

(2)

i = 1, ..., L,

with Di (∂1 , ..., ∂n ) ∈ k[∂1 , ..., ∂n ] exactly consist of (usually inﬁnite) “linear combination” of exponential polynomial functions Pλ (x)eλx , where x = (x1 , ..., xn ) and λ = (λ1 , ..., λn ) ranges over the algebraic set Vf = λ ∈ k n Di (λ) = 0, i = 1, ..., L . 2.2. Proposition [Zei1] With notation as above, the following are equivalent. (i) f is a solution of the system (2) with a ﬁnite expression f=

R

Pr (x)eλr x .

r=1

(ii) The vector space k[∂1 , ..., ∂n ]f = k-Span ∂1β1 · · · ∂nβn f (β1 , ..., βn ) ∈ IN n is ﬁnite dimensional. (iii) k[∂1 , ..., ∂n ]/(If ∩ k[∂1 , ..., ∂n ]) is a ﬁnite dimensional k-space. (iv) The algebraic set Vf = λ ∈ k n D(λ) = 0 for D(∂1 , ..., ∂n ) ∈ If ∩ k[∂1 , ..., ∂n ] is zero-dimensional (i.e., Vf is ﬁnite). 2 A function f satisfying the equivalent statements of Proposition 2.2 is called a multi-C-ﬁnite function. Recall from CH.VI §1 that a left An (k)-module M is called a holonomic module, if GK.dimM = n (note that GK.dimAn (k) = 2n). Holonomic module theory over Weyl algebras was introduced by I.N. Bernstein and M. Kashiwara in the

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159

algebraic study of the solutions of linear diﬀerential equations (i.e., D-module theory). The earliest grand application of holonomic module theory was given by Bernstein in ([Ber] 1971) for an elementary algebraic proof of a famous conjecture of Gelfand concerning the existence of a meromorphic extension of the distribution valued complex function λ → P λ , where P is a polynomial in several variables over IRn . One of the key properties of modules over An (k) is that the Bernstein inequality holds: GK.dimM ≥ n for any nonzero An (k)-module. Using Bernstein inequality and Proposition 2.2(iii), Zeilberger ﬁrst observed the following fact. 2.3. Proposition If f is a multi-C-ﬁnite function, then the left An (k)-module An (k)/If is holonomic. 2 (II) P-ﬁnite functions and sequences Let Bn (k) = k(x1 , ..., xn )[∂1 , ..., ∂n ] be the k-algebra of linear partial diﬀerential operators with rational function coeﬃcients (§1 Example (i)). Then An (k) is clearly a subalgebra of Bn (k). Let f be a function and Jf = D ∈ Bn (k) Df = 0 , the (left) annihilator ideal of f in Bn (k). 2.4. Proposition ([CS], [Lip1–2], [Stan], [Zei1], a more general case is considered in §4) With notation as above, the following are equivalent. (i) The family {∂1α1 · · · ∂nαn f | (α1 , ..., αn ) ∈ IN n } spans a ﬁnite dimensional k(x1 , ..., xn )-space. (ii) Jf ∩k[x1 , ..., xn ][∂i ] = {0}, i = 1, ..., n, i.e., f satisﬁes a system of “ordinary” equations with polynomial coeﬃcients: Di (∂i )f = 0,

i = 1, ..., n.

(iii) The left Bn (k)-module Bn (k)/Jf is a ﬁnite dimensional k(x1 , ..., xn )-space. 2 A function f is called P-ﬁnite if it satisﬁes the equivalent statements of Proposition 2.4. Write α = (α1 , ..., αn ) ∈ IN n , as before. Let u = (uα )α∈IN n be a sequence deﬁned on IN n , J a nonempty subset of {1, ..., n}. For each i ∈ J and each ai ∈ IN , deﬁne

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VII. (∂-)Holonomic Modules and Functions

a. a section of u as any subsequence of u obtained by considering only the terms of u whose indices α = (α1 , ..., αn ) satisfy αi = ai for all i ∈ J, i.e., any subsequence obtained by setting at least one index to a given value; b. an s-section of u as any section of u deﬁned as previously by J and some ai , with additional constraint that ai < s for all i ∈ J, where s ∈ IN . A sequence u = (uα )α∈IN n is called P-recursive ([Lip2]) if there exists s ∈ IN such that (i) (1) for each i = 1, ..., n, there exists polynomial pα (ni ) such that (i) pβ (αi )u(α−β) = 0 β∈{0,...,s}n

when α satisﬁes αi ≥ s for all i ∈ {1, ..., n}; (2) if n > 1 then each s-sections of u satisﬁes (i) with respect to s. Let A = k[x][E] be the k-algebra of linear partial shift (recurrence) operators in one variable (see §1). Note that the above deﬁnition entails that if u = (uα )α∈IN , then u is P-recursive if and only if there exists P (E) ∈ A such that P (E)u = 0. In one variable case, “P-ﬁnite functions”and “P-recursive sequences” were introduced and studied by Stanley in [Stan]. One of the main results of [Stan] is to fuse both notions, as stated below. 2.5. Theorem (P.R. Stanley 1980) A sequence u = (un )n∈IN is P-ﬁnite if and only if its generating function f (x) = un xn n∈IN

is P-ﬁnite. 2 In the case of several variables a similar result was obtained in [Lip2]. 2.6. Theorem (L. Lipshitz 1989) A sequence u = (uα )α∈IN n is P-recursive if and only if its generating function αn 1 f (x1 , ..., xn ) = uα xα , where xα = xα 1 · · · xn , α∈IN n

is P-ﬁnite. 2

2. Holonomic Functions

161

Example (i) Legendre polynomials {F (n, x)}n∈IN are deﬁned as:

[n/2] −n

F (n, x) = 2

k=0

k

(−1)

n k

2(n − k) k

xn−2k .

∂ F (n, x), EF (n, x) = F (n + 1, x). Then F (n, x) satisﬁes a Write ∂F (n, x) = ∂x linear diﬀerential equation and a linear recurrence equation: + * 2 2 )∂ − 2x∂ + n(n + 1) F (n,+x) = 0 (1 − x * 2 (n + 2)E − (2n + 3)xE + (n + 1) F (n, x) = 0.

Hence F (n, x) is P-ﬁnite and u = {F (n, x)}n∈IN is P-recursive. (III) Holonomic functions The next theorem reveals the relation between holonomic An (k)-modules and Bn (k)-modules which are ﬁnite dimensional k(x1 , ..., xn )-spaces. The proof of the “only if” part is given by Bernstein in [Ber] and the proof of the “if” part follows from a result of Kashiwara [Kas] concerning holonomic D-modules (an elementary proof was given by Takayama in [Tak3] but this depends again on the nature of holonomic modules). 2.7. Theorem (Bernstein-Kashiwara) With notation as above, let J be a left ideal of Bn (k). Then M = An (k)/(J ∩ An (k)) is a holonomic An (k)-module if and only if Bn (k)/J is a ﬁnite dimensional vector space over k(x1 , ..., xn ). 2 It follows from Proposition 2.4 and Theorem 2.7 that we have the following holonomic module characterization of P-ﬁnite functions (sequences). 2.8. Proposition With notation as before, a function f is P-ﬁnite if and only if An (k)/(Jf ∩ An (k)) is a holonomic An (k)-module. 2 Note that If ⊂ Jf . Bernstein inequality, Proposition 2.8 and Proposition 2.3 immediately yield the following corollary. 2.9. Corollary Any multi-C-ﬁnite function f is P-ﬁnite. 2 Summing up, the above recalled results lead to the following pure algebraic uniﬁcation of multi-C-ﬁnite function, P-ﬁnite function, and P-recursive sequence, as Zeilberger said in [Zei1]:

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In the interest of thawing the cold war between the discrete and the continuous, I have decided to combine these two names into one. 2.10. Deﬁnition ([Zei1] 1990) Let f be a nonzero member of a family F on which the Weyl algebra An (k) acts naturally, and let If be the (left) annihilator ideal of f in An (k). f is said to be a holonomic function if the An (k)-module An (k)/If has Gelfand-Kirillov dimension n, i.e., GK.dim(An (k)/If ) = n, or in other words, if An (k)/If is a holonomic An (k)-module. Example (ii) The functions listed below are holonomic (see e.g., [Bj], [CS], [Lip1–2], [Stan], [WZ], [Zei1]). (1) Rational functions 1/P where P is a nonzero polynomial in n ≥ 1 variable(s). (2) All algebraic functions. (3) All special functions falling in the Askey’s scheme (J. Labelle, Tableau d’Askey, in: Polynˆ omes Orthogonauxet Applications, Bar-le-Duc, C. Brezinski et al., Eds., LNM. 1171, Springer-Verlag, 1985).

3. Automatic Proving of Holonomic Function Identities Roughly speaking, automatic proving of function identities implies that the following tasks are carried out on computer: • Find a system of (a ﬁnite number of) operator equations satisﬁed by the given function(s) or sequence(s). • Use the system obtained above and (a ﬁnite number of) initial conditions to identify the given function(s) or sequence(s) and consequently to prove (disprove) the human-posed (human-conjectured) function identities, or to ﬁnd the generating function of a given sequence. • Find new function identities by performing the operations well-deﬁned on the given functions. Let f be a nonzero member of a family F on which the Weyl algebra An (k) acts naturally. As shown in §2, in the light of Bernstein-Kashiwara theorem for Weyl algebras, the P-ﬁniteness of f is equivalent to the holonomicity of the left An (k)-module An (k)/If , where If is the (left) annihilator ideal of f in An (k). In this section we describe, brieﬂy, how the holonomicity of modules over Weyl algebras makes the automatic proving of holonomic function identities feasible, following [CS], [Tak1–2], [Zei1], and CH.V.

3. Holonomic Function Identities

163

(i) How to verify holonomicity • For a given function f , ﬁnd the (left) annihilator ideal If in An (k) and produce a Gr¨ obner basis for If . This can be directly realized by using one of the Maple packages or one of the computer algebra systems developed by experts in computational algebra (see the websites listed in the introduction). • Compute the GK-dimension of An (k)/If by using the Gr¨ obner basis obtained above. (In the early work of Zeilberger and others there was no systematic algorithmic method for computing GK-dimension. Now we have a quite easy algorithmic method, as shown in CH.V.) (ii) How to verify holonomic function identities • The holonomicity is preserved by many algebraic operations, in particular, all holonomic functions form a k-algebra in the usual sense. This big advantage ﬁrst enables us to recognize many holonomic functions; and moreover, if f and g are holonomic functions, in order to see whether or not f and g are equivalent, it is suﬃcient to see whether or not f − g is equivalent to 0. • A holonomic function f can be recovered from a ﬁnite amount of information. More precisely, Proposition 2.4(ii), or more generally, the elimination lemma for Weyl algebras (Lemma 3.1 below) guarantees the existence of a “canonical holonomic representation” for f from which it is always possible to know when such a representation is equivalent to 0, and thus it is possible to know when two diﬀerent canonical representations represent the same function. A canonical holonomic representation can be established by ﬁnding n “ordinary” operators (1)

Pi (∂i , x1 , ..., xn ),

i = 1, ..., n,

in An (k) that annihilate f , where each Pi is of degree αi (in ∂i ), and by ﬁxing the “initial conditions”: (2)

∂1i1 · · · ∂nin f (x0 ),

0 ≤ i1 < α1 , ..., 0 ≤ in < αn ,

where x0 is any point that is not on the “characteristic set” of the system (1) (the characteristic set of a system (1) is the set of common zeros of the leading coeﬃcients of the operators Pi ). This procedure was ﬁrst recognized by a noncommutative version of Sylvester’s dialytic elimination algorithm given by Zeilberger in [Zei1] and later in [WZ] more eﬀectively. Moreover, a good “canonical holonomic representation” may be established by applying left Gr¨ obner bases to the elimination ideals If ∩ k[x1 , ..., xn ][∂i ], as shown in [CS] and [Tak1–2].

i = 1, ..., n,

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VII. (∂-)Holonomic Modules and Functions

3.1. Lemma (elimination Lemma for Weyl algebras, [Zei1] Lemma 4.1; existence proof is due to I.N. Bernstein) Let L be a left ideal in An (k) such that An (k)/L is a holonomic An (k)-module. For every n + 1 generators selected from the 2n generators {x1 , ..., xn , ∂1 , ..., ∂n } of An (k) there is a nonzero member of L that only depends on these n + 1 generators. In particular, for every i = 1, ..., n, L contains a nonzero element of the subalgebra k[x1 , ..., xn , ∂i ] ⊂ An (k). 2 Elimination lemma for Weyl algebras is nowadays known as the “fundamental lemma” in the automatic proving of holonomic function identities (cf [WZ]). Based on this lemma, eﬀective automatic proving of holonomic function identities has been carried out, and a large class of special function identities including all terminating hypergeometric (alias binomial coeﬃcient) identities has been identiﬁed (see “A = B”). Using left Gr¨ obner bases, the idea of [Zei1] has been generalized to the automatic proving of multivariate identities in the context of certain iterated Ore extensions (e.g., [CS]), where the holonomicity was replaced by a ∂-ﬁniteness (see §4 for the details). We ﬁnish this section by quoting two simple examples from [Zei1]. 2

Example (i) Let h = e−x + e−x . A canonical holonomic representation of h is given by D(∂) = (−2x + 1)∂ 2 + (−4x2 + 3)∂ + (−4x2 + 2x + 2), h(0) = 2, h (0) = −1. (ii) Let F (n, x) be the Legendre polynomial as in §2 Example (i). A canonical holonomic representation is given by D(∂) = (1 − x2 )∂ 2 − 2x∂ + n(n − 1), F (0, 0) = 1, F (0, 0) = 0, F (1, 0) = 0, F (1, 0) = 1.

4. Extension/Contraction of the ∂-Finiteness After seeing how essential the “holonomicity” is in the automatic proving of Pﬁnite function identities, we further explore the Extension/Contraction problem proposed in [CS], in order to have a decent holonomicity over other operator algebras that are diﬀerent from Weyl algebras. At this stage, the elimination lemma for quadric solvable polynomial algebras (CH.V Lemma 7.6) plays a key role. To be modest, we restrict our consideration to a class of speciﬁc quadric solvable polynomial algebras which we now deﬁne, as follows.

4. Extension/Contraction of the ∂-Finiteness

165

Given the sets {x1 , ..., xn }, {∂1 , ..., ∂m } of symbols with n, m ≥ 1, the quadric solvable polynomial algebra A = k[x1 , ..., xn , ∂1 , ..., ∂m ] we are considering is the one with the associated (left) admissible system (A, B, gr ), where ∂m gr ∂m−1 gr · · · gr ∂1 gr xn gr xn−1 gr · · · gr x1 , n+m αn β1 βm 1 , (α · · · x ∂ · · · ∂ , ..., α , β , ..., β ) ∈ I N B = xα 1 n 1 m n m 1 1 and the generators of A satisfy  xi xj = xj xi , 1 ≤ i < j ≤ n,      ∂ xi = λi xi ∂ + λp  i xp ∂ + Ei + λ,   where λi , λp , λ ∈ k, λi = 0, 1 ≤ i ≤ n, p < i, 1 ≤ ≤ m, i (3)  ∈ k-Span{x , ..., xn , ∂ }, and E i 1       ∂h ∂ = λh ∂ ∂h + fh ,  where 0 = λh ∈ k, 1 ≤ < h ≤ m, fh ∈ A, ∂ ∂h gr LM(fh ). Obviously, A contains the commutative k-algebra k[x1 , ..., xn ] as a subalgebra and every element D ∈ A may be written as αn β1 βm 1 D = c(α,β) xα 1 · · · xn ∂1 · · · ∂m βm = Pβ (x1 , ..., xn )∂1β1 · · · ∂m , where (α, β) = (α1 , ..., αn , β1 , ..., βm ) ∈ IN n+m and Pβ (x1 , ..., xn ) ∈ k[x1 , ..., xn ]. Hence, A can also be written as A = k[x1 , ..., xn ][∂1 , ..., ∂m ]. We further assume that A satisﬁes the following condition: (Q) S = k[x1 , ..., xn ] − {0} forms a left and right Ore set in A, i.e., for any given s ∈ S, f ∈ A, there are s , s ∈ S and f , f ∈ A such that s f = f s and f s = sf . Except for the Weyl algebra, one easily ﬁnds other quadric solvable polynomial algebras satisfying the condition (Q), in particular, the algebras listed in §1 with polynomial coeﬃcients are such algebras. From the condition (Q) we know that the localization of A at the Ore set S = k[x1 , ..., xn ] − {0}, denoted S −1 A, exists. Thus, we may view A as a subring of S −1 A and write A ⊂ S −1 A. Bearing this in mind, every element D ∈ S −1 A may be written as βm , Qβ ∈ k(x1 , ..., xn ), D= Qβ ∂1β1 · · · ∂m where k(x1 , ..., xn ) is the rational function ﬁeld in variables x1 , ..., xn , i.e., S −1 A is indeed a K(x1 , ..., xn )-algebra with generating set {∂1 , ..., ∂m }. So we may write S −1 A = k(x1 , ..., xn )[∂1 , ..., ∂m ].

166

VII. (∂-)Holonomic Modules and Functions

With the preliminary as above, instead of deﬁning a holonomic-like function, we ﬁrst introduce a holonomic-like module. 4.1. Deﬁnition A left ideal J of S −1 A is said to be ∂-ﬁnite if S −1 A/J is a ﬁnite dimensional vector space over k(x1 , ..., xn ). 4.2. Proposition Let J be a left ideal of S −1 A. J is ∂-ﬁnite if and only if for each i = 1, ..., m, J ∩ k[x1 , ..., xn ][∂i ] = {0}. Proof If J is ∂-ﬁnite, then for each i, {1, ∂i , ∂i2 , ...} spans a ﬁnite dimensional vector space over k(x1 , ..., xn ) in S −1 A/J . It follows that there is a nonzero element Pi (∂i ) ∈ k[x1 , ..., xn ][∂i ] ∩ J . Conversely, suppose that for each i = 1, ..., m there is a nonzero Pi (∂i ) ∈ k[x1 , ..., xn ][∂i ] ∩ J with degree hi . Note f that since S is an Ore set in A, if ∈ k(x1 , ..., xn ) and αi ≥ 1, then there exist g s ∈ S, a ∈ A such that (1)

s∂iαi = ag in A,

(2)

af ∂iαi f · = in S −1 A. 1 g s

The relations in above (3) and (1) above entail that a is a polynomial in ∂i with coeﬃcients in k[x1 , ..., xn ], and the degree of a with respect to ∂i is equal to αi . af is a polynomial Thus, the relations in above (3) and (2) above entail that s in ∂i with coeﬃcients in k(x1 , ..., xn ) and of degree αi . Paying attention to the multiplication in S −1 A, a formal division by {P1 (∂1 ), ..., Pm (∂m )} in S −1 A yields αm that S −1 A/J is spanned by {∂1α1 · · · ∂m }0≤αi
4. Extension/Contraction of the ∂-Finiteness

167

Let A be an algebra of the type we are considering, and S −1 A the corresponding localization of A at S = k[x1 , ..., xn ] − {0}. If J is a left ideal of S −1 A, then the left ideal J c = J ∩ A of A is called the contraction of J in A; if L is a left ideal of A, then the left ideal Le = S −1 AL of S −1 A is called the extension of L in S −1 A. Note that J ce = J and Lec is usually larger than L. Since the ∂-ﬁniteness is deﬁned at the level of S −1 A, in order to extract the elimination information contained in J in terms of Gr¨ obner bases, it is natural c to pass to the contraction ideal J because A is a quadric solvable polynomial algebra. And it turns out that the next theorem actually provides us a way of dealing with extension/contraction problem with respect to the ∂-ﬁniteness. For a subset U = {xi1 , ..., xir , ∂j1 , ..., ∂jt } ⊂ {x1 , ..., xn , ∂1 , ..., ∂m } with i1 < i2 < · · · < ir < j1 < j2 < · · · < jt , as in CH.V §7 we write V(U ) for the k-vector space spanned by βt αr β1 r+t 1 xα . (α · · · x ∂ · · · ∂ , ..., α , β , ..., β ) ∈ I N 1 r 1 t i1 ir j1 jt 4.3. Theorem With notation as before, let L be a left ideal of A, and let J be a proper left ideal of S −1 A. (i) Let U = {xi1 , ..., xir , ∂j1 , ..., ∂jt } ⊂ {x1 , ..., xn , ∂1 , ..., ∂m } be as above. If L ∩ V(U ) = {0}, then GK.dim(A/L) ≥ r + t. (ii) If GK.dim(A/L) = n, then for each i = 1, ..., m, U = {x1 , ..., xn , ∂i }, L ∩ V(U ) = {0}. Hence, the extension left ideal Le of L in S −1 A is ∂-ﬁnite. (iii) GK.dim(A/J c ) ≥ n. (iv) If GK.dim(A/L) = n and L ∩ k[x1 , ..., xn ] = {0}, then GK.dim(A/Lec ) = n. Consequently, if GK.dim(A/J c ) = n then J is ∂-ﬁnite in S −1 A. Proof (i) and (ii) follow from CH.V Theorem 5.7 (or its version for general quadric solvable polynomial algebras) and Lemma 7.5 (elimination lemma). Since J is a proper left ideal of S −1 A and the localization (or the extension) of the contraction left ideal J c is equal to J , it is clear that J c ∩k[x1 , ..., xn ] = {0}. Hence (iii) follows from (i). Finally, (iv) follows from (ii), (iii) and the natural A-module epimorphism A/L → A/Lec because Le is now a proper left ideal of S −1 A by the assumption on L. 2

168

VII. (∂-)Holonomic Modules and Functions

5. The ∂-Holonomicity Let A be a quadric solvable polynomial algebra as in last section. By the discussion of previous sections, we are now ready to introduce the ∂-holonomicity for both “functions” and modules over A. Notation are maintained as before. Let S −1 A be the localization of A at S = k[x1 , ..., xn ] − {0}, and let F be a family of “functions” including k[x1 , ..., xn ]. Suppose that S −1 A (and hence A) acts on F naturally, or that F forms a left S −1 A-module (hence an A-module). For a nonzero member f ∈ F, we consider the (left) annihilator ideal of f in S −1 A, respectively in A: Jf = D ∈ S −1 A Df = 0 , If = D ∈ A Df = 0 . It is clear that Jf is a proper left ideal of S −1 A, Jf ∩ k[x1 , ..., xn ] = {0}, and If ∩ k[x1 , ..., xn ] = {0}. In particular, since Jfc = Jf ∩ A = If , it follows from Theorem 4.3 that GK.dim(A/Jfc ) = GK.dim(A/If ) ≥ n. In view of Theorem 4.3 and the above remark, it is natural to deﬁne ∂-ﬁnite functions, ∂-holonomic functions, ∂-ﬁnite modules and ∂-holonomic modules as follows. 5.1. Deﬁnition With notation as above, let f be a nonzero member of F, and let M be a ﬁnitely generated A-module. (i) f is said to be a ∂-ﬁnite function if Jf is a ∂-ﬁnite left ideal of S −1 A in the sense of Deﬁnition 4.1, i.e., S −1 A/Jf is a ﬁnite dimensional k(x1 , ..., xn )-space. (ii) f is said to be a ∂-holonomic function if GK.dim(A/If ) = n. (iii) M is called a ∂-ﬁnite A-module if the localization of M at S, denoted S −1 M , is ﬁnite dimensional over k(x1 , ..., xn ). (iv) M is called ∂-holonomic if GK.dimM = n. Remark (i) (compare with Deﬁnition 2.10.) From Theorem 4.3 and Deﬁnition 5.1 it is clear that the ∂-holonomicity of f implies the ∂-ﬁniteness of f ; but the converse is not necessarily true for algebras of type A (the author believes this but failed to have a counterexample). Some discussion on the equivalence of ∂-holonomicity and ∂-ﬁniteness is given in connection with the generalization of Bernstein-kashiwara theorem (Theorem 2.7) after Proposition 5.2 below. (ii) Since for a left ideal L of a quadric solvable polynomial algebra A the Gelfand-Kirillov dimension of the A-module A/L is always computable in terms of Gr¨ obner basis (CH.V), we see that the ∂-holonomicity deﬁned above is algorithmically recognizable.

5. The ∂-Holonomicity

169

Next we show that the automatic proving of ∂-ﬁnite function identities, respectively of ∂-holonomic function identities, is feasible in the following sense: if f and g are ∂-ﬁnite functions, respectively ∂-holonomic functions such that f − g is the same type of function, then, to prove f and g are the same function, it is suﬃcient to show that f − g is equivalent to the zero function by using the ∂-ﬁniteness, respectively the ∂-holonomicity, as one did in the context of Weyl algebras. In particular, the automatic proving of ∂-holonomic function identities can be done at the level of manipulating polynomial function coeﬃcients. 5.2. Proposition With notation as above, let f, g ∈ F be ∂-ﬁnite, respectively ∂-holonomic functions in the sense of deﬁnition 5.1. Then f + g is a ∂-ﬁnite, respectively a ∂-holonomic function. Proof If f, g ∈ F are ∂-ﬁnite, respectively ∂-holonomic functions, then we have the S −1 A-module S −1 Af ⊕S −1 Ag, respectively the A-module Af ⊕Ag. Considering the element (f, g) in both modules and the S −1 A-submodule S −1 A(f, g), respectively the A-submodule A(f, g), we obtain the following exact sequences of modules: 0 −→

0

−→

S −1 A J(f,g)

−→

S −1 A S −1 A Jf Jg

S −1 A J(f,g)

−→

S −1 A Jf +g

−→

A A If Ig

A I(f,g) A I(f,g)

−→

A If +g

−→ 0

−→ 0

where J(f,g) , respectively I(f,g) is the (left) annihilator ideal of (f, g) in S −1 A, respectively in A. Since

−1 −1

S A A S A A dimk(x1 ,...,xn ) < ∞, GK.dim = n, Jf Jg If Ig it follows from Theorem 4.3(iii) that: dimk(x1 ,...,xn ) as desired.

S −1 A < ∞, If +g

GK.dim

A = n, If +g 2

Finally, we discuss the possibility of extending the Bernstein-Kashiwara theorem to algebras of type A.

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VII. (∂-)Holonomic Modules and Functions

It follows from the foregoing remark that if the ∂-ﬁniteness always implied ∂holonomicity, then we might replace ∂-ﬁniteness by ∂-holonomicity, and consequently the automatic proving of ∂-ﬁnite function identities could be reduced completely from manipulating rational function coeﬃcients to manipulating polynomial function coeﬃcients. More clearly, we might expect that the equality in Theorem 4.3(iii) holds, and hence we would have an analogue of the Bernstein-Kashiwara theorem for algebras of type A. By CH.V Theorem 7.4 the most obvious case where the equality GK.dim(A/J c ) = n holds is when J c β contains elements of the form ∂j j + fj of A, where the fj are elements of A β

β

with ∂j j = LM(∂j j ) gr LM(fj ), j = 1, ..., m. However, it seems to the author that the equality of Theorem 4.3(iii) does not hold for an arbitrary ∂-ﬁnite left ideal in S −1 A. Nevertheless, we prove that for certain algebras of type A (different from the Weyl algebras) the equality in Proposition 4.3(iii) may hold for arbitrary ∂-ﬁnite left ideals of S −1 A. 5.3. Proposition If A is generated by n + 1 elements x1 , ..., xn , ∂, i.e., A = k[x1 , ..., xn , ∂], and J is any proper left ideal of S −1 A, then GK.dim(A/J c ) = n. Moreover, every proper left ideal in S −1 A if ∂-ﬁnite. Proof That GK.dim(A/J c ) = n follows from Theorem 4.3 and CH.V Theorem 5.7. The ∂-ﬁniteness of J follows from a formal division as in the proof of Proposition 4.2 by considering the polynomial P (∂) in ∂ with coeﬃcients in k[x1 , ..., xn ] which is contained in J and has the smallest degree. 2 Another example is obtained by considering the algebra A = k[x1 , ..., xn , ∂1 , ..., ∂m ] with n, m > 1 but we assume that in S −1 A:  1 f ∂j − δ(f )   , j = 1, ..., m, > 0,  ∂j · = f f 2 (∗)    where f, δ(f ) ∈ k[x1 , ..., xn ] with degδ(f ) ≤ degf. It is not diﬃcult to ﬁnd such algebras (including Weyl algebras as a special case). 5.4. Proposition Let A be a k-algebra as above and S −1 A the localization of A at S = k[x1 , ..., xn ] − {0}. If J is a ∂-ﬁnite proper left ideal of S −1 A, then GK.dim(A/J c ) = n. Proof By Theorem 4.3(iii), we only have to show that GK.dim(A/J c ) ≤ n. Let s1 , ..., st be a basis of the k(x1 , ..., xn )-space S −1 A/J . We may assume that β β the sj are classes of the monomials sj = ∂1 j1 · · · ∂mjm in S −1 A/J and that s1 = 1. Since {s1 = 1, s2 , ..., st } is a basis, there exists a p ∈ k[x1 , ..., xn ] and

5. The ∂-Holonomicity

171

u ∈ k[x1 , ..., xn ], 1 ≤ u ≤ m, 1 ≤ v, j ≤ t, such that qvj

p∂u sv =

(∗∗)

t

u qvj sj .

j=1

Let p be the polynomial we ﬁxed in (∗∗), and consider the k-subspace M of S −1 A/J which is deﬁned as follows: M=

t

k[x1 , ..., xn , p−1 ]sj .

j=1

Putting E=

max

1≤u≤m, 1≤v,j≤t

u degp + 1, degqvj ,

T = 2E, M has a ﬁltration consisting of k-subspaces:   t   gj sj gj ∈ k[x1 , ..., xn ], deggj ≤ wT , w ≥ 0, Fw M = p−w   j=1

wT + n which is a polynomial in w of degree n n. If we consider the ﬁltration on (A + J )/J ∼ = A/J c induced by the standard ﬁltration F A on A, it follows from CH.V Theorem 5.2 and the formula (∇) before CH.V Corollary 5.4 (or their version for general quadric solvable polynomial algebras) that we can ﬁnish the proof by showing that and moreover dimk Fw M = t ·

Fw (A/J c ) ⊂ F2w M,

w ≥ 0.

βm αn β1 c 1 Indeed, let D = xα 1 · · · xn ∂1 · · · ∂m be a monomial in A/J such that |α| + |β| ≤ w, where |α| = α1 + · · · + αn , |β| = β1 + · · · + βm . If we start with ∂m in the foregoing (∗∗):

p∂m = p∂m s1 =

t j=1

1 m = q sj , p j=1 1j t

m q1j sj

implying ∂m

172

VII. (∂-)Holonomic Modules and Functions

then by the assumption (∗) we obtain 1 m q sj p j=1 1j t

2 =∂ ∂ ∂m m m

=

∂m ·

=

t 1 m (p∂m − δ(p)) q1j sj p2 j=1

=

t 1 m m p q1j ∂m + δ(q1j ) sj + δ(p)sj 2 p j=1

=

t 1 m m )sj + δ(p)sj q p∂m sj + pδ(q1j p2 j=1 1j

" " t # # t 1 m m m q qjh sh + pδ(q1j )sj + δ(p)sj p2 j=1 1j

=

h=1

t 1 qj sj p2 j=1

=

with qj ∈ k[x1 , ..., xn ] and degqj ≤ 2E. A repetition of this procedure yields: βm βm −1 ∂m = ∂m ∂m =

t

1 p

2βm −1

qj sj with qj ∈ k[x1 , ..., xn ], degqj ≤ 2βm −1 · E,

j=1

.. . ∂1β1

βm · · · ∂m

=

1

t

p2|β|−1 j=1

qj sj with qj ∈ k[x1 , ..., xn ], degqj ≤ 2|β|−1 · E,

and hence βm αn β1 1 D = xα 1 · · · xn ∂1 · · · ∂m

=

=

1

t

p

2|β|−1

p

2|α|+|β|

αn 1 xα 1 · · · xn qj sj

j=1

1

t j=1

p2

|α|+|β|

−2|β|−1 α1 x1

n · · · xα n qj sj .

5. The ∂-Holonomicity

173

Here

|α|+|β| |β|−1 −2 αn 1 xα · · · x q deg p2 n j 1

≤

(2|α|+|β| − 2|β|−1 )degp + |α| + 2|β|−1 · E

≤ ≤ ≤ = ≤

(2|α|+|β| − 2|β|−1 )degp +2|α| + 2|β|−1 · E |α| |α|+|β| · E + 2 + 2|β| · E 2 |α|+|β| 2·2 ·E 2|α|+|β| · T 2w · T.

This shows that D ∈ F|α|+|β| M ⊂ F2w M , as desired.

2.

CHAPTER VIII Regularity and K0-group of Quadric Solvable Polynomial Algebras

From CH.III §2 we have seen that the class of quadric solvable polynomial algebras includes many popular algebras that are Auslander regular (see CH.VI §1 for the deﬁnition) and have K0 -group ZZ (hence every ﬁnitely generated module has a ﬁnite free resolution, or in other words, a noncommutative version of Hilbert’s syzygy theorem holds), e.g., the Weyl algebra and its additive and multiplicative analogues, enveloping algebras of ﬁnite dimensional Lie algebras, q-Heisenberg algebras, ... . To qualify the name of a quadric solvable polynomial algebra A (comparing with a commutative polynomial algebra), the results obtained in previous chapters naturally motivate us further to explore the regularity of A (at least at the level of having ﬁnite global dimension and K0 -group ZZ). In the case where a given quadric solvable polynomial algebra A is an iterated skew polynomial algebra starting with the ground ﬁeld k, the regularity and K0 -group of A are well-known. However, enveloping algebras of Lie algebras and examples constructed in CH.III §2 show that quadric solvable polynomial algebras are not always iterated skew polynomial algebras starting with the ground ﬁeld k. In this ﬁnal chapter, we ﬁrst derive that every tame quadric solvable polynomial algebra A is Auslander regular with K0 (A) ∼ = ZZ. This is achieved by a closer look at the associated graded algebra G(A) of A with respect to the standard ﬁltration F A. After introducing the gr -ﬁltration on modules, we prove, by

H. Li: LNM 1795, pp. 175–186, 2002. c Springer-Verlag Berlin Heidelberg 2002

176

VIII. Regularity and K0 -group

passing to the associated IN n -graded algebra GF (A) of A with respect to its gr -ﬁltration FA, that every quadric solvable polynomial algebra A is of ﬁnite global dimension. Backing to the standard ﬁltration again in the ﬁnal section, it is proved that K0 (A) ∼ = ZZ holds for every quadric solvable polynomial algebra A. At this stage, we may say that every quadric solvable polynomial algebra is regular in the classical sense. Yet, the author strongly believes that the following proposition is true, though he himself failed to prove it in general. Proposition Every quadric solvable polynomial algebra A is Auslander regular. Note that since every quadric solvable polynomial algebra is a left and right Noetherian domain (with 1) over a ﬁeld, the invariant basis property holds for such algebras, and consequently, there is no problem to talk about global dimension and K0 -group of such algebras. We refer to [Rot] for general homological algebra, and refer to [Bas] for general algebraic K-theory.

1. Tame Case: A is Auslander Regular with K0 (A) ∼ = ZZ Let A = k[a1 , ..., an ] be a tame quadric solvable polynomial algebra in the sense of CH.III Deﬁnition 2.1, and (A, B, gr ) the associated (left) admissible system, where n αn 1 α2 B = {aα 1 a2 · · · an | (α1 , ..., αn ) ∈ IN }, and gr is a graded monomial ordering on B. Then it follows from CH.III Proposition 2.3 that A is completely constructable. In this section, we prove that A and its associated graded structures with respect to the standard ﬁltration F A on A are Auslander regular with K0 -group ZZ. Notation is retained as in previous chapters. 1.1. Theorem Let A be as above, and let G(A) and TA be the associated graded algebra and Rees algebra of A with respect to the standard ﬁltration F A on A, respectively. Then A, G(A), and TA are Auslander regular domains with K0 -group ZZ. Proof By the deﬁnition of a tame quadric solvable polynomial algebra and CH.IV Theorem 4.1, the generators of A satisfy aj ai = λji ai aj + λk λh ah + cji , 1 ≤ i < j ≤ n, ji ak a + k≤<j

2. gr -ﬁltration on Modules

177

where λji = 0, λk ji , λh , cji ∈ k, the generators of G(A) = k[σ(a1 ), ..., σ(an )] satisfy the quadratic relations σ(aj )σ(ai ) = λji σ(ai )σ(aj ) + λk ji σ(ak )σ(a ), 1 ≤ i < j ≤ n, k,<j

and G(A) has the k-basis σ(B) = σ(a1 )α1 σ(a2 )α2 · · · σ(an )αn (α1 , ..., αn ) ∈ IN n . Consequently, the above deﬁning relations determine an iterated skew polynomial algebra structure starting with the polynomial algebra k[σ(x1 )]. Therefore, G(A) is an Auslander regular domain. It follows from [LVO1] and [LVO2] that A and TA are Auslander regular domains, and it follows from the K0 -part of Quillen’s theorem ([Qui] Theorem 7) that K0 (F0 A) ZZ

∼ =

= =

−→ K0 (k) −→ K0 (G(A)0 ) =

K0 (TA0 )

∼ =

K0 (A)

∼ =

K0 (G(A))

∼ =

K0 (TA)

as desired.

2

2. The gr -ﬁltration on Modules From CH.III Deﬁnition 2.1 and the examples constructed in CH.III §2 we know that not every quadric solvable polynommial algebra is a tame quadric solvable polynomial algebra. To study the regularity and K0 -group of an arbitrary quadric solvable polynomial algebra A = k[a1 , ..., an ], in this section we introduce the gr -ﬁltration on A-modules and discuss the gr -ﬁltered homomorphisms and the associated IN n -graded homomorphisms. Let (A, B, gr ) be the (left) admissible system associated with A, where n αn 1 B = aα = aα , α = (α . . . a , ..., α ) ∈ I N 1 n n 1 and gr is a graded monomial ordering on B. Let FA be the gr -ﬁltration FA on A and GF (A) the associated IN n -graded algebra of A with respect to FA, as deﬁned in CH.V §6.

178

VIII. Regularity and K0 -group

2.1. Deﬁnition Let M be a (left) A-module. M is said to be a gr -ﬁltered Amodule if there is a family FM = {Fα M }α∈IN n consisting of k-subspaces Fα M of M such that (FM1) ∪α∈IN n Fα M = M , (FM2) Fβ M ⊂ Fα M if α gr β, and (FM3) (Fα A)(Fβ M ) ⊂ Fα+β M for all α, β ∈ IN n . FM is called a gr -ﬁltration on M . Since for each α ∈ IN n there is α∗ = max{γ ∈ IN n | α gr γ}, to be convenient, for the least element 0 = (0, ..., 0) ∈ IN n , we set F0∗ M = {0} in every FM . If M is a gr -ﬁltered A-module with gr -ﬁltration FM , then the associated graded GF (A)-modules of M is deﬁned as the IN n -graded additive group GF (M )α with GF (M )α = Fα M/Fα∗ M GF (M ) = α∈IN n

on which the module action of GF (A) is given by GF (A)α × GF (M )β (f , m)

−→ GF (M )α+β →

fm

where, if f ∈ Fα A, respectively if m ∈ Fβ M , then f stands for the image of f ∈ GF (A)α = Fα A/Fα∗ A, respectively m stands for the image of m in GF (M )β = Fβ M/Fβ ∗ M . A gr -ﬁltration FM has the property that if 0 = m ∈ M , then there is α ∈ IN n such that m ∈ Fα M − Fα∗ M . In this case we call α the degree of m and write σ(m) for its corresponding homogeneous element in GF (M )α . Before dealing with the associated IN n -graded GF (A)-module GF (M ) of a gr ﬁltered A-module M with gr -ﬁltration FM , we ﬁrst note that, for α, β ∈ IN n with α gr β, the equation α = β + x does not necessarily have a solution in IN n . In particular, even if for α gr β and α∗ gr β, by the deﬁnition of α∗ , the equations α = β + x and α∗ = β + y may not have solutions in IN n simultaneously. This makes the gr -ﬁltrations behave quite diﬀerent from ZZ-ﬁltrations. To remedy this defect, let us put [0, α] = γ ∈ IN n α gr γ . Then clearly, α∗ = max{[0, α] − {α}}. 2.2. Lemma Let α, η ∈ IN n be such that α = η + γ for some γ ∈ IN n . For any β ∈ [0, α∗ ], if β = η + δ for some δ ∈ IN n , then γ ∗ gr δ; and if β = α∗ , then δ = γ∗.

2. gr -ﬁltration on Modules

179

Proof Note that gr is a monomial ordering. The ﬁrst conclusion is then clear by the deﬁnition of a ∗-element. Suppose α∗ = η + δ. Then, γ gr γ ∗ implies α = η + γ gr η + γ ∗ . This, in turn, implies η + δ = α∗ gr η + γ ∗ , and hence δ gr γ ∗ . Combining the ﬁrst conclusion, we conclude that δ = γ ∗ . 2 2.3. Proposition Let M be an A-module. (i) If M has a gr -ﬁltration FM such that GF (M ) = i∈J GF (A)σ(ξi ) with ξi ∈ M and degσ(ξi ) = α(i) ∈ IN n , then M = i∈J Aξi . In particular, if GF (M ) is ﬁnitely generated then so is M . (ii) If M is ﬁnitely generated, then M has a gr -ﬁltration FM such that GF (M ) is ﬁnitely generated over GF (A). Proof (i) Since GF (M ) = i∈J GF (A)σ(ξi ) with ξi ∈ M and degσ(ξi ) = α(i) ∈ IN n , we have GF (M )α = GF (A)β(i) σ(ξi ), α ∈ IN n . i∈J

β(i)+α(i)=α

Thus, for any m ∈ Fα M , m = aβ(i) ξi + m , where aβ(i) ∈ Fβ(i) A with β(i) + α(i) = α, m ∈ Fα∗ M . Similarly we have m = aγ(i) ξi + m , where ∗ aγ(i) ∈ Fγ(i) A with γ(i) + α(i) = α , m ∈ Fα∗∗ M . Since α gr α∗ gr α∗∗ and gr is a graded monomial ordering, after a ﬁnite number of repetition of the above procedure, we arrive at   m∈

  

i∈J

 Fγ(i) A  ξi

γ∈[0,α]

γ(i)+α(i)=γ

and it follows that

 Fα M =



   i∈J

 Fγ(i) A  ξi

γ∈[0,α]

γ(i)+α(i)=γ

because ξi ∈ Fα(i) M , i ∈ J. Hence M = i∈J Aξi . s (ii) Suppose M = i=1 Aξi and {ξ1 , ..., ξs } is a minimal set of generators for M . Choose α(1), ..., α(s) ∈ IN n arbitrarily and set Fγ(i) A = {0} γ∈[0,α]

γ(i)+α(i)=γ

if γ = α(i) + x has no solution for any γ ∈ [0, α].

Then, it is easy to see that

180

VIII. Regularity and K0 -group

the family FM consisting of  Fα M =

s i=1



  

 Fγ(i) A  ξi ,

α ∈ IN n ,

γ∈[0,α]

γ(i)+α(i)=γ

is a gr -ﬁltration on M , where ξi ∈ Fα(i) M − Fα(i)∗ M , i.e., degξi = α(i), i = 1, ..., s. And by Lemma 2.2, it can be veriﬁed directly that GF (M ) = ⊕si=1 GF (A)σ(ξi ) with GF (M )α = GF (A)γ(i) σ(ξi ), α ∈ IN n . 1≤i≤s

γ(i)+α(i)=α

2 Let M and N be gr -ﬁltered A-modules with gr -ﬁltrations FM and FN , respectively. An A-module homomorphism ϕ: M → N is said to be a gr ﬁltered homomorphism, if ϕ(Fα M ) ⊂ Fα N for all α ∈ IN n . A gr -ﬁltered homomorphism ϕ is said to be strict if ϕ(Fα M ) = ϕ(M ) ∩ Fα N,

α ∈ IN n .

If M is a gr -ﬁltered A-module with gr -ﬁltration FM , and if N ⊂ M is an A-submodule of M , then N has the gr -ﬁltration FN consisting of Fα N = Fα M ∩ N,

α ∈ IN n ,

and the quotient A-module M/N has the gr -ﬁltration F(M/N ) consisting of Fα (M/N ) = (Fα M + N )/N,

α ∈ IN n .

The gr -ﬁltrations FN and F(M/N ) deﬁned above are called the induced gr ﬁltration on N and M/N , respectively. With respect to the induced ﬁltration on N and M/N , the inclusion map N → M and the natural map M → M/N are strict gr -ﬁltered homomorphisms. If ϕ: M → N is a gr -ﬁltered A-homomorphism, then ϕ induces naturally a IN n -graded GF (A)-module homomorphism: GF (ϕ) : GF (M ) = GF (M )α −→ GF (N )α = GF (N ) α∈IN n

α∈IN n

→

m

2.4. Proposition Let (∗)

ϕ

ψ

K −→ M −→ N

ϕ(m)

3. gl.dimA ≤ n

181

be a sequence of gr -ﬁltered A-modules and gr -ﬁltered homomorphisms such that ψ ◦ ϕ = 0. Then GF (∗)

GF (ϕ)

GF (ψ)

GF (K) −→ GF (M ) −→ GF (N )

is an exact sequence of IN n -graded GF (A)-modules and IN n -graded homomorphisms if and only if (∗) is exact and ϕ, ψ are strict. Proof First suppose that (∗) is exact and ϕ, ψ are strict. If GF (ψ)(m) = 0 with m ∈ Fα M − Fα∗ M , then 0 = ψ(m) ∈ GF (N )α , i.e., ψ(m) ∈ Fα∗ N ∩ ψ(M ) = ψ(Fα∗ M ). Thus, ψ(m) = ψ(m ) for some m ∈ Fα∗ M , and hence m − m ∈ Kerψ ∩ Fα M = ϕ(K) ∩ Fα M = ϕ(Fα K). Let m − m = ϕ(k) for some k ∈ Fα K. Then m = m − m = ϕ(k) = GF (ϕ)(k). This shows that KerGF (ψ) = GF (ϕ)(GF (K)), i.e., the graded sequence is exact. Conversely, suppose that the graded sequence GF (∗) is exact. To show the strictness of ψ, let f ∈ Fα N ∩ ψ(M ) and f ∈ Fα∗ N . Then f = ψ(m) for some m ∈ Fβ M where β gr α. If β = α, then f = ψ(m) ∈ ψ(Fα M ). If β gr α, then since f ∈ Fα N , we have GF (ψ)(m) = ψ(m) = 0 in GF (N ). By the exactness, m = GF (ϕ)(k) = ϕ(k) for some k ∈ Fβ K. Put m = m − ϕ(k). Then m ∈ Fβ ∗ M , and ψ(m ) = ψ(m − ϕ(k)) = ψ(m) = f . Note that the chain β gr β ∗ gr β ∗∗ gr · · · gr α has ﬁnite length in IN n . It follows that, after a ﬁnite number of repetition of the above procedure, we have f = ψ(mα ) ∈ ψ(Fα M ). This shows that Fα N ∩ ψ(M ) ⊂ ψ(Fα M ), i.e, ψ is strict. A similar argument does reach the strictness of ϕ and the exactness of (∗). 2 2.5. Corollary Let ϕ: M → N be a gr -ﬁltered A-homomorphism. Then GF (ϕ) is injective, respectively surjective, if and only if ϕ is injective, respectively surjective, and ϕ is strict.

3. General Case: gl.dimA ≤ n Let A = k[a1 , ..., an ] be an arbitrary quadric solvable polynomial algebra, and let FA be the gr -ﬁltration on A, as before. With the preparation made in §2, we proceed to show gl.dimA ≤ n in the present section. First recall a well-known result concerning graded projective modules over a (semi)group-graded ring (e.g., [NVO]). Let G be an additive (semi)group and S = ⊕g∈G Sg a G-graded ring. A graded free S-module is a free S-module T = ⊕i∈J Sei on the basis {ei }i∈J , which is also G-graded such that each ei is

182

VIII. Regularity and K0 -group

homogeneous, i.e, if deg(ei ) = gi , then T = ⊕g∈G Tg with Tg = ⊕hi +gi =g Shi ei . For any graded S-module M = ⊕g∈G Mg , there is a graded free S-module T = ⊕g∈G Tg and a graded surjective S-homomorrphism ϕ: T → M . If T is a graded free S-module and P is a graded S-module such that T = P ⊕Q for some graded S-module Q with the property that Tg = Pg + Qg for all g ∈ G, then P is called a graded projective S-module. 3.1. Proposition Let G be a (semi)group, S a G-graded ring and P a graded (left) S-module. The following are equivalent. (i) P is a graded projective S-module. (ii) Given any exact sequence of graded S-modules and graded S-homomorphisms ψ α M −→N → 0, if P −→N is a graded S-homomorphism, then there exists a unique ϕ graded homomorphism P −→M making the following diagram commute:

ϕ

M

−→ ψ

P   α  N →

0

(iii) P is projective as an (ungraded) S-module. 2 Return to modules over the quadric solvable polynomial algebra A. Let L = ⊕i∈J Aei be a free A-module on the basis {ei }i∈J . In view of Lemma 2.2 and the proof of Proposition 2.3, chosen α(i) ∈ IN n arbitrarily for i ∈ J, we may deﬁne a gr -ﬁltration FL on L:   Fα L =

   i∈J

 Fγ(i) A  ei ,

α ∈ IN n ,

γ∈[0,α]

γ(i)+α(i)=γ

where [0, α] = {γ ∈ IN n | α gr γ} as deﬁned before Lemma 2.2, and Fγ(i) A = {0} γ∈[0,α]

γ(i)+α(i)=γ

if γ = α(i) + x has no solution in IN n for any γ ∈ [0, α]. 3.2. Observation In the construction of FL made above, the following properties may be veriﬁed directly by using the monomial ordering gr . (i) For each α ∈ IN n and each i ∈ J, Fγ(i) A = {0} or Fγ(i) A = F A either γ (i) γ∈[0,α]

γ∈[0,α]

γ(i)+α(i)=γ

γ(i)+α(i)=γ

3. gl.dimA ≤ n

183

where γ (i) = max{γ(i) ∈ IN n | γ(i) + α(i) = γ for some γ ∈ [0, α]}. (ii) For each i ∈ J, ei ∈ Fα(i) L − Fα(i)∗ L, i.e., each ei is of degree α(i). 3.3. Deﬁnition Write FL = {Fα L; α(i), i ∈ J}α∈IN n for the gr -ﬁltration on L as deﬁned above. L is called a gr -ﬁltered free A-module with the gr ﬁltration FL. 3.4. Proposition With notation as above, the following holds. (i) If L is a gr -ﬁltered free A-module with the gr -ﬁltration FL, then GF (L) is a IN n -graded free GF (A)-module. (ii) If L is a IN n -graded free GF (A)-module, then L ∼ = GF (L) for some gr ﬁltered free A-module L. (iii) If L is a gr -ﬁltered free A-module with the gr -ﬁltration FL, N is a gr ﬁltered A-module with gr -ﬁltration FN and ϕ: GF (L) → GF (N ) is a graded surjection, then ϕ = GF (ψ) for some strict gr -ﬁltered surjection ψ: L → N . Proof Let FL = {Fα L; α(i), i ∈ J}α∈IN n be the gr -ﬁltration on the free A-module L = ⊕i∈J Aei . By Lemma 2.2, it can be veriﬁed directly that, for α ∈ IN n , GF (L)α = GF (A)β(i) σ(ei ), i∈J

β(i)+α(i)=α

where each σ(ei ) is a homogeneous element of degree α(i) and {σ(ei )}i∈J forms a free GF (A)-basis for GF (L). This proves (i), and then (ii) follows immediately. (iii) Let L = ⊕i∈J Aei with the gr -ﬁltration FL = {Fα L; α(i), i ∈ J}α∈IN n . For each i, choose xi ∈ Fα(i) N such that ϕ(σ(ei )) = xi , where xi is the homogeneous element in GF (N )α(i) represented by xi . Now ψ: L → N may be constructed by putting ψ ai ei = ai xi , where ai ei ∈ L. Clearly, ψ is a gr -ﬁltered homomorphism and GF (ψ) = ϕ since they agree on 2 generators. By Corollary 2.5, ψ is a strict gr -ﬁltered surjection. 3.5. Proposition Let P be a gr -ﬁltered A-module with gr -ﬁltration FP . The following holds. (i) If GF (P ) is a projective GF (A)-module, then P is a projective A-module. (ii) If GF (P ) is a IN n -graded free GF (A)-module, then P is a free A-module. Proof (i) By Proposition 3.4, let ϕ: GF (L) → GF (P ) be a graded surjection, where L is a gr -ﬁltered free A-module and hence GF (L) is a graded free GF (A)module. Again by Proposition 3.4, ϕ = GF (ψ) for some strict gr -ﬁltered surjection ψ: L → P . Let K = Kerψ and FK the gr -ﬁltration on K induced

184

VIII. Regularity and K0 -group

by FL: Fα K = K ∩ Fα L, α ∈ IN n . There is the short exact sequence

ψ

0 → K −→ L −→ P → 0 and it follows from Proposition 2.4 and Corollary 2.5 that the sequence GF ()

GF (ψ)

0 → GF (K) −→ GF (L) −→ GF (P ) → 0 is exact. By Proposition 3.1, this sequence splits by graded GF (A)-homomorphisms. Consequently, GF (L) = GF (P ) ⊕ GF (K) with GF (L)α = GF (P )α ⊕ GF (K)α , α ∈ IN n , and there is a graded surjection γ: GF (L) → GF (K) such that γ ◦ GF () = 1GF (K) . By Proposition 3.4(iii), γ = GF (β) for some strict gr ﬁltered surjection β: L → K. Note that GF (β) ◦ GF () = GF (β ◦ ) = 1GF (k) . It follows from Corollory 2.5 that β ◦ is an automorphism of K, and hence L∼ = K ⊕ P . This shows that P is projective. (ii) Suppose GF (P ) = ⊕i∈J GF (A)σ(ξi ), where each ξi ∈ P has degree α(i) and {σ(ξi )}i∈J is a IN n -graded free basis for GF (P ) over GF (A). Then, by Proposition 2.3, P = i∈J Aξi with   Fα P =

   i∈J

 Fγ(i) A  ξi ,

α ∈ IN n .

γ∈[0,α]

γ(i)+α(i)=γ

We claim that {ξi }i∈J is a free basis for P over A. To see this, construct the gr -ﬁltered free A-module L = ⊕i∈J Aei with the gr -ﬁltration FL = {Fα L; α(i), i ∈ J}α∈IN n as before, such that each ei has the same degree α(i) as ξi does. Then we have an exact sequence of gr -ﬁltered A-modules and strict gr -ﬁltered A-homomorphisms ϕ

0 −→ K −→ L −→ P −→ 0 where K has the gr -ﬁltration induced by FL, and it follows from Proposition 2.4 that this sequence yields an exact sequence GF (ϕ)

0 −→ GF (K) −→ GF (L) −→ GF (P ) −→ 0 However, GF (ϕ) is an isomorphism. Hence GF (K) = {0} and then K = {0}. This proves that ϕ is an isomorphism, or in other words, P is free. 2 3.6. Proposition Let M be a gr -ﬁltered A-module with gr -ﬁltration FM , and let (1)

0 → K → Ln → · · · → L0 → GF (M ) → 0

be an exact sequence of IN n -graded GF (A)-modules and graded homomorphisms, where the Li are graded free GF (A)-modules. The following holds.

3. gl.dimA ≤ n

185

(i) There exists a corresponding exact sequence of gr -ﬁltered A-modules and strict gr -ﬁltered homomorphisms 0 → K → Ln → · · · → L0 → M → 0

(2)

in which the Li are gr -ﬁltered free A-modules. Moreover, we have the isomorphism of chain complexes 0→

0→

→ Ln → ··· → K      ∼ ∼ = = F F G (K) → G (Ln ) → · · · →

L0 → GF (M )     ∼ = = F F G (L0 ) → G (M )

→0

→0

(ii) If K is a projective GF (A)-module, then K is a projective A-module; If K is a IN n -graded free GF (A)-module, then K is a free A-module. (iii) If the modules in (1) are ﬁnitely generated over GF (A), then the modules in (2) are ﬁnitely generated over A. Proof (i) By Proposition 3.4, the homomorphism L0 → GF (M ) in (1) has the form GF (β) for some strict gr -ﬁltered surjection β: L0 → M , where L0 ∼ = GF (L0 ) and L0 is a gr -ﬁltered free A-module. Let K0 = Kerβ with the gr ﬁltration FK0 induced by FL0 . Then we have the exact diagram of graded GF (A)-modules and graded homomorphisms · · · → L2

→

L1

→

0

→

GF (K0 )

→

L0 → GF (M )     ∼ = = F F G (L0 ) → G (M )

→0

→0

Note that the square involved in the above diagram commutes. Hence the homomorphism L1 → L0 factors through GF (K0 ), i.e., there is the graded exact sequence L1 → GF (K0 ) → 0. Starting with GF (K0 ), the foregoing construction can be repeated for ﬁnishing the proof of (i). (ii) and (iii) follow immediately from Proposition 3.5 and Proposition 3.6, respectively. 2 We are ready to mention the ﬁniteness of global dimension for A. 3.7. Theorem Let A = k[a1 , ..., an ] be an arbitrary quadric solvable polynomial algebra with the gr -ﬁltration FA. Write p.dim for projective diemnsion and write gl.dim for global dimension. The following holds. (i) If M is a gr -ﬁltered A-module with gr -ﬁltration FM , then p.dimM ≤ p.dimGF (M ) ≤ n. (ii) gl.dimA ≤ gl.dimGF (A) = n.

186

VIII. Regularity and K0 -group

Proof Let FA be the gr -ﬁltration FA on A as deﬁned in CH.V §6 and GF (A) the associated IN n -graded algebra of A. Then, by CH.V Proposition 6.1, GF (A) is a homogeneous solvable polynomial algebra with respect to gr . Hence, GF (A) is an iterated skew polynomial algebra over the polynomial algebra k[t] with all σ-derivations being 0, and consequently GF (A) is an Auslander regular domain of global dimension n. Note that every A-module M has a gr -ﬁltration FM . Therefore, (i) and (ii) follow from Proposition 3.6.

4. General Case: K0 (A) ∼ = ZZ We put the result as stated by the above title in this separate and ﬁnal section just for emphasizing that we are backing to use the standard ﬁltration again. 4.1. Theorem Let A = k[a1 , ..., an ] be an arbitrary quadric solvable polynomial algebra with its standard ﬁltration F A. Let G(A) and TA be the associated graded algebra and Rees algebra of A with respect to F A, respectively. Then ZZ ∼ = K0 (A) = K0 (G(A)) = K0 (TA). Proof By CH.IV Theorem 4.1–4.2, G(A) = k[σ(a1 ), ..., σ(an )] and TA = k[Ta1 , ..., Tan , X] are quadric solvable polynomial algebras with respect to some gr , respectively. It follows from Theorem 3.7 that gl.dimG(A) ≤ n and gl.dimTA ≤ n + 1. Now, it follows from the K0 -part of Quillen’s theorem ([Qui] Theorem 7) that K0 (F0 A) ZZ

∼ =

= =

−→ K0 (k) −→ K0 (G(A)0 ) =

K0 (TA0 )

∼ =

K0 (A)

∼ =

K0 (G(A))

∼ =

K0 (TA) 2

Final remark Concerning the K-theory of iterated skew polynomial rings, the reader is referred to, e.g., [Art], [MR] and [Pas].

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Index

A admissible system 35 associated graded module 18 associated graded ring 16 Auslander regular algebra 135 B B-standard ﬁltration 92 Bernstein inequality 159 Bernstein-Kashiwara theorem 161 Buchberger’s algorithm 51 C canonical element 16 C-ﬁnite function 157 coordinate subspace 136 CM (G, J) 140 D deﬁning relations 8 deformations of U (sl2 ) 75 deformations of An (k) 78 deformations of Heisenberg enveloping algebra 79 degree of w, d(w) 15, 34, 92 degree function 12 dehomogenization 25 ∂-ﬁnite ideal 166 ∂-holonomic function 168 ∂-holonomic module 168 Dickson’s lemma 59

Dickson quasi-ordering 57 Dickson system 57 diﬀerence operator 155 diﬀerential operator 62, 154 division algorithm 40 division leading monomial 39 down-up algebra 32 E elimination lemma 119, 123, 129, 164 enveloping algebra 30 extension/contraction 164 F ﬁltered A-homomorphism 20 ﬁltered A-module 18 free k-algebra 7 G GK-dimension 115 G-Noetherian algebra 54 graded A-homomorphism 15 graded A-module 14 graded lexicographic ordering 36, 38 graded monomial ordering 35 graded ring homomorphism 15 graded submodule 14 grading ﬁltration 20 Grassmann algebra 31 Gr¨ obner basis 44

196 H Hayashi algebra 31 Hilbert function 110 Hilbert polynomial 115, 120, 129 holonomic function 162 holonomic module 135, 162 homogeneous element 14 homogeneous solvable polynomial algebra 73 homogenization 26 homogenized algebra 87 homogenized enveloping algebra 82 I independent (modulo L) 116, 122 induced ﬁltration 20 J Jacobi sum 72 L leading coeﬃcient LC(f ) 35 leading monomial LM(f ) 35 leading term LT(f ) 35, 109 left admissible system 35 left Dickson system 57 left division leading monomial 39 left Gr¨ obner basis 44 left monomial ordering 35 left quasi-zero element 39 lexicographic ordering 36 left G-Noetherian 54 linear solvable polynomial algebra 73 M M (G, J) 140 mixed gradation 15 monomial left ideal 109 monomial ordering 35 multi-C-ﬁnite function 158 multiplicity e(M ) 134 N natural gradation 15

Index O order function v 19 P PBW k-basis 68 P-ﬁnite function 159 P-ﬁnite sequence 159 positively ﬁltered ring 16 positively graded ring 14 Q q-diﬀerential operator 29, 155 q-dilation operator 155 q-enveloping algebra 80 q-Heisenberg algebra 30 quadric solvable polynomial algebra 73 quasi-zero element 39 R reduced (left) Gr¨ obner basis 44 remainder of f 42 Rees module 18 Rees ring 16 S S-element 52 σ-derivation 10 shift operator 154 σ(a), σ(m), 19 skew polynomial ring 10 solvable polynomial algebra 60 special function 156 standard basis 84 standard ﬁltration 21 strict ﬁltered A-homomorphism 20 strictly ﬁltered basis B 92 gr -ﬁltration 124, 178 gr -ﬁltered free module 183 T tame quadric solvable polynomial algebra 73 a, m, 19 translate 136

Index U u is divisible by w 39 W weak Gr¨ obner representation 69 weakly independent (modulo) L 118, 122 Weyl algebra 28 Z ZZ-ﬁltered ring 16 ZZ-graded ring 13

197