[PPT] - GELFAI\D- KIRI LL OV DIMENS I ON AND POI NCARE S ERIES Martin PowerPoint Presentation

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GELFAI\D- KIRI LL OV DIMENS I ON AND POI NCARE S ERIES

Martin Lorenz D epartment of Mathematical Sciences Northern Illinois University DeKalb, Illinois 60115 U.S.A.

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Agtor: Martin Lorenz Department

f Mathematical

Sciences Northern IlIionois University DeKalb, Illinois 60115

U. S. A.

Conité E-iecutivo:

E. Aznar.

J. L. Bueso.

P. Jara

( Pres id.ente ) . Departamento de Algebra. Facultad de Ciencias. Universidad de Granada. 18071=Granad.a. España. AMS Subject Classification ( 19BO) : 16A55, 16A64, 16A33, 16A05, 16A03. Reeibido en Octubre de 1.987 UNIVERSIDAD DE GRANADA. CUADERNOS DE ALGEBRA NA 7-1.988. Depósito Lega}:

GR. Ltz/7985.

ISSN : OZ 13-1 42O. Inrprime: Secretariado de PubI icaciones. Colegio Máximo de Cartr,rja. [Jniversidad de Granada. Granada. Printer in Spain Impreso en España

nll ^r

F

ur L,nrlsune

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CONTENTS Preface Acknowledgements Notations and Conventions Chapter I: Introduction to Gelfand-Kirillov dimension $ I Definition and Basic Properties $ 2 Some Examples $ 3 Further Properties of GK-dimension Chapte r II: Factors modulo Nilpote nt Ideals $ I The Problem $ 2 Strongly Finircly Presented Algebras $ 3 Application Noetherian PI-algebras Chapter III: On Graded Algebras a¡¡d Modules and Their Poincard Series $ 1 Rational Power Series $ 2 Poincarl Series $ 3 A Non-commutative Hilbert Serre Theorem $ 4 Application to Certain Filrcred Algebras Chapter [V: On Associate d Graded Rings and Modules $ I Generalized Rees R'ngs and Modules $ 2 Examples References

,

6

n n

t

13 l9 26 26 29 34 36 36 42 46 49

54 54 58

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PR^EFAOE

These notes fairly faithfully reflect the conte nts of a series

f lectures given by the author at

the University of Granada, Spain, in April 1987.

The purpose of these lectures was twofold. Firstly, our aim was to give an introduction to Gelfand-Kirillov (GK-) dimension which rapidly leads to advanced material and some open problems. Secondly, we have tried to collect some wide-spread "folklore" results about growth and power series. An effort has been made to render the present set of notes reasonably self- contained. So, in particular, the basic facts about GK-dimension are presented in full detail in Chapter I. On the other ha¡rd, we have included a number

f paragraphs surveying,

without proofs, certain a¡eas which a¡e of vital interest to the subject but are not pursued further in these notes. Needless to say that we do not intend to give a complete account of the state of the a¡t of GK-dimension. In fact, the overlap with other standa¡d texts on the subject, such as the book [K-L] by Krause and Lenagan, is rather small a¡rd essentially consists of the basic material ir Qgp'1q¡_! , although

ur

presentation focusses

n modules

rather than algebras more than it is usually the case. Two general themes have served as a guideline for selecting our material: (1) integrality of GK-dimension, and

(2) the "exa¿tness" question concerning the GK-dimensions of modules in short exactsequences 0 - M'-* M - M" -O. In classical situations, the Gl(-dimension, d(M),

f a module M can be interpreted as the
rder of the pole at t :l
f a certain rational Poincará series

Pu(t) that can be associated with M. In this case, "exactness" of GK-dimension as well as integrality are trivial consequences. Thus our approach, especially in the second half of these notes, is to study GK-dimension via

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/

Poincare series. This leads to additional information which, we feel, is interesting in its own right. In contrast with Chapt". I, QL_Ut"._Il concentrat¿s

n algebras and ideals rather

tha¡r modules, although the original motivation here is a problem about modules (the exactness question). At the cost of having to introduce further terminology, most

f the material

discussed in this chapter can be put in a module theoretic framework, and this is done in [Lo 3l . The present treatment hinges on the notion of a "strongly finitely presented algebra" that was introduced, in a slightly more restrictive form in [Lo 3] . The main result in this chapter,

riginally due to Lorenz and Small [Lo - Sm] and Lenagan [L], proves integrality and exactness
f GK-dimension

for Noetherian Pl-algebras. This result is fairly quickly derived from a result

n strongly finitely presented algebras which possibly admits further

applications once the class

f strongly finitely presented algebras will be better understood.

Chapter III has a more classical flavor. After presenting some more or less sbandard results about rational power series, we prove a rationality result of Hilbert^Serre type for Poincare' series of modules

ver certain (non-commutative)

graded algebras (also taken from [Lo 3] ). Elaborating

n some

remarks in [Lo 3] , we then apply this result to obtain a notion

f

multiplicity and an exactness result, for modules over a certain type of filte..d algebra (Theorem (III. 4.4)). This material applies in particular to enveloping algebras of finite-dimensional (strictly) positively graded Lie-algebras and to positively graded affine Pl-algebras. The final a!gpt"_I_lY is formally nearly independent

f the rest of

these notes. GK- dimension is not even explicitly mentioned in this chapter, but the motivation for the material presented here comes from the earlier results on GK-dimension and Poinca¡J ,..i.. which all use certain finiteness assumptions on (associated) graded rings and modules. In $ 1, we study a construction

f associated graded rings and modules

which is closely related to but different from the usual construction and seems to be more perspective in some respects. This construction is not new: When applied to the I-adic filtration of a ring R , it yields the so-called

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Rees ring of the ideal 1, and it has also been used by Quillen [Q, proof Schapira [S, p.57] , and possibly

thers. Thus some of the results discussed

in may very well be known, in some form, but the construction probably popula.rized among non-commutative ring theorists. In $ 2, we present some

f associated graded rings having bad properties. We also mention the

computations of growth series for certain groups. Bonn, June 1.987

M. Lorenz
f Theorem 7],

$ 1 of Chapter [V deserves to be explicit examples results of some

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AcknowledFements. This work was supported by the Deutsche Forschungsgemeinschaft/Heisenberg Programm (Lo 2Sll2-2), by the Max-Planck-Institut für Mathematik in Bonn, a^nd by the University of Granada, Spain. It is a pleasure to tha¡rk these

institutions. I especially

would like to express my thanks to Jose-Luis Bueso, Pascual Jara, a.nd Blas Torrecillas for making my stay at Granada so enlryable, to George Bergmm, Lance Small, and Toby Sta^fford for enlighte ning letters, preprints and conversations concerning the subject of these notes (Toby Stafford has in particular contributed Example (IV.2.3). ) , and to Lisa Thompson for her expert job on the word processor.

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NOTATIONS AND CONVENTIONS

All rings and algebras considered in these notes are associative and have a I which is inherited by subrings, etc. When not explicitly specified otherwise, modules will be right modules. Throughout, /c denotes a commutative field which will be the base field for all algebras and vector spaces under consideration. Vector space dimensions, dim ., and GK-dimensions, d(.), refer to ,t as base field. The subspace generated by a collection of elements d,9,...in a given vector space will be denoted by 1o,8,...)*. If tr/ is asubspare

f a,t-algebra

R, then we put V': 1u¡a2'...'aolo;e V>¡ C R, 14") : 1a1'a2.....a^l* 4 n, a; e Vyo C R, with the convention thalV": l"l: (0) for n ( 0and lfl: l4ol: &. The subalgebraof /? that is generated by lz wiH be denoted by ,t IV], so

¡lvl - ¡ l") -c n.

Furthermore, for a given vector space V, k {I/} denotes the free algebra on a basis of V, that is the fensor algebra of v. similarly, ,t {x,'{\ denobes the free algebra on {X, Y} , ele . Finally, we shall use the following notations

hJ : {0, r,2,...\ , lN+: {I,2,8,...}, Q+:{ceqlq>o}, #E : cordinolíty of the set E A reference of bhe form Lemma (III.1.2) refers to the lemma in Section (1.2) of Chapter

III. References

within a given chapter

nly mention the relevant

section.

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Chapter I: INTRODUCTION TO GELFAND-KIRILLOV DIMENSION g 1. Definition a¡rd Basic Properbies (1.1) a:g-!h_Sgglgj!. Ler f :N* R be a function which is eventually positive. We are especially interested in the case where the values f (n) are eventually monotone increasing a.nd a¡e dominated by the powers nP for some p € R which we would like to choose as small as possible.

Lemrna. Letl:iN*Rbeeaentuallypositiae,andletD(f)

denotetheaetolallp€Rsuchthat, for some positiue constant C :Co, l(") S C'no holds lor almost all n. Then inf D(/): lim log'/(n). (Here, logo denotes the logarithm to the base n, and inl 0 : a.) Pr@f. Put s ::Hlog^/(n). Suppose Lhat p e D(f), so that Í(") < C.np holds for a suitable C > O and all sufficiently large n. Then log^ Í (") < p llognO and so e I lim (p + log" C\: p. Hence s l infD(/) (trivial for D(l):0). For the reverse inequality, we may assume that 6 < oo. Fix e ) 0. Then, for all large enough n, logn/(n) < s *e and so .f (n) { n'*'. Thus, for all e ) 0, we have s *e € D(/) from which we deduce that s 2 i"Í D(l). U

Definition. If /:N * R is eventually

positive, then we put

d(f),: inf D(Í): lim loso f @) e [0,*], where D(/) is as in the lemma. We will call d(/) the degree of growth of / (especially if / is eventually monotone increasing) .

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Thus d(/) :- holds precisely if D(Í):0. If I is bounded, then d(f ):0, because 0 € D(/) in ttris case. But /(n) : logn also satisfie" d(Í ) :0. We also note that if p e D(/) then, clearly, o € D(/) for all o )_ p. Thus, if /1 and f z te functions a.s above, üen D(/') c D(Í z) or D( Ír) c D(I ). Clearly, if /r( ") < lz(fn*c) for suitable

, ú € lN+
nd almoat

all n , then D(Í) C D(Ít) , andhence d(Ír) S d(f ,) . In particular, D(f t + f ) : D(Ít) ) D(f ,) and so d(f t + Í) : max {d( f '),d(Í r)}. (t.z)Thisisjustanasidewhichwillnotbeusedinthesequel but which perhaps adds some interest to the above definition of growth degrees. Suppose that ,f :lñ* R is positive-valued and such that X /(z)

diverges. Consider the Dirichlet

seriea

i, ¡ 1n¡ 1n"

n:1

for o € CI. Then, putting F(n):: i f (*), d(F,) is the cóscüeo

f conuergencc
f the series

m:l

D f (")fn',

i.e., the series converges (absolutely) for all s € CI with .Re(r) > d(F), and it

¡=

diverges for all o € (D with Re(t) < d(F). (See [2, $ 1].) (1.3) D efinition of Gelfa¡rd-Kirillov dimension (Gelfand and Kirillov IG-Ki] , Bernstein [Bern]). Let.R be a t-algebra and let M be a nonzero right.R-module. lf U I M and 'V 9 R are fixed finite-dimensional /c-subspaces, then we put du,v :: lt* logn dím IJ'I^"),, the degree of growth of the function n - dím U'IA") (n € [J). The Gelfand-Kiríllou (GK-) dimension

l M is then defined bY

d(M) :d(Mn) : : sup {du,v lU C M, V C R fínite- dimensional k- subspaces). (For M :(0),

ne usually puts d(M) :
co.) Of course,

the GK-dimension of left modules can be defined analogously. (See also (3.1).) The Geffand-Kiríllov (GK-) dimension

/R is the

special case d(R):: d(Bn). (f .+) Rema¡ks. Let r9 a¡rd M be as in (1.3). Let (J, U' C M and V, V, C R be finite- dimensional subspaces, a¡rd assume that tJ' c tl'14'l a¡rd V' C I^t) hold, for suitable s, ú € lN+. Then it follows from the last paragraph

f (1.1) that d.u,,y,

I du.y. ln particular': IÍ R : k[V] is affine, generated by V C.R, and M : U.R ís generated by the finite-dimensional subspace U CM,then d(Mr): dr,v.In general, d(Ma) : tup {d(Ns) | S an al fine aubalgebrc

l R, N a linitely generated

S- submodule

l Ms\ .

In the special case where M : R : ktvl we may üake, U : k, and we deduce ühat d(R) : dr,v : lim los" d¡m /"). Thus: Il R: klvl ü affine, then tt(ft) : Tiñ-losn dim Á") In general, d(fi) -- sap {d(S) | S an af fine subalsebra

f R\.

This shows in particular that the definition of d(n) is right left symmetric. (Indeed, we will see in (3.1) that d(,R) equals the GK-dimension of R as (.R,R)-bimodule.) (t.S) pr &lg_lgb_gl (survey). Let R : k [f] be a¡ afne algebra a¡rd let M -- U.R be a finitely generated,R-module. Then U'lA"\ C U I^¡+r) holds for all n. lf U'lA"\ : U'f z+r) for some n then, by induction, we conclude that U.V1") --U.14-) holds for all m t n.

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Consequently, either M ü f inite- dimenaional, in which caae d(Mp) : du,v: O,

r

dim U. tz(") Z n lor all n, a.nd ao d(Mp): du,y Z 1. For general algebras R and .R-modules M , we deduce that d(Mn) :0 if and only if , for all alfine subalgebras S C R , Ms ü tocolly finite- dimensíonal, and' d(Mn) Z t athentt'ise. Warfield [Wa] has explicitly constructed, for each real number p 2 l, a finitely generated module M

ver an affine (PI-) algebra R with d(Mn)

:

p. In contrast with this, G. Bergman

fBerg 1] (cf. also [K-L]) has shown that there is no algebra.R whose GK-dimension is strictly between I and 2. On the other hand, for each real number p > 2, there exists an affine (PI-) algebra,R with d(R):p [Bo-Kr]. We shall however see later that, for many importa.nt classes of alsebras R, one has d(R) € [Iu {*} . (See (2.1), (2.3), (2.4), (II.3.1), (III.2.1),

(ilr.4.4).) By the foregoing, d(n) : 0 holds precisely when the algebra E is locally finite-dimensional

(i.e., all affine subalgebras

f -R are finite-dimensional)

. Small, Staflord, a¡rd Wa¡field ISm-St- Wa] have shown that if R is affine with d(n) : 1, then -R must be PI' (1.6) In the following proposition, we collect most of the basic properties

f GK-dimension

that will be needed in these nor€s. Occasionally they will be used without further comment Iater

n.

Propcitian. Let R be a k -algebra, and let M be a right R -module. (o) IÍ O-Mt'M'M"'0 tr d(M\ ) max {d(M'), d(M")\.

&n eract Eequence "f R -modules, then

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(b) IÍ, :^Pntr lor aubmodulee M¡ C M, then d(M) : sup {d(M¡) l\ € A}. (,) d(M) < d(R). (d) Let $: ^9 * R be a k-algebra homomorphisrn. Then, aiewing M as S-module aia g, one hasd(Ms) < d(Mn).EqaolityholdailR iaafinitelygeneratedrightS-moduleuiag. (t) Asaume that M is finitely generated and that a € End(Mp) r'c injectiue. Then d(M la(M)) < d(M) - r. hof. (") and the inequality d(Ms) < d(M*) in (d) are immediate from rhe definirion of d('). (b) Put s '.: sup {d(Mr) l\ € ^}. Inasmuch as d(M¡) 3 d(M), by (a), we certainly have s < d(M). For the reverse inequality, fix finite-dimensional subspaces U 9 M and V C R. Then U e ,&oUr for a suitable finite subset A0 C A and finite-dimensional subspaces Ux9M* Hence U.14") ! ^&oUn.lo) ,ttd tlim U.IA") < ^?noOO^ Ur.IÁ"). The rema¡ks at the end of (f .t) now imply that du,v l max {dyr,u l\ € A6} ( s, where the latter inequality is of course obvious. This proves that d( M) < t. (.) By (b), any free -R-module F satisfies d(F):d(R). Now choose F so that M is a¡r image of l' and apply (a). (d) It suffices to show that if ,S is a subalgebra

f R such that r?5 is finitely generated,

then d(Mt) > d(Mn). For this, fix a finite-dimensional subspace G C R such that 1€ G a¡rd .R :G'.9. Then, for any finite-dimensional subspace V C R, there exists a finite-dimensional subspace Vt 9 S with I/' G 9 G'71. Hence'v'l.") C G'VIü holds for all n and so, íf U C M is finite-dimensional and Ut:U'G I M , then we have U''Á") C tlr'Vl") . Therefore, du,v 1 dur,nrS d(Ms), whence d(Mn) 3 ¿@s).

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(.) Let -: M - Mlo(M) denote the canonical map, and let U I M and V C finite-dimensional subspaces. It suffices to show lhat d¡,, < d(M) - l. After enlarging V if necessaxy, we may assume that M - U'R and that a( U) C U'V. For each n, denot¿ a fixed vector space complement ol a(M) n U'Á") in U'Á"). Thus dim Co : dim U'1"), a¡rd since C, ) d(M) : (O) and a is injective, it easily follows that the sum C, + o(C") +

l o^(C.) i. direct for all rn. Therefore,

7nn

9^ "r(C,) g F^"i(u./ü) : z^ai(u¡ ' tAü

14 jn J4 n

C DU.Ví.IA") C U.IAz").

-i

J_

Since o is injective, it follows that diln U.V2") > (n+l) . dim Co : ( n+1) .dim Ú.IAü from which we deduce lhat du,v ) I * dO,v. Part (e) follows. tr (1.7) Corollar,y. Let R be a k -algebra. (o) Il S is a subalgebra

f R, then d(S) < d(R).

Equality holds if R is finitely generated as (right or left) S -module. (b) Let I be an ideal of R. Then d(R lI) < d(R). If I contains a (right or left) regular element

f R, then

d(R lI) < d(rq)- l. (t) Let I¡ ( j:1,...,¿) be finitelv mony ideals of R and put , : q ,, Then d(R lI): max {d(R lI)) Prof. (") First, by Proposition (1.6) (.), (.) and (d), d(^9) : d(Ss) : d(Rt) < d(Rn) :d(R). If Rs is finitely generated, then d(fis) : d(Rn), by (d) again. Rbe U a¡rd let Co

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(b) The first assertion is clea¡. Let c € f be rightregular. Then, by Proposition (1.6) (d), (a) and (e), d(R /I):d((RlI)n) < d((RlcR)") < d(.R)- 1. (.) We may a.ssume that f : (0). Put d:: mar( {d(Rlli\l By part (b), d(n) 2 r. Moreover, R¡ embeds into ó-n ¡t, via r - (r + I¡)¡. So d(,R) < s, by Proposition (1.6) i={ (r), (b) and ( d). E

g 2. Some Examples

(Z.t) We firsü deal with the polynomial algebra R: klXr,...,4) in r commuting variables -{. Letting I/ denote the subspace

f .R

generated by the {.'e, we have d(n) :lim- logo dim llo), by (1.4). Here, l"\ is the subspace of R generate d by the monomials in the {'e

f degree at most n. Each such

monomial ca¡r be represented by a string consisting of r slashes I and n sta¡s * , where the ith I corresponds to X¡ and the number of *'s immediately following it is the exponent of X¡ (e.g., ** | * | ** | corresponds to XrX] e V15) g klxr,X2,Xsl). This makes it clea¡ that dim Á"): (tll , apolynomial of degree r in n. Hence, clearly, d(klXL,...,l4l) : d(dim Il')) : .. Now let R be a¡r arbitrary affine commutative É-algebra. Then, by the Noether Normalization Lemma, R is a finitely generated module over a subalgebra .9 which is isomorphic to a polynorrrial algebra t[Xt,...,X,]. Here, d(S) : r : cl. Kdim(^9) : c/. Kdím(R), where cl. Kdim (.) denotes the classical Krull dimension,

r

prime length, a¡rd cl. Kdim(S): ct. Kdim(,R) follows from Going Up and Incomparability (see, e.g., [AuM]). By Corolla^ry (1.7) (a), we conclude that d(R) -- ct. Kdirn(R).

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_ 14 _ In particular, d(R ) € N . The latter fact can also be deduced from the Hilbert-Serre theorem as follows. Fix any finite-dimensional generating subspace V of R and consider the graded algebra

Gv(R) ': ,9ol')x' g fitxl

Then Gy( R) is affine: generators are X and the elements uX, where ? runs through a ba.sis

f
V. All are homogeneous
f degree l. By the Hilbert-Serre theorem (e.g., [At-M, Corolla.ry

11.2] ), it follows that, for sufficiently large n, the dimension of the n-th homogeneous component of Gy( R) , dim ¡"), is a polynomial in n (with rational coefficients) . As above, d(.R) is the degree

f this polynomial.

Finally, rf M (#(O)) is a finitely generated B;module, then d(Mn): d(Rfannx(M)). Indeed, putting I : annn(M), we have d(Mp\: d(Mnt ) < d( RII), by (d) a¡rd (.) . On the other hand, if M is generated by ffi1, .

t

/ : n annp(m¡) and so d(R lI) - m?x {d(R lann¡(r"¡))} < d(Mn), j:r inequality follows from the fact thai each R lannp(m¡) embeds rIannp(m¡)

m¡r.

tn.1,...,mo

f M

with onnp (M): /

nnp(^¡). Right Noetherian rings

i:r condition are identical with üe so-called rightFBN-rings. These include, e.g Pl-rings. For all this, see [Ch-H, Section 7]. The argument in the la.stparagraph

f (2.1) can be copied literally to yield

(2.2) Right FBN-rings and Gabriel's H-condition. Recall that a ring B satisfies Gabriel's H- cond,ition if, for each finitely generated right,R-module M , there exist finitely many elements

Proposition ( 1.6) . , ffit, S&Y, then where the latter into M p via

satisfying the H- , right Noetherian the following

lo -

Lemma. Let R be a k -algebra which aatiafica Gabriel'a H -condition, and fi.nitely generated right R -module. Then d(Mn) :d(R f annp(M)) For a simila¡ equality in the context of bimodules, see Proposition (3.1).

(2.3) Free subalgebras (survey). Let R : k {X,Y} be the free algebra be the subspace of r? that is generated by X and )/. Since there monomials of length j with factors X a¡rd Y, we see that

nX

are 2i let M I (0) üe c a.nd Y, a¡rd let V formally distinct dim /") : D 2i :2"+r- !. i{ l)rus, clearly, D(dim Il")) : 0 a¡rd so d(dim I^4) :

o. Therefore,

d(k{x,)Z}) : *, and the same conclusion holds if ,R is any /c-algebra which contains a non-commutative free

subalgebra. The converse is not true. There even exist groups G with two generators so that

the group algebra /cG contains no non-commutative free algebra but d( kG) :

o IGr] . The

dichotomv

eitherd(R)<oo

r R contains a non- commutatiue f ree mbalgebra

is however known to hold for the following classes of algebras: (") finitely presented monomial algebras, i.e., algebras of the form B : k{Xt, ... ,Xr}lI, where 1 is generated by finitely many monomials in Xr, . . . , X. [A].

(b) group algebrae kG, where G is a finit¿ly generated solvable-by-finitt group lRo, Theorems 4.7 and 4.12], or G is afinitely generated linea¡ group ITi]. (c\ díaision algebras D which are generated, as division fr-algebras, by some polycyclic-by- finite subgroup G < D' [Lol, Theorem 2.3] or by some finiie-dimensional Lie subalgebra C C Dl,¡over/c , where & is algebraicallyclosed, char f :0 [Lol, Lemma3.6], [M].

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Moreover, in each of these cases,

ne actually has d(E)€

[f if d(ft) < oo' Finally, we mention affine pl-algebras and almost commutative algebras (that is, homomorphic ima8es of enveloping algebras of finite-dimensional Lie algebras) as examples of algebras satisfying d(n) < oo . ([Ber], [8"-K.], see also [K-L])' (Z ¿) fug!-ulgg-b[g1 (survey, see a]so (III'2'3))' Let G be a group a¡rd let [G denote the group algebra

f G over ft. Then d(&G) turns out to be equal to the growth

d(G) of G, a notion due to Milnor [Mi 11. It is defined as follows: For ea¿h finite subset E g G, let n@ C G denote the set of all products e1'e2'...'e^ wilh e; €. E and rn ( n, and let dB denote the degree

f growth of the function n '

fff,(")' Then the growth ol G is defined bY d(G): : suP {dsl E a linite sabaet

! G\'

The equality d(kc) : d(G) is trivial to verify, a¡rd rema¡ks analogous to (1.4) can also be made for the numbers dB artd their dependence upon E. In the .r" *h.r. G : ¡t 1M is the fundamental group of a Riemannian manifold M , lhe growth of G is linked to properties of the curvature oÍ M (cf ' [Mi 1]). The following theorem combines results of Gromov [Gro] and Bass [B]. Tlreo¡em (Gromov, Ba.ss). Let G be a finítely generateil group' Then d(G) < e il and only i! G has a nilpotent enbgroup

l finite
inder. In this case,

il N s G ia nilpotent and has fi'nite inder in G, one has d( G) : d(N) :

E,r"rlc(M/N'*')'

whereNl:N,N¡+t:[NJ,N]('>l)isthedcscendíngcentralseriesofN,and"r(N'/¡(*l) denotea the rank ol thc finitely generated

bclian group N;f N;¡.'
t7-

In particula.r, one sees that, for any group algebra kG , d(KG) € [f U {oo},

(z.s) Q¡edsCi!

bras and modules.

Let R : ^?oRo be aZ-graded, &-algebra. Here the .R' a.re &-subspaces

f .& with ,R.'Ro C R^*o.

One easily checks that 1€ Bo, and hence k c Ro. Now let M : ^9o*", Mn.R^ 9 Mo*^, be a graded right R-module. Then each Mo is a i-vector space, and we assume that dirnMo <oo (n€T'). Note that this assumption is automatically satisfied if M is finitely generated over R and R satisfies dim Ro < oo (n e Z). For, if ffi1, . . ,mr € M are homogeneous generators for M

f degrees

dr, . ., d¿, respectively, then Mo :?*rR^-d, , whence Mo is finite-dimensional. Furthermore, dim Rn { oo (n € T,) cerlunly holds in the sta¡rdard situation where .R is affine and positively graded (i.e., ,R, :(0) for n ( 0) with Ro:k. To see this, consider a finite set of homogeneous algebra generators

f ft having positive

degrees. In the above situation, we ca.n define D(M) to be the degree of growth of the function n + t d'im M¡, so

n(¡'(¡

D(M\:lim log, I dim M; ,

n+m líl<o

by Lemma (1.1). Lemrn¿. Let R :,9o Ro be a Z-graded k'algebra, and let, :,@- Mo be a graded R' module with dim Mn 1crc(n € T,). Then d(M) < D(M). Equality holds wheneuer the algebra R ís affine and M is finitely generated as R'module.

Prof. PutM( n)'.=he.Mr CM andfi(n) ::,,.,@=."r

gn. If u cM and vCR are finite-dimensional subspaces, then U g M(c) and V C R(f) for suitable s,¿ € N+' Hence U./") C M(s*ún) a.nd so, putting l(n),: dimM("), we have dim U'lA") 3 /(c+tn)'

SLIDE 12

18-

comparing growrh degrees (t.r) we get du,v < d(f ):D(M). Therefsre, d(M) s D(M). If R is a.ffine and M is finitely generated over R, we may a.ssume lhat U:: M(s) generates M alrrd that V:: R(s) generates the algebra B, for a suitably chosen s € lN+. We will show that u.lA") : M(r),R(s)(') > M(") (n e z). This will imply thx dr,v ) D(M), whence d(M) > D(M). Arguing by induction on n and by symmetry, the problem is reduced to showing that u.,v1") ) Mo (n € [I+). Certainly, Mn C U.ún for some r. Hence each 0 I * e Mo can be written as a sum of nonzero monomials of the form a :ao'ut'...'ac with deg(a) :n a¡rd with homogeneous factors ase. [J, a; e V (i > 1), and g ( r. By shortening the expression, if necessary, we may assume that aovt4 U and a;u;a1

Q. V. As deg(r):n

) 0, at least

ne of the factors u¡

must have positive degree. But then all do, because if deg(o¡) and deg(a¡*1) have opposite sign then

ldes(a¡u¡+) | S *r* {ld"g(u¡)1, )des(r¡*r) l} I t,

a¡rd so aiaj+t € n(s) : V (aivi+t e M(t) : U if i: 0), contradicting our choice. It follows that n : d"C(r¡ : {, a"g(ud) > g * 1, whence u e U'l^c) C (l'1^ü, as required' ft

Finally, we rema¡k that if M is any finitely generated right module over an affine &-algebra R, then there exists a positively graded affine É-algebra,s:"904 with ,56:/c and a finitely generated graded ^9-module N :o@oN" with

r(Mn):d(Ns). Indeed, fix finite-dimensional subspaces v t R a¡rd u I M with R :k[ v] and M :u'R

and define

t9 -

S : erv(n ) ' : $o'v1") ¡1"- rl , N : sru,r(M) , : ,9o,'/ü lU'IÁ"-t) ,

the usual graded algebra a¡rd module associated with the filtrations {l^"]'ln € n\

f R and

{U'l^"1ln e nl '

f

M . Then ,rrondO* N; : dímU'IA"l and so, clearly, D(N) : du,r: d(Mn). Thus, since S is affine, generated by S,, and N is generated as S- module by No, the above lemmayields d(Nr) : D(N) : d(Mn). S 3. Further Properüiea

f GK-dimension

(3.1) Bimodules. Let R and,S be fr-algebras. In dealing with (-8,,9)-bimodules M, we will implicitly assume that the /c-operations

n both sides coincide. Thus (R,,S)-bimodules are

identical with right R"o ? ^5-modules and fib in the framework considered so far. In particular, d(nMt) :d(Mn,o * ") is defined, as well * d(" M) :d(Mn",).

Propciticr. Let R and S be k-algebras and let pMg Ms it finitely generated. Then be an (R,S)-bimodule. Assume that

d(Rf ann¡(M)) : d(*M) < d(Ms): d(oMs) < d(^e). Prof. Put T:R'o?t. Then bhere a.re obvious maps RoP

? and S -

T, a¡d Proposition ( I 6) (d) implies d(Mr) ) max{d(rM), d(Ms)). Moreover, by Proposition ( f .O) (.) and the symmetry

f

d(R) ( 1 .4) , d(R lannp(M)) > d(nM ) and d( Ms) < d(s). Let U 9 M a¡rd V C T be finite-dimensional subspaces. Choose a finite-dimensional subspace G C M with U g G a¡3 M:G.5, and choose finite-dimensional subspaces

SLIDE 13

20-

VrC R'p and V2C S with V a Vt?V2andVrG C G'Vz. Then

U.tÁ") c VÍü .G.v[ü g c.v!z"t

21 -

[G is a Noetherian domain of finite GK-dimension (2.a). But it D : Q(kG) denotes the classical division ring of fractions of kG, then d(D) < m if a¡rd

nly if G is abelian, in which

case d( D) : d(kC) : rank(C) ([Lo1, Corollary 2.4]). Examples of localizations ^RC-1 with d(n) < d(RC-t) < oo ca¡¡ also be constructed ([Lo2, Lemma 2.4 and Example 2.5]). For these reasons, another invaria.nt, now usually called GK-transcendence degree, was introduced by Gelfand and Kirillov in IG-I(i] which, by definition, does not increase under localizations but has other disadvantages. Besides being quite ha¡d to compute in practice, there is no apparent general relationship between the GK-transcendence degrees of a¡r algebra and its subalgebras. Thus this invariant has been much less widely used ühan

rdinary GK-dimension,

a¡rd we will concentrate on the latter in these notes. The following result goes back to Borho a¡rd Kraft [Bo-Kr] and still is the best result asserting the inva¡ia¡rce of GK-dimension under a special type of (Ore-) localization. The localizations considered here include central localizations as a relatively trivial special case. Tlreorem (Borho and Kraft) . Let R be a k-algebra and let C be a multiplicatiuely closed subset

l R

cons'isting

f regular etements
f R .

Assume that C 'is commutatiue and ,is generated by elernents c€R such that the inner deriaatíon adc'.R *R,r+cr--rc, is locally nilpotent. Then: (o) C sat'isfies the left and right Ore conditíons in R. (b) For any right R -module M , d(MC-t rs-r) < d(MC t""_,) : d(Mr). Pror'. (u) . By symmetry, itsuffices to check the d(Mn). Thus, íf M has no C-torsion, then Il ce C andr€R,then right Ore condition, that is: cR a rC *0.

Comparing growth degrees, and so equality must hold. Finally, annp(M) : n r * annp(M) * (r9), gives ( 1.6) (b), this shows that proposition. tr

d(R) : d(¿Re) This proves that d(R): d(eRe). Finally, if I IR n Re:IRe :IeRe :1. Consequently, if R¡

we obtain du,v 1 dc,vr 3 d(Mt). Therefore, d(M7) < d(Ms) annp(g), where g runs through a basis of G, and so the map an embedding

f R f annp(M)

into ¡4ünG. In view of Proposition d(R lannp(M)) < d(rM), thereby completing the proof

f the

We rema¡k that the equality d(Ms): d(nMs), applied to M: R -- S, shows in particular that d(F) : d(nRn) Another application is given in the following corolla^I'y which has also been observed by S. Montgomery. C-o¡olla¡y. Let e : e2 be a nonzero idempotent ol the k-algebra R. f Re ú finítely generateil as ri,ght eRe-module, then d(eRe) :d(R f l.ann¡(n')). This holds in particular tf R is ríght Noetherian. Prof. PutR :: Rfl.ann¡(Rr). Since ¿Re ) l.annp(nt) : (0), we ha'te eRe C rR and so d(eRe) < d(R) is clea.r. Conversely, by the proposition applied to ¡Re.¿., we have

< d(eRe).

Re is an efie-submodule

f Re,

then Noetherian then so is Re"¡r. E

q

is

(3.2) Localization. Gl(-dimension does not behave well with respect to localization. For example, if /cG is the group algebra of a finitely generated torsion-free nilpotent group G, then

Letting \, (resp. p ") (ad c)t : (\. - p,)' : denote left (resp. right)

t /t\

,gt,J\i-'(- 1)'pi in End(R).

multiplication with c,

ne

has Hence

SLIDE 14

rr-

It (ad r)'(") : 0, then we obtain 0: (

L)trct + cr' with r' :

:ij(il- t)icü-í- trcí e R. Thus {ci li > 0} satisfies the Ore condition il ad. c is locally nilpotent. Now, a general c e C has the form c - c;'...'c, with c; € C and od c; localiy

nilpotent. Arguing by induction on E,

we pirt s':¿¡'...'ct-r a¡¡d assume that c'r': rd' for suitable r' e. R, d'e C. As cr-tÍtt : r'd" for suitable r" e R, d" e C, we conclude that

¿¡'t : ctcr-1r't : cirtd't : rdtd" : ril with d : d'd" e C, as required. (b). After replacing M by its canonical image in MC- r, we may assume thaL M I UC-'. Now let U C MC-l and V C RC-l be finite-dimensional subspaces. Choose a "common denominator" c€ Csothat Uc C M andVc I fi. Write c: c.'...'crwibh c¡€ C andadc¡ locally nilpotent

n R , and let D denote the multiplicatively

closed subset of C generated by cr, ., ce-

r. By part (a) , D is a left and right Ore subset of R and, since C is commutative,

ad c, yields a locally nilpotent derivation on ,S: : RD- l. Arguing inductively, we may assume rhar d(Mp) : d(Nr), where .A/:: MD-t, and we have to show that d(¡/r) : d(NE tr, ,) with E: {tilt > 0}. Since U g NE- I and V C SE-t, this will prove du,, 3 d(Ns) : d(M^), *d hence Part (b)' In other words, we axe reduced üo dealing with the case where C :{t'li > 0} and ¿d c is locally nilpotent

n

R. Start afresh with finite-dimensional subspaces U C MC- I and V C RC- 1, and choose ú with Ut'.:Uct C M and Vct C R. Since ad c is locally nilpotent, there exists an ad c-stable finite-dimensional subspace W g R with c,ct e W and Vct C W. Pick m large enough so that (ad c)^(W): (0). For a given u e W, Put ui: (ad c)i(w) e W so that ui+t-- cwi - w¡c. Multiplying

n both

sides with c-r, we

btain

(ad c)'(,) :,i(Nt-tf ;ct-;r c; (r € R).

23-

c-rwr: ü¡c-l

c-rw;¡1c-1.

Itcrating this, stalting with i :0, and using the fact that w^: 0 we get c-tu

utc-| -

lrtc-2 * w2c- !. w^-tc-^ . Thus c- 1W C Wc-L + Wc-2 + ... *Wc-'and, more generally,

c-'W 9 Wc-'+ Wc-'-t +... I Wc-'-^ (s € N). It follows that ,-,rwr-trw.....r-r,W C Wrc-, + Wrc_r_l + ... +Wrc_r_*, where r :El * ... * s,. Therefore tJ.l,A") C (J1c t . (Wc-r)(o) :

Ur. (r-t14t¡ft) . r-t

C (Jr'(Wt"lc-* + 14t(ür-*-' + ... + Wb)r-nt-tm¡ ,-t

UL.(Wt"lc'

+ yy(n)rnm- t + ... + W(')) ,-(t+^)n-t C (Jt. yry(@+t)") c-(+m)n-t. consequently, ttim (J'Á^) a dim u:.1,y((m+r)ü and so du,v 1dur,* 1d(Mil part (b) follows from this. tr (3.3) Filtrations. Let ¡9 be a &-algebra with an increasing filtrarion F : {nlüln e Z } bv e- subspaces ¡(n). Thus I 6 ¡(o), ¿(") c ¡(n+t), p(").p(^) C pb+^), md ,? : g R('). Furthermore,IeL M be a right ¡?-module with filtration G : {M0)ln € Z }, where the M(") a¡e i-subspaces with ¡,7G\ c ¡4(n+rl, ¡4b) 'p(.) q ¡|(n+m), and M : L) Ut"'1. As usual, we let

sr(R) : err(A), : 6P(") ¡P("-r\

and

SLIDE 15

24-

cr(M) : crc(M): : OM(¡) lh¡{"-t¡

denote the associated graded algebra and the associated graded module

ver gr(r?).

For a.ny subspace V C R, we put

cr(v),- g( v. R''l)l(va pb- ')) g c"(a),

a¡rd simila¡ly for subspaces

f M . Note that, clea,rly

, dim V ) dím Cr(V). Lemrna., (notations as above). 1a/ d(gr(M )c,(n)) < d(Mn). (b) Assume that atl Ml") are finite-dimensional

uer le and that )

Irtb): (0) . Assume further that gr(R) is affine antl that cr(M) is finitely generateil aa gr(R)-"*odulr. Then d(gr(M)c,(n)) : d(Mr) : drgree

f growth of n -

¡lfp l4(n). Prof. (t) Let V C gr(R)

d

U C gr(M ) be finite-dimensional subspaces. We can choose finite-dimensional subspaces Vt g R and U1 C M with V c gr(Vl) and U c gr(U ¡). Then U.úü c cr(Ur) . cr(V,)(') 9 cr(U) . sr(V{ü) C cr(U t'Vl"t ¡. Hence, tlim U.14") a dim U1.Vf') undso dy,v 1du,vr1d(Mp). Thisproves(a). (b) By Lemma (2.5) , d(gr(M) s,(n) equals ihe degree of growth of the function n - ,.P. dim M("\lMb- t) ::p(n). Our assumptions

n {M(")} imply that Mb\:

(0) for

li l( ¡, small n . So we have p(n) : dim Ml") for all sufficiently large n, a¡rd hence d(sr(M)c"(n)) : d(dim Mt")) ::6(M). Now let U g M and V C R be finite-dimensional subspaces. Then, for some t, U C M{t) and V e R(,), a¡rd so

25-

U.tl") g UO. ¡(nt) C ¡4\.n+t) Taking growth degrees, we obtain du,v 16(M), and hence d(Ma) < 6(M). In view of (a), this completes the proof of (b). D

SLIDE 16

Chapter II: FACTORS

26-

MODULO NILPOTENT IDEALS $ 1. The Problem

(l.l)

Bxactness. Let R be a t-algebra

and let C denote a class of right R-modules. If, for all short exact sequences O - Mt- M - M"

0 with

M,M', and M't belonging to C, the equality d(M)=: m¡x {d(M'),d(M")} is satisfied, then one says that GK-dimension ís exact for the modules in C. Unfortunately, and in strong contrast with (Gabriel-Rentschler-) Krull dimension, exactness usually fails for the class of all R-modules (see (t.2) below) . The following lemma is implicit in IL]. Lem¡ne. Consider the following conditions

For all ideals I of R and all t ) r, d(R lI) : d(RlI'). Forallídealsl andJ olR, d(RIIJ): max {d(RlI),d(RlJ)). GK-dímension is eract for all R -modules.

Cne has

(o) (z) :> (2) <:> (1) (b) If R is risht FBN (1.2.2), then (1), (2), and (3) are equiualent

Prof. (a). If GK-dimension is exact, then

d(R lIt): max {d(Ii lIi+\) lr:0,...,t*t } S d(R lI), because "u"6 ¡i ¡¡i+l is a module over R f

I. Since d(

R lI) < d(R llt) is clear, we have (3) :> (t). The implication (2) :> (1) follows by induction on ú. For (1) :> (2), note that (t ¡ l)2 9lJ 9l ) J andso

(1) (2)

(s)

27 -

d(Rl(I n 42) > d(RIIJ) ¿ d(RlI n /) : max {d(a lI),d(RlJ)) where the latter equality holds by Corolla.ry (I.t.Z) (c). If (1) holds, then all ) must be equalities.

(b). Since.R is right Noetherian, exadtness for al finitely generated E-modules clearly implies exactness for all -R-modules. Moreover, by Lemma(1.2.2), d(M): d(Rlannp(M)) holds forallfinitelygenerated.R-modules M #(O). Now, if 0--* Mt-M +l14tt *0isa¡r exact sequence

f (nonzero) right B-modules, then cnn¡(U"r'annp(M')

C ann¡(M), and so (2) implies that l(M) : d(R lannp(M)) S d(R lannp(M")'annp(M'))

mrx {d(M'),d(M")).

By Proposition (I.f .0) (a), equality must hold here. tr

(1.2) Counterexamples. By (1.1), exactness of GK-dimension for all l?modules surely implies that

d(R) : d(R lI) lor all n'ilpotent ideala I oÍ R' The simplest counterexample to this equality is obtained by taking R:Rt::,t{X, Y)l. v|)' Here, d(R,) : t . Indeed, putting V::<X,Y>r C Rr, a basis for a complement of V{o-l) in trl") is provided by all formally disünct monomials in X and Y having length n and containing at most t- | Y-

factors. There *.

(?)such monomials containing exactly f Y-fáctors, and hence

ttim Áü : dim IAo-,\. ,1(i)

Inductively, it follows that ttim V") is a polynomial of degree t in n, which implies that d(R) -

t. Now, if ,I denotes

the ideal of -R¡ generated by the image of f, then.I is nilpotent

SLIDE 17

28-

and R lI = k tX] : R1 has GK-dimension ¡. More elaborate examples have been constructed by G.M. Bergma^n. In [Berg 2] , he consf,ructs (t) an affine Pl-algebra R having a¡r ideal 1 of square (0) such that I is cyclic as right ideal, i.e., 1 :r¡9 for some r € R, but d(n) : S, d(R lI) : 2. (b) an affine algebra R having an ideal / of square (0) which is Noetherian (and cyclic) as left and right,R-module, but d(E) : a, d(R lI) < 6. (f .f) Our main goal in this chapter will be to discuss conditions on the nilpotent ideal 1 of .R, a¡rd

n the fa¡tor fr/1, which ensure that the equality d(E) : d(R lI) holds. We will also be

interested in the question when equality holds in the following Lemm¿. Let I be an ideal

l the

k -algebra

R. Then

d(R) > max {d( R lI), d(nIn)\. Proof. The exact sequence O

I. - R + R lI -0
f

(,R,.R)-bimodules gives d(aRn) > max{d(*(n /I)R), d(nIn)}. By Propostion (L3.1) , d(RRR) :d(R) and

d(a@ lr)a) :d(R lr). tr

Equality surely holds in the lemma whenever .I contains a (right or left) regular element of R. For, in this case, B embeds into 1 as (right

r left) ¡R-submodule.

Equality also holds when f2: (0) and B : I@ S for some subalgebra S of R (see [Lo 3, Example Z.+(a)]). In general, however, the above inequality will be strict. As an explicit example, take

R: k{x,Y)l(Y)n: klr,vl and let 1: (v)'c

R. Then d(n):

4, d(RlI):2, by (1.2), but d( nIn) :3 (see [Lo 3, Example 2.4(b)]).

29-

$ 2. Strongly Finitely Prcaentcd Algebras

(2.1) Definition qI ¡!f" Slightly extending the terminology introduced in [Lo 3], we will say that É-algebra R ís atrongly finitely pretented if R ha^s a¡r increasing filtration (0) : ¡(-t) c ¡:¡(o) 9... C B(') g ¿(n+t) g...9 R: U ¿(n) such that gr(R) : O pft\ ¡p("-t) in I;nit"ty presenteil.' The algebra ft will be called locolly strongly finitely presented if every affine subalgebra

f R is

contained in a strongly finite ly presented subalgebra

f R.

(2.2\ Part (b) of the following technical lemma shows in pa^rticular that strongly finitely presented algebras are indeed finitely presented.

Lemma. Let R be a k -algebra

with a fltration

(0) : ¿(-t) c r: R(0) c... c ¿(o) c... g n: U B(o).

(o) If cr(R) 'is

affine, then there eriety; e R and d; € [{+ (i:l,...,u) such that, lor all n, ¡(n) : 1!¡l!¡2.....!;, I I < i l r, r."*d;, 1 n)*. In particular, il V:: ( at, .,!o)h C R, thenR(") -C l"\ for oll n, anil so -R : lc[iz] m affine. (b) Assumethatgr(R) ú f.nitelyprescnted, andletó:S

R

beanepimorphismwithS an affi.ne k-algebra, cay S :k [Wl Pur I:: Keró. Then there ecüt s,ú € ['l+ euch that, for all n,

I.W$)t t W(i) .(/n Ytr,)¡.1,Y(i).

i+i< ü't¡

In parücular, I ís a finitely generated ideal of S.

SLIDE 18

30-

hoof. (a) . By assumption, gr(R) has finitely ma,ny algebra generators 11, ...,xowhich may be taken to be homogeneous

f

positive degree, say deg(t¡): d; € N+. Write x;: li + R(d'- l) with y; € ,?(d') ' we claim that.R(o) is spa^nned by all monomials yi,.y;r.....a;, with É d¡. 1 n' Clearly, these monomials belong b R(o), and the assertion is true for n :0.

j:'J

(The

empty monomial, by convention, represents 1.) If g € R(o) , yS.RG- t) , then r i -! * Rb-t) e cr(R)o ca¡r be written as a linear combination

f monomials
f the form

x;r'x;r'...'x¡, *itn É ai:n. Therefore, modulo R(n-l), y is equal to the corresponding linear combination

f monomials

in the yi',e. By induction, we conclude that y has the required form. (b). Keep the above notation and fix elements a; e S with,/(

;):

V; (í:1,...,a). Then ^9: So *.I, where So: k'¡a1,...,ao). Thus we can pick elements co*1, . . . ,ew € 1 with l/ S so -f (d¡+r, ,arlh. Put d;:1 for u*l ( i 1w and filter ^9 by defining s(") to

be the l-linear span of all monomials a;r.o¡r.....a;, with d(Sf "t¡ : ¡(n) and so

I

D d, 4 n.

Then, clea.rly,

cr(R) = sr(S)lsr(I), where gr(I):: @ (1n s('))/(/ n ir('-t)). since cr(s) is affine, with senerators ,, * 1to'-t) ( i:L,...,w), and gr(.R) is finitely generated. Therefore, there exists p € lN+ with 1n ^9(') c X Sttl .(/ n g(r)¡ . g(i) (n € [r).

i+ j1n

(Look at homogeneous generators for gr(.I), or see [N-V.O., Proposition IV,3, p.2g+].) Finally, for suitable Q,r € N+, we have W I 5{t) and o¡ e WG) (i:1,...,u). Therefore, W(") g g(r")

¿

g(o) c W(-), and hence

InWl") C/ng(en) C D W{,i).UnW(n)¡.WGi)

í+j1qn

g X 7,Y$)'(/nw(')¡ 'w(i),

d* j( ta

with f - rg a,nd s :

rp. This finishes the proof.

tr

3l -

(2.3) I know no general method to decide whether or not a given finitely presented algebra R is strongly finitely presented. Moreover, even if R is a priori known to be strongly finitely presented, say by Lemma (Z.e) below, ühe filtration {ft(')} must be ca^refully chosen in order to ensure that 9r(R) is finitely presented. (See Example (fV.2.2).) (2.4) Examples

f strongly finitely presented

algebras. (t) All finítely presented positívely graded algebras " :"9, Rn with Eo : ft are obviously strongly finitely presented. Indeed, filtering fi by ¡(') : : @ R- we have gr(R) = R

:'"

(b) Almost commutatiae and aomewhat commutotiue algebras. A &-algebra rR is almost commutative, by defin,tion, if R has a finite-dimensional generating subspace Y such that

grv(R) ::@l')1'v1"-r\

n

is commutative. Since Crv(R) h^ algebra generators u+Il0) E flr)¡l(o), where u runs through a basis

f V, gry(.R) is afEne

and commutative, a¡¡d hence is finitely presented. Thus Lhe V-fittration R,") : Á"1 has the desired properties. McConnell a¡rd Robson fMc-Rl have introduced a wider class of algebras which they call somewhat commutative al6ebras. By definition, an algebra R is somewhat commutatiae if R has a filtration.

(0) : R(-t) c fr: R(o) q...g.R(") g ¿(n+t) c... c E: ¡ R(") such that Cr(R) :@ R(")lR(o-l) ¿'s af line commutaüue.

Of course, such algebras a¡e also strongly finitely presented. (c) Affine algebras R that are finitely generated modules

uer their centers

are sirongly finitely

presented. For, by the Artin-Tate Lemma, the center of R is also affine in this case, and so

the assertion is a consequence

f the following slightly more general

Lemm¡. Assume that R: i Sr, lor some subalgebra S having a finite-dirnensional generating

í:l

SLIDE 19

space w w;th

1U¡W, !p!qlall j,p,q)¡ g frWr,

t'{

crw(S) : @1'y'") ¡ry("-t) kfi Noetherian anil linítety presenteit ThenV:: (W,!t, ,at}¡ leneratesR, anilcrv(R) isleftNoetherianandfinitelypresented. h'oof' Clea.rly, V generates R and gry(R) is generated by the ,,leading terms,, $ 1- w + I^o) (w e W) and gd:: yi + l0). Since ty(n) C llo), there is a¡r obvious map

f gr¡a,(^9)
nto the subalsebra T, : g (Wft¡ +V1"-t))/Á"-r)

: f [W] of Crv(R). Since crw(S) is left Noetherian, the kernel of this map is finitely generated, a¡rd so ? is finitelv presented a¡rd left Noetherian, us grw(,s) is. Moreover, by assumption on w.,

<l¡W,TrW Iatl i,p,q)* C t -*n

l:l

holds in Crv(R ). It follows thar

crv(R):r*ft6.

d:l

Thus grv(A) is finitely generated, and hence finitely presented a.s left ?-module. Therefore, c'v(R ) is left Noetherian, and a finite algebra presentation

f crv(R) also follows easily.

tr For some explicit applications

f this Iemma, see

(rv.2.1) a'd (IV.2.2). (2.5) Propeition. LetR beak-algebraandletl beanideatof.R. SupposethatRll isafinitety generated ríght tnodule

aer

sorne subalgebra which is locally strongly finitety presented. Ihen

d(R) : max {d(fi lI), d(Rlil\. If, in addition, I 'ia finitely generated as right ideal

f

R then, lor all t ) I,

33-

d(R) : max {d(fi lI), d(Ih)}. In particular, il I ís also nilpotent, then

d(n) : d(R lr).

Prof. In ea¡h case, it suffices to show that d(R) is bounded above by the right hand side. After replacing r? by a suitable subalgebra containing .f , we may assume that R /I is locally strongly ñnitely presented (Corollary (I.1.7)(a)). Letg : R - Rf I d,enotn the canonical map. Ler V C R be a finite-dimensional subspace. By assumption, /( /) is contained in a strongly finitely presented subalgebra S of R lI. Choose an affine subalgebra ?" of R with v c T and/(") : S andwrite T-- kfW] with ]l¡finite-dimensional, V c W. ByLemma (2.2)(b), there exist e ,t € lN+ such that

(*) I.I^") gIn 1,yG)c D ry(;).(Inw@¡.yr(i) (n€N).

i+iStn

the notation li,:WEI+f EWgT"p$^T a¡rd viewing IaT as right ?-module, we can rewrite the last expression ," (/ n wk)) . 14tun). Therefore, ttirn Á") : dim ó(vfa * dim I ¡ ,v1") 3 dim ó(v¡ta i dirn (r n ñA1 . 1ryUn) Since the growth degrees of the last two terms, as functions of n, a.re bounded by d(.9) < d(R lI) and d(r(I a r)r) < d(nln), respectively, w€ conclude thar d(R) < max{d( R lI), d(nIn)}. Thus, in view of Lemma (1.3), equaliry musr hold. Now, keeping the above notations, assume that 1 : G ' R holds for some finite-dimensional subspace G of R with G f 1n I4z('). Fix t > l. Choose afihite-dimensional subspace Wl of B with V + W 9 W, and llz.G g G.Wy Then (*) above yields I n /") C G.1,y¡t") (n e N). Hence, putting Vr:WIt), we obtain

Using TOP E t

SLIDE 20

34-

I n V1") C G.yf ") (n € N). Continuing in this f ashion, we can successively dete rmine finite-dimensional subspaces V:Vo C Yr C ... 9 V,

f R such

rhat

I n v!') c C'V,.9 (n € [I)'

For each n, fix subspaces

{.,o

(r:O,1,...,ú) of %(") *ith (f n Vrf ')) @X.., :y.("). Then and so rA") c Xo,o + G' xr,o+ ... + Gr-|. xr_t.o I Gt . v,("\. Since d¿rn X,o : dim g(%)(") < diln ó(V)G), we conclude rhar dim Á") 1 s . dim S(V,)b) * dim F.Vr("), where gt:l+dim G+...*dimGt-r and F::G'gI'. This proves that d(fr) < moc {d( R /I), d(rh)\, thereby finishins ühe proof of the proposirion. D $ 3. Application to Noett¡erian Pl-algebras (3.1) Theorem. Let R be a Noetherian Pl-algebra. Then (a) (Lorenz and Small [Lo-Sm]). d(R): d(R/P) lor some prime ideat p of R. In particular, d(R) € NU {oo}. (ü/ (Lenagan [L]). GK-dimension is ecact for att R -modules . Prof. (u) It suffices to show that d(n) S d(RlP) for some prime ideat P of R. Afrer replacing R by a suitable homomorphic image, we may assume Lhat d(R lI) t d(fi) holds for all nonzero ideals I of R. By Corollary (I.f .7)(.), R is irreducible, i.e., any bwo nonzero ideals

f ,B intersect non-trivially. By [Go l, Corollary 2.4) and [Go 2, Theorem L], R has a¡r

Artinian classical ring of fractions, A : a(R). Let N denote the nilpotent radical of ¡R . Then NQ is the radical of Q, and letting-: Q - QINQ denote the canonical map, we have

d: q@) : é. e(RtP;),

' i:l

where Pr, . .. ,P, a"" the minimal primes of R. Furthermore, g is a finitely generated module

ver its center' because each Q( R lP;) is a simple Pl-algebra, and hence is finite-dimensional
ver its center, by Kaplansky's Theorem.

Since commutative algebras a,re clearly locally strongly finitely presented, Proposition (2.b) implies that d(Q) : d(q) But d(O) : ma"r {d( a(RlP))}, by Corollary (Lt.z)(c), a.nd d(e(Rlp)): d(Rlp;) , by Theorem (I.3.2), because each Q( R lP;) is a central localization

f R /p¡. Therefore,

d(,q) < d(Q) : max {d( e(R lp;))}: *e" {d(R lp)) , whence equality must hold. Finally, since each Q@ lP;) is a finitely generated module over its center, we also see that d(R) € [tU {oo}. , (b) By Lemma (1.f ), it suffices to show thaü, for all ideals I of R and all ¿ € lñ+, we have d(RlI): d(RlI'). Apply part, (a) tD Rllt b obrain a prime ideal p or a *,,n p ) /, and d(RlP):d(Rllt). clearly, P ) I ) It and so d(R/p) < d(R/r) < d(R/It). Thererore, equality holds throughout, and (b) follows. tr (3.2) It would be interesting to extend Theorem (3.1) to right Noetherian pl-algebras .R. What is needed here is rhe equaliry d(F) : d(R/p)

f parr (a). Of course,

rhis would follow immediately from Proposition (2.5) if semiprime right Noetherian PI-algebras were known to be locally strongly finitnly presented, at least up to a finite module extension.

SLIDE 21

36-

Chapter III: ON GRADED ALGEBRAS AND MODITLES AND f¡f nfn POINCARE SERIES

$ 1. Rational Power Series (1.1) We will be concerned with integral power series

f the form

P(r) : Ddot" ezlftll.

¡)o Later on, do will usually be the dimension of the n-th homogeneous component of some graded alsebra

r module. The rirts Zllt)l

is a subring "f q(ú)) : qtt¿]llú-Il, the field of Laurent por.ver series over Q , and the latter contains the field I ú) of rational functions over Q. The power series P(t) is called rational, if P( ú) € q ú). The above notations P(t) and do will be kept throughout this paragraph. (1.2) Criteria for rationality. The following Iemma is classical. Lemma- Equíaalent are: (ü P(t) ís rational; (iü P(t) : Í(t)lg(t) with l(t\,cU) e T,lt) and c(0) : 1; (iii) There eviata a linear reeurrence relation do : atdo-t * azdo-z+ . . . I ardr-, for euítable a; e Z whích is aalül for n large enough. (ia) There ex,íst polynomials p¡(r) e -Q["] (Q: algebraíc closure

f Q) and algebraic

integera a¡ (i:l,...,s) auch that, lor all sufficiently large n,

37 -

d, : D n¡fu)ai.

hoof. (i) :>

(ii). See [P-Sz, #156,p.Laz] (ii) :> (iii). Write C(t) : ,\O,t'with g; e'2, to:

1. Then comparing coefficients
f

fn in s(¿)'p( t) : f(t) gives ,ior,O,-i : 0 for ¿ ) drc(f). Take ai:

s; (i:l,...,r).

(ilrl__=>_ÍlL Suppose that do : atdo_1 *... I a,do_, holds for all n ) N and pur g(ú) : l- clt - ... - e,t' . Then the power series | (t) ,:g(r) . P(t) has vanishing coefficients in desrees ) N, and hence f (t) e e[r] and p(t):Í(t)lc(r) e er). (ii) :> (iv). Write g(¿) :(1-o,ú)'r.....(l-o,ü)" with algebraic integers o¡ (roors of t'- al'- at: trg1lt) ) and e1*...te": drC(C). We may express the rational function P(t) : l(t)lC(ú) in terms of partial fractions,

I

d

{

1

P(t) : Éi ^-e-.-,,

;:ri:r ( r- d¡t J' where the p¡.to are appropriate consta¡rts € -Q. Using the formula

(*)

--I-:

t''fntttl).o".¿o \ / (l-at¡i

fto\ i-t

, "

we obtain

P(t):fÉryn)ofto,

¿ )0d:l

where we have set p;( n) - P:rr,,("f:¡t) , . polynomial in n of degree at most e;- 1, with coefficients ( Q . (iv) :> (i). Assume that, for suitable polynomials pi(t) e Q[c] and constants a¡ € Q we have

SLIDE 22

38-

d,:Dp|rn)oi (, >N).

'-t

J-t

We will show that P(t) : ,Dr_odoro belongs to ü¿) This will imply that P(ú) € (¿) n zlltll c qü), as required. After adding a suitable polynomial ta P(t\, we may assume that the above equality holds for all ¿ 2 0. Writins P(t):f,,.e,@,t) with P¡( t): P^p¡(n)t" € -Attúll, itsuffices to show

i4

'

¡)0

that earh P¡( ú) belongs t" -q ü). Thus we may a^ssume that P(t) : ,\on(") ú' for some polynomial p(X):p,x'*...*po. If r:0,then P(t):e, +isrational. If r )0then, using formula (*) above, we get

P(t)

''o'(.h- : ,e

q(n)t",

where q(c) is a polynomial of degree ( r. The result now follows by induction on r. fl (1.3) Lemma- Let P(r):,\_od"f enlltll be giaen, with o < dn < dn¡1 for all sufficiently Iarge

n. Suppose

lurther that for some polynomial there ec';st functions

P(ú) : p(t)1rt0-f*')

i:1

p( ü) € Zltl and positiae integers 6,t r, ,kr. Put k : l.c.m. {k;}. Then c¡ : Z, lkZ + Q (l :0,1,...,9) with co constant ) O , such that

d,: f, c¡(n+kZ)nl

¡={

holds for n large enough. The integer g is the degree

f growth
f {d") , and
39-

C + l:

rder
f the

pole

Í P(t) at t:

l.

Prof. As in Lemma (1.2), proof of (ii) :> (iv), we ca¡r write d^ : i.p¡(n)ai (n large enough)

J:

for suitable polynomials pi(t) and algebraic integers a¡ which, in the present situation, are roots of unity. Indeed, o,lt :l and so, if * : : l.c.m. {k¡), then each or? depends only on the residue class nlkZ. Rearranging the above expression for do according to powers of n, we

bcain a new expression

d^ : fr,(n+kL,)nt (n large enough)

where c¡(n*kZ) e Q, since d, € Q. Since do+t ) dn ) 0 holds for all sufficiently lalge n, the highest

ccuring

coefficient function co :Z lkZ,

Q must

be constant > 0. For,

therwise we could pick positive integers n {

m with co(m*kZ ) < ro(n*kZ ). Replacing n a¡rd rn by n*kz,

resp. m*kz,

for a large enough z, this would imply d^ {

do. Clearly, g is

the degree of growth of {d,}. It remains to show that 9 : s- d-I, where tl is the order ol aaníshing of p(t) at ú: 1. For this, we use an idea of K. Bongartz. One has where q (ú) :

c(t)'P(,) :"lo

t

II.(1+ú+...+r Dot" e nllt))

¡

h-r

"' ') : t c-úr with h

' j=O h

: X&;-s

and a; € Iñ+. Write with Do :,1ordn-;. For large enough n, we have dn 1 Do ( ñ'ma>< {o;}',1".

So g is also the degree of growth of {D,},

d

hence we can assume that all &i: l. Writing p(t) : (l- ú) n p,,(t), we can further assume that

SLIDE 23

40-

P(^: pr(t). with ( 1- ü)'- o pr(¿):ó6+ ó,(l-¿) +... + ó.(1-ú)", bo:pt(L) *o. Using formula (*) i.t Lemma (1.2), proof of (ii) :; (iv), we obtain

Prú) : i--1-r=r : É,, "!(';l; !;:1t7*,

Ía0- t)'-

and so

,. : á0,(";:;!;: L') : Tr# ,n nc- d-, * do,,

a polynomial in n of degree I a- d-I. This shows that {d,} has growth degree where do' s- d- l.

(1.4) Growth behavior. The power series P(t) is called algebraic if P(¿), viewed as an element

f the field Q(t)),

is algebraic

ver the subñeld Qú). The following lemmapa.rtly explains

the interest of rationality, or algebraicity, results in connection with Poinca¡e series of graded

algebras. Not¿ that the a.ssumptions
n {d"} are satisfied, for example, if do:

dím flo) fot some finite-dimensional subspace V of a &-algebra. (See also (2.1) below.)

Lemma. Asaumc

that P(r) :,lod" f e n Ilt]] * algebraíc and that 0 ( do 1 dn+t and d¡o ( df*" hotrls lor all sufficientty large n anil lc, where c € lñ is fired. Pat r :: (lim- d,:l\-t (th" rad,ius

l conaergence
l P(t)).

Then r 1 | and

If r {l,thenthereecistsaconstantC

> | suchthatdo2Co holdsloralllargeenoughn ( "erponential growth ").

Il r:

L, then P(t) ;a rationol

l the

formP(ü) : p(t)|fr.(r- r*') , where e(t) e Z,ltl and t,krr . . ,l, € 1N+. (Thus the do are deacribed by Lemma (1'9): "polynomial growth".) is EI

47-

Prof. Since Ddo diverges, we clearly have r ( l. Case 1: r (

1. Following

[Mi 1] , we show that, for some constant C ) L, do ) Co holds for all sufficientJy large n. Fix n¿ (large enough so that our assumptions

n {d,}

apply) . Consideralargeenough n a¡rdput/c: i3] + l,where ["] : max{z eZlz 1x}. Then 'm'

and so d:/" S large enough,

1 do

I d*^ 1 dh*, a d#u:,t, dlli+t¡". This shows that lim ¿]/" S'jrr A¡/i*t/": d|ry, and hence, for M

lim dnrlo < ;nf {d}ti¡ S tím nl d}/i : tim in! d}t^.

n+m m>M m+m m+ó

Therefore, thesequence {a}/'¡converges to !t l andsothere exists C ) I with d}/'> C r for all sufficiently large n, as we have claimed. CgS_2._t_=_1r. This part follows [C, proof of Theorem 8.5]. By [P-Sz , #167, p.143], r: t implies rhat, P(ú) is rarional. Write P(r) : f (t)lg(ü) wiü Í(t),s(t) e T,ft) a¡rd 9(0) : 1, as in Lemma (1.2). If g. denotes the leading coefficients

f 9(ú), then we have

s(t) : c^(t-a,)( ú- o,).....( t- o,)(ú- 4,)( t- pt)-...-(r-)), where oL,ltt. ,o.d" axe the non-real roots and pt, .,pr ñe the real roots of g(f). Thus, since 9(0) : 1 and g^ € Z,

r >

+:

lo rl, ..... lo,1, . lprl..... lp,l.

lg^ | As r: l, we have lo,l,lp¡l > r andso la,l: :lo"l:lprl:...:lp,l:r.Thusthe

inverses of the roots of g(t) are algebraic integers all of whose conjugates

ver Qhave absolute

value 1. By a theorem of Kronecker (cf. [I-R], p.215]), they axe roots of unity, and hence so a¡e the roots of g(t). The assertion follows from this. D

SLIDE 24

42-

$ 2. Poinca,re'series (z.t) P"r".*"""rl"r

t er

Let -R : graded ,t-algebra, where the Ro a¡e finite-dimensional [-subspaces of R In this ca.se, \¡r'e ca¡r define the Poincaré (or Hübert) serúea PnQ) of ,R by

'$o"o

" u

with R- 'Ro

positively C R^*o'

Pn(t) : : I dim Ro. t" e zllt)\.

n)0

Lernrne. Let R be as aboae. Assume,

in additíon, that R 'is affine and that Pn(t) is algebraic. Then d(R) e NU {o"}. In case d(R) € N, Pn(t\ ís rational

f the

fonn

PnU) : f (t) /.n (t- r*')

foraome Í(t) e Zltl and s,&r, ,k, € IN+, and d(R):

rder ol the pole of Pn(t) at ú:

l. Prof. Since R is affine, it follows from Lemma (I.2.5) that d(n) : D(R), the degree

f

growth of the function n - do: -- D,odim R-. We are going to apply Lemma (1.4) with P{t)::Ddoún. Note¿hat

r)0

P(ú) : PnU)(1 +ú + t2 +...) : Pn(t)lQ- t),

so that P( ü) is algebraic. Moreover, we clearly have O I d, S do+r for all n € lN A(n) : :P,"- (:0 for t¿ < 0), so thai do:dim n(n). For some c, R(c) generates /c-algebra. We claim that, for all m and n,

Put r? as

R(n*ml q in (m+i) ' R (n- í').

r{

Arguing by induction on rn, the case m:O being clear, the problem is reduced to showing that

d- I

Ro+' C -DoP,n*r.fto-,. holds for all m a¡rd n. To prove this, fix homogeneous algebra generators x; (i:1 ,...,s)

f

R with deg(r;) 3 c. Lel ¡.t: Í;r'x;r""'ü;u be a nonzero

43-

monomial with d"g(tt) : n+*, and pick u 1 a minimal such that the initial segment u:t¡r't;"'...'ri satisfies deg(v) ) m. Then, elearly,

m l deg(u) < m+d,eg(ri)-f 1m*c- l, a¡rd F € R¿,01u)' Ro*^- des(). This proves ühe above inclusion. It follows that R(n+m) C n(ni-c)'R( m), and hence do*^ 1 d,+"'d^ holds for all rn a¡rd n.- Consequently, dxo 1di*., and so Lemma (1.a) applies. We conclude that either do ) C" for some constant C > | a¡rd all sufficiently lar:ge n, in which case clearly D(R): €,

r P(ú) has the form

P(t) : Í (t) I i1.(1- ¿*,) ,

a¡rd D(R) +l:

rder of the pole oÍ P(t) at t:

t, by Lemma (1.3). In the latter case, Pn(t): P(r).(t- t) also has the required form, and D(n) equals the order of the pole of Pn(t) at t: L This complebes the proof. tr Various examples of rational Poinca¡J series Pn(t) will be discussed

below. Here we just

mention the following facts:

(u) Govorov [Gov] has shown that Pp(ü) is rational for any finitely presented monomial algebra (cf. (I.z.r)( a)). (b) There exist fini0ely presented positively graded algebras .R rational ISh]. (.) Formanek lF] has determined a rational expression for the gradation) of the trace ring T^,o of nr generic n Xn-matrices. The interested reader is referred to tRl for further informations guide to the literature.

such ühat Pn(t) is not Poincard series (in multi-

n Poinca¡J series a¡rd a

SLIDE 25

44-

(2.2) Application to filtered alcebras. Let R be a,t-algebra with an increasing fiitration F:{n(") ln e T,\ be finite-dimensional fr-subspaces .R(") such that .R : [J R('t and ¿{') : (0) (n < 0). Assume further that S :: grr(B) is affine. fhese *.r,lprion, *" clearly satisfied in the important special case where R : k IIz] is affine and is endowed with the V_filtration .R(o) : ty'").

In the above situation, Lemma (L3.3)(b) yields d(n) : d(S). The Poinca¡J series

f 5

will be denoted by P¡(ü) (or Pv(t) if F is the V-filtrationl?(o) : I/(

)¡,

i.e., P¡(t):: Pr,¡n1(ú) : D dím(nlü¡n(r-t)¡¿n ezllt)1.

n)0

Suppose that, for some reason, Pr( ú) is known to be rational,

r at least algebraic.

Then Lemma (2.1) implies that d(R) e NU {oo}. Furthermore:

(u) If d(R ) < oo, then P¡( ú) t : l, and there exists I g [I+ that c¿ is constant ) 0 and

,, : I/4ln (r- ¿^') , d(n)

' ' i:l

and functioDs c¡ : Z,lkZ,

d

is the order of the pole of P¡(ú) at + Q (l :0,1,...,d : d(n)) so dim R("): Dr,(n*kZ)nt (n large enough).

¿{

(b) If d(fi) : m, then there exists a consbant C > 1 with dim Rlü > C" (n large enough). Case (a) applies in particula,r to almost commutative algebras .R. Recall that in this case, by definition, R has a finite-dimensional generating subspace V such that S : grv(n) : g lü ¡1"-t) is commutative. Thus ^9 is an affine, positively graded commutative algebra, with ^90 : #, and so rationality ol Py(t) : Ps( ü) is guaranteed by the classical Hilbert Serre theorem [AuM, Theorem 11.1]. For further rationality results see (2.3) and (3.3) below. 45- (survey). Let G be a group a¡rd let E be a fixed finit¿ subset of define the growth sedes

f G associated

with E

G. Le¡ El") C G be defined as in (I.2.4), and

by

pB(t) :: ¿ ( #n(")

#E{"-

r))¿' € nll\l.

Inasmuch x ffTft): d.ún lln), where V is the subspace

f the group algebra lG that is

spanned by,E, we see that Pe(t) : Pv(t), and so the present situation is covered by (2.2). The following theorem summarizes some known results. Part (a) goes back to work of Milnor [Mi 2], Wolf [Wo], and Tits [Ti]. Part(b) is due to Benson [Be].

Tlreorern. Let G be a group

and letE be a finite subut ol G which generatec G as a monoid. 1c/ (Milnor, Tits, Wolf) Il G is soluable-by-finíte,

r linear, the either

#E("1 3 p(") lor some polynomial p or ffBftl > C" for some conctant C > | antl all sufficiently large n. /ó/ (Benson) Il G is abelian-by-fi.nite, then PB(t) is rational (and hence PB(t) has the form deacribed in (2.2) (a), with d(R) : rank ol sn abelian eubgroup

l finite index in G

). Grigorchuk IGr] has shown that the dichotomy in part (a) above does- not hold for general groups G. Results a,nologous t" (b) exist for certain isometry groups of hyperbolic space IC] and for Coxeter groups, at least for a canonical specific choice of the generating set E lSo] , [Bou 1, p.230-231, exerc. 10]. See also (IV.2.4). It would be interesting to exiend (b) t" finitely generated nilpotent by-finite groups. Also, a ring theoretic proof of (b), viewing PE(t) ^ Py( ú) as above, would be most welcome. Note that, for G abelian, (b) is an immediate consequence

f the Hilbert-Sene theorem (cf

. (Z.Z)). For a survey about growth series

f groups see [W].

SLIDE 26

46-

5 3. A Non-commutative HilberLSerre Theor.em

(3.1) We will be concerned with positively graded *-algebras fi : @ r?n having the following ¿)0 [hree properties: R is affine over /c and connected (i.e., fio: t), and d(P) < m. Every graded ideal 1 of R contains a finite product of primes P )

I. These can be

chosen to be graded, because if P is prime then so is Ps' : : @ (P n R"). (3) If P is a graded prime ideal of R, P ?. Rn::'90 ",, rhen n : n fP contains a nonzero normal element c which is homogeneous

f positive degree.

Here, eg-R is called nortnal ii rR : R¡. These properties a¡e inheribed by all graded images of R. Instead of assuming Ro: fr in (l), it would actually suffice to have fre finite-dimensional

ver /c a¡d central in R.

The connected case, however,

ccurs naturally in most applications.

Note that (1) (also in its weakened form) impiies that di¡¡t Bo {

o for all n.

More generally, if , :9rMo is a finitely generated graded R-module, then dtm M, (

o for all n (cf. (I.2.5)).

So we can define Lhe Poincarí series of M by

P¡a(t) :: I dim M,. tn e Zlltl).

¡)0

Assumption (2) above is cerbainly satisfied if the maximum condition for 2-sided ideals holds in R. (3.2) Examples. The main examples thai we have in mind are as follows. Note that, in these examples, the normal element z € R in (3) ca¡ even be chosen to be central (*d homogeneous, because the center of a graded algebra is a graded subalgebra) .

(1) (2)

47-

(r) Affine positively graded connected PI-algebras. For these, d(R) < oo is due to Berele fBer] (or see [K-L, Corollary 10.7]). (2) is a consequence of Braun's theorem [Br], and (3) follows from the fact that each nonzero ideal of a prime,

r even semiprime,

Pl-algebra r? intersects the center of ¡? nontrivially IRow, Theorem 1.6.271 .

(b) Enveloping algebras U: U(9) of finite-dimensional graded Lie-algebras C: é g,, where the g¡ are subspaces with lg,,g¡) C g;+i (: (0) for i+7.) f) and U is graded as in I J, Secrion 8.2]. Here, d(rJ): dim g (lK-L, Theorem 6.10]), (2) is clear, since Lr is Noetherian, and (3) follows from the fact that [1 is polycentral, because g is nilpotent (lD, Proposition 4.7.1]). (S.3) Theaem. Assume that the graded algebra " :9, Ro sat'isfies the hypotheses (1), (2), and (S) ín (s.l). Then, for any Noetherian graded R-module , :\oM, , Pr(t) 'is a rational function of the form

Pu(t): I(t) li.f.-,0'),

where f (t) e 7' lt) and E ,k t, . , k, € lN*. In particular, ' d(M):

rder of the pole ol Pu(t) at t:

l. Moreoaer,iÍO-M'-M-Mtt-0r¡anecactsequenceofNoetheriangradedR-modules(so that the maps are also graded), then d(L,f ) : max {d( M'),d(M")\ . Prof. We first show that, Pu(t) has the required form. If d(n) : 0, then R and M are finite-dimensional and so Pu(t) € T'lt). Suppose that d ( ,q ) > 0 and that the assertion about Pu(t) is true for all graded homomorphic images R of .R which satisfy d(R) < d(R)- 1. By assumption (2), there are

SLIDE 27

48-

graded prime ideals P; (i : 1,...,r) of rg (not necessa¡ily distinct ) so that M .Pr.P2- ... P, : (o) Setting Mo: M, M;: M;,,4 (i > 1) we obtain a decreasing sequence of graded submodules

f M with 1tr[,

: (0). since pu(t) :,i- pr,-rrr,(r) , it suffices ro show thaü each Pu,-r¡u,(ú) has the required form. Thus, after replacing M by M;-1fM; znd E by R fp;, we may assume that R is a prime ring. Nowlet r € R be normal asin (3.t) (a), with r e R^(m > 0), say. Then ¡,? isagraded ideal of ,? with d(R /zR) < d(E)- r, because c is regular in R (corollary (I.1.7)(b)). Moreover, K :: ann¡a(c) and L :: M /M't ^re graded subquotients

f M, and hence K

and 'L are Noetherian. In fact, K and 1, are modules over RlxR and so, by assumption, p¡r(t) and P¿(ú) have the desired form. Furthermore, for each n, we have an exact sequence

f

vector spaces

O - Ko : K f\ Mo - Mo

Mr*^
Lo+^ :

Mo+^lMr.x +

Therefore,

dim Mo*^

dím Mo :

dim L,a^

dím Ko

(n g [I ) . Multiplying with t"+' and summin8

ver ail n € [tr in

Zltt]], we obtain

(r - t^)PM(¿) : Pt(t) - t^Px(¿) + ¿(r) for some polynomial ¿ ( ,) € Z' ltl of degree { rn. This implies the result about pu(t).

By Lemma (I.2'5) , d(M) equals the degree of growth

f

rhe function n + rnn,:,f, dímM;. Setting P(t):: ,!o motn €Zlltl], we have P( ú) : Pu(| . (t+r+r2+... ) : pu(t) /(r - t\, and so, by the foregoing, P(t) has the form described in Lemma(1.3). We conclude that _49_ d(M)+l :

rd¿a

P(t) :

rd¡:¡ PM(t) + l,

where ord¿:1 sta¡rds for "order of the pole at f:1". This proves the formula for d(M). Finally, let o + M': M i ¡4" * 0 be an exact sequence of Noetherian graded R- modules, and assumes that Á¡ has degree g and n has degree p, i.e., p(M^,) ! Mr*o and "(M") C M"n+0. Then Pu(t): tq.Pu,(t) + f ,.Pu,(ú), and so

rd¿:y

Pu(,t) ( max Thus, by the foregoing and d(M): max {d(M'), d(M")i. g {ord¡:1 Pu,(t), ord¡4 Pu,(t)\. Proposition ( I.1.6) (a) ,

we conclude that

There are examples of affine positively graded connected PI-algebras R having non-integral GK-dimension [Bo-Kr, satz 2.10]. In this case,.R has properries (9.1)(1), (2), ffrd (3), but the regular r?-module R¡ does not satisfy the conclusion of the theorem. Thus the Noetherianness assumption

n M , or some other conciition,

is needed here.

$ a. Application to Cert¿in Filtered Algebras

(¿.t) LetB be a ft-algebrawith d(.R) < oo, and assume üat -R has an increasing filtration

(0): ¡(-t) c ft: ¡(o) e R(tl c... c R: ¡ R(") so that Cr(R) :6 pfülR("-t) ¿c alf ine, right Nocthuían, and aat'isf¿s (3.1)(3). Note that Cr(R) can be generated by homogeneous elements having positive degrees. It follows that all ¡(n) are finite-dimensional, and so Lemma (L3.3)(b) implies that d{gr(R)) : d(R) < o". Thus sr(.R) satisfies (3.1)(l),(3), and (3). Note also that our assumptions force R to be affine and right Noetherian.

SLIDE 28

50-

(¿.2) B**pt"" *a n"-.rtr Almost commutative algebras.B fit in the framework described in (+.1). Here we can take ühe V-filtration ¡(') : ln) for some finite-dimensional generating subspace V of R such that Crv(R) :9 fln) ¡lln-t) i" commutative. More generally, somewhat commutative algebras (2.4a) satisfy all our requirements. Jantzen, in IJ, Kapitel 8] , considers the situation where .R has an increasing filbration (0) : ¿(-t) c *: ft(o) g. g,?: ¡ R(') with sr(R)= U(g) for some finite-

dimensional graded Lie-algebra g

By (3.2)(b), this case is also covered by (a.t). Possibly all affine right Noetherian Pl-algebras satisfy our assumptions in (a.1). In view of

ur remarks in (3.2)(a)

, this would follow if the following question has a positive answer.

:9. gt'

Question. Let R be an affine

filtration (0) : ¡(-t) c ¡ : ¡(o) Noetherian? right Noetherian PI-algebra. Does there exist an increasing C... glq: U ,9(n) such that gr(.R) is affine and right

Finally, we remark that it is an open question whether every affine right Noetherian algebra is finitely presented (cf. IBerg 2, g 11]). (4.3) Multiplicities: Let R be as in (4.1), and let M be a filtered right R-module, with filtration (0) : ¡4(r) g ¡ztol 9 ... ut¿ c ¡4(n+r) c ... c M : ¿ urü. Assume that gr(M) : @ M@ l¡4("-r\ is f initety generated

aer

gr(R).

Since all .R(o) are finite-dimensional, it follows that all M("1 ar" finite-dimensional. Theorem (3.3) applies to gr(M), and we conclude that

5l -

Pp(t) :: I dim ¡4(a).¿" : Po,1u) (t)/(t- t) ¡)0

has the form described in Lemma (1.3). Therefore, for some /c € [J+ a¡rd suitable functions c¡:V, lkZ

q

(/:0,1,...,d), with c¿ ) 0 consta¡rt, we have

(t

dim M$) : D c¿(n+kT,)nt (n large enough).

¿{

He re, d : d(MR) is the GK-dimension

f M , by Lemma

(I.3.3)(b). We define Lhe maltiplicity of M, (sometimes also called Bentstein number of M) bv e1tul ¡ : "(Mn) :: d!", € Q+. (The factor d! ensures that e(M) is in fart a posilive integer in some important cases, e.g. for modules over almost, commutative algebras [K-L, Chapter 7] .) This definition is independent

f the particular choice of the filtration Ml")
f M,

6long as Cr(M) is finitely generated

ver

gr(R). Indeed, suppose thab (0) : M-t) c Mo) g ... C M: y M', is anorher filtrarion wiüh sr,( M): @Mi)/Mn-t) finitely generated over gr(-R). Then gr'( M) : (oa9a, M")/M"-t)) ' c"(R) for some s, and we deduce that M") g M') ..R(') holds for all n. Now, for some ú, M') g M(ú) and so M') C Mlr) .¡(') C ¡4ln+t). Thus d;m ñ") <. dim ¡4ln+t), and writing dim ñ.ü as above, using functions c/:ZlktT,

q

(/:0,I,...,d:d(M) ) with c¿')O constant, we easily conclude lhal c¿' ( c¿. By symmetry, we must have equalily here, as required. Note that our assumptions on the filtration of M routinely imply Lhat M is finitely generated

ver R.

Conversely, if M is finitely generated over R , say M: U'R for some finite- dimensional subspace U of M, then fhe foregoing applies to lhe filtratioin M("1 ,: ¿7'6(") o¡ M. Thus we have a notion

f multiplicity

for any finitely generated R-module which is independent

f the particular choice of the finite-dimensional

generating subspace U of M.

SLIDE 29

52-

(4.4) Theonem. LetR: U R(') 2 ... 2 R(ol: fr I ¡(-t): (0) üc a filtered k-alsebra as in (1.1), i.e., d(R) < * on) ,r1n) is affine, rightNoetherian, anrl satisfies (5.1)(9). Then, lor any right R-module M , d{M) e N . Moreoaer, ü O-M,-M-M"-O ,is an eract sequence

!

R-motlules, then d(M) - max {d(M,),d(M,,)\. IÍ M is finitely generated (anrl hence M' and M" also), then e(M) if d(M,) : d(M") if d(M') > d(M") if d(M") > d(M') Prof. In order to show that d( M) e [.[ , we may assume that M is finitely generated. Fix a finite-dimensional generating space U 9 M and put M("\ , : ¡¡ 'p(n). Then .,r(M):@M@l¡4ft-r) is a finitely generated graded module over gr(.R), a¡rd Lemma (I.3.3)(b) a¡rd Theorem (3.3) tosether imply that d( Mn) : d(sr(M)e'(E)) € lÑ' Now let O - M, - M - M" -r 0 be 'an exact sequence

f right R-modules. We must

show that d(M) ( m¿¡r {d(ry'),d(M")} *d, for this, it clearly suffices to consider the case where M is finite ly generated. Let U g M be a finite-dimensional generating subspace for M , arrd let (Jt' C M" be the image of U in M". We consider the filtrations M(o) 1: lJ'ftln), (U,¡O 1:1,4(ü a M', and (M")(') ' : grt'p(nl. Then the associated graded modules are related by the short exact sequence O * N' | : gr(M') -N : : gr(M)+ N" : : 7t(M")'O' Here, N is finitely generated

ver gr(R) and, since gr(R) is right Noetherian, all modules in

rhe sequence axe Noerherian. Thus, by Lemma (I.3.3)(b) again, d(Mn) : d(Nc'(n)) arl¿ similarly for M'and Mt'. Theorem (3'3) gives

53-

Finally, to determin e e(M) , we can use the above filtrations for M, M', andM,,. Thus dim M{ü : dim (U,¡tA * dim (Mtt)(n) (n € ñt), and so the formulas for e(M) follow by comparing leading terms in the polynomial expressions

f these

functions. tr By a result due to Tauvel, the equality d(M): max {d(M'), d(M")} ("exactness") holds under more general circumsta¡lces. Indeed, the assumptions d(n) < oo and (3.1)(3) are not needed for this part (see IT], or [K-L, Theorem 6.14], or [Lo 3, Corollary 1.5]). which yields the result. d(No"rnl) : max {d(No'1n¡') , d(No,1n;")i ,

SLIDE 30

54-

Chapter IY: ON ASSOCIATED GRADED RINGS AND MODULES

$ 1. C*neralised Rees Rings and Modr¡les (l 1) In this section, we will study certain graded rings G(.R) and modules G(M) that ca¡ be associated with a given filtration

n a ring R , or module M , and which have some advantages
ver the usual associated graded rings and modules.

Throughout, we consider a C-algebra -R, where C is a central subring of R , üogether with a filtration ¡' : {R(') ln € Z}such that

(0) : ¿(-t) c...! R(,) c ¡(n+t) g. c R: U Rfn), g ¡ p(o). ,R(").p(.) q p(n+^). Furthermore,letMbeafilteredrightr?-modulewithfiltrationH:{MlnrlneZ}satisfying, for some m9, (o) : ,ln") g .. c ¡4(n) c ¡¡("+t\ c ¡4ft).pln) ¡ 1r1ft+n).

..cM

tl

\J 1

¡¡\n)

(1.2) D"fi"iti"" "f e."utrli

and modules. In addition to the usual associated graded rings

a¡rd modules gr(R) : 7rr(R) : @ p(ü ¡p("-t) a¡rd

Cr(M) : grH(M) :g ¡afü¡U{n-r), we will consider the graded subring

G(R) : Gr(fi) ,:@R(")X"

f the

polynomial ring r? [X] and the graded G(.R)-module

G(M) - GH(M), :@ ¡7(") ¡*.

DO-

We will call G(R) and G(M) the generalized Rees ring, resp. module, associated with the given filtrations (Schapira IS] calls G(R) the formal graded ring associated to the filtered ring R.) Of course, G(n) and G(M) could have been defined without explicitly refering to the "variable" X, in analogy with gr(R) and gr(M), but X will be a convenient notational device in the following. Note that X is a central element of G(R ). (t.S) Co*p".i.on "f the .. The following lemma shows that, as far as finiteness condiüions and Poincare senes axe concerned, the construction s G (M ) *d gr(M ) are essentially equivalent. However, as the proof will demonstrate, the relationship

f G( M)

to bobh M and 9r(M) is quibe perspective, thereby making G(M) a useful link belween M a¡rd gr(M). For related results, see lS, p.57] Lemrna ( notations ( 1.1) ) (o) G(M) ís finitely generated (finítely presented; Noetherían) as G(R)-module if and only if gr(M) is finitely generated (finitely presented; Noetherian) as gr(R)-module. In this case, M is finitely generated (finitely presented; Noetherian) as R -module. (b) G(R) 'is affine (finitely presented; right Noetherian) as C-algebra if and

nly if the same

holds for gr(R) . In this case, R is affine (finitely presented; right Noetherían) as C -algebra. (r) Assu¡ne that all lúl"l are finitely generated as C-modules and let\, be an additiue integer- ualued lunction on the class of all finitely generated C -modules. Put

p"tríú) ,: !x1,u(')¡r', po4u¡(r) ,: Dx(vtn)lM(n-t))tn

Then, in Z((t)), we haae

(r- ú) P"@ít) : Po,lu¡(t). Pr@f. Part(c) isobvious,and(b) issimilarto(a) and,forthemostpart,followsfromthe

SLIDE 31

56-

proof of (a). So we will concentrate

n (a).

We shall use the following trivial fact: If S is any ring, f is an ideal of S, and M is a¡r .9- module which is finitely generated (fin. presented; Noetherian), then M /MI v M Q ^g// is

¡

likewise, as module over S lI. Consider the following ideals 1 a¡¡d J of G(R) :

I : : c(,q).x : @¡(n-t)¡9, J : : c(R).(X_l). Then G( M).1 : G(M) x:O r4'"-t)yn, c(R)lr= sr(R), and G(M)lG(M).r = cr(M).

In view of the above remark, this proves the implications.(:)" i¡ (r) . Similarly, C(R)lJ=R via DroF -Drn,andG(M)|G(M).J=M via Dmo]fl"-}ffi,, which proves the assertions about M as R-module in (a) . As to the implications " q:", we consider the filtrations G(M)f") : : @ ¡¡(nia(n'n))y, G(R)(') . : 6p(m-(m,n))-ym

f G(M) and G(R).

Itis readilychecked rhar G( M)@.G(n)(-) g G(M)(o+-) and, clearly,

c(u¡{ü¡c(M)b-" = 3" #* = # ? "trl.

Jl

gr(G(M))
fin. gen. (fin. pres.;

Noetherian) over gr(G(R))

v

G(M) fin. gen. (fin. pres.; Noetherian) over G(B). This completes the proof of ( a). tr (1.4).Vai"ti"n

f tn" nttr*

Another advantage

f G(M) over gr(M),

besides its more

bvious connection wiü M, comes from the fact thaü va¡iaüions
f the filtration ff a¡e more

easily dealt with using G¡7(M) rather than gry(M). The proof of the lemma below should serve as an example. We will use the following notation. If V g R is a C-submodule generating R as C-algebra, then we write Gv(R) :G¡(n) where F : {I^ü} is the V- filtration of ,R. Lemma- Let V be a finitely generated C-submodule

l R generating

R as C-algebra and. put W: I^d),lor some fired (t > l. Then Gv(R) ü (rtght) Noetherian if ancl onty if Gw(R) iE 6ChU Noethe¡ian.

Prof. Since

lo) 9 Wft) : fa"¡ holds for all n, we have inclusions

f C-algebras

.9::@ w(n)ydn c Gr(R) :@ l")y 9 Gw(n) :@ w(')x'.

Here, ^9 is isomorphic to Gw(R) via X - Xd. Moreover, setting

A :: e rAdY 9 Gv(R), we have

0(m(d-l

Gv(R):S.A:A.5. So Gy(r?) is finitely generated as (left and right) S-module, and "<:')

follows. For the

converse, note that II

t

T

{

!

I¡

,

Therefore,

cr(G(M))> cr(M) I clx),

a¡rd simila¡lv

sr(G(,?)) = cr(R) q CV)

U

Using the Hilbert Basis Theorem and [Bou 2, Chap.3, $ 2 nog, Cor. l], we obtain the following implications : Cr(M) fin. generated (fin. presented; Noetherian) over gr(R) =) Gv(R) : SOsBr with B: Thus, for any right ideal 1 of ,9,

@ IA^) X^ .

dl^

SLIDE 32

58-

r.Gv(.R) nS:1,

and simila¡ly for left ideals. E Examples (Z.f) and (2.2) below show that the above lemma fails to hold in general if V is more freely varied. Note that the last part of the argument works for general filtrations F : {R(')} and shows rhar if G(R) : Gr(R) is (risht) Noetherian then so is G¿(R) :: Gpo(R), where F¿ :: {nta')1. We also rema¡k that if all p(o) a¡e finitely generated C-modules a.nd P61¿¡(ú) is defined as in Lemma (1.3)(c), then it is not hard to show tl¡at rationality of P6J¡¡(ú) implies rationality of P6r1p¡(ú).

$ 2. F,xamplee.

f infinite uniform

dimension.

59-

On the other ha¡rd, in -R : kG, the elements y; (o < t 1n), yit (o <, ( n-l) , xyi:y-ix (tS¡<"-t) , cyís:y-i (l<t<"-2) ar:e linearly independent and belong b Il'). It follows that

n

I am S;: dim ú") , 4n-2.

j=O n, ó [. maps onto é S; , we conclude that these spaces must both have dimension 4n-2. t={ i<) This establishes the claimed isomorphism S = T. We also see that the Poinca¡e series of ^9 is given by

ll

@

i>l

/¡, where earh

Let G : 4c ,y l r' : (ry)' : 1> be the group algebra

f G over the field fr. Then R

a¡rd

y. We claim that

group and let R : kG be the I/ is ttre subspace generated by e

S : : crv(R) = r {X,Y\¡*,YXY)

: T

Indeed, putting ( : : c * l and rl' : y * k in,5, we see that x2 : I implies €t : 0. Simila,rly, yxy : yy-tx : z gives q€rt :

O. Thus T maps onto 5.

Now ? is graded by total degree in the images of X a¡rd Y in ?, and the n-th component, {n, is spanned by the images

f

{Y ,X7*-t , Y-|X , XY-2X¡ (" > 2).

I¡ : : €rti€. S : €zt€ . k : S . €rti€ is an ideal of .9. Therefore, S has infinite uniform dimension, even a.s (^9,S)-bimodule. Note üat R is a finitely generated module over its enter i[y+y-']- Thus, by Lemma (II.2.4), we know that grlt¡(.R) must be Noetheria¡r for eom¿ finite-dimensional generating space W ol R. Indeed, taking W :1t,A,!- l)* and arguing a.s above,

ne checks

that grw(R) =* {X, Y,Yr)l(YYt, YLY, *, YX- XYr, ytx- Xy). This is a finitely generated (left a.nd right) module over the commutative subalgebra generated by the images of Y and Y1 , and so gr¡r{^R) is Noetherian. For completeness, we note that the Poinca¡e series

f gr'¡(R) is given by

Pv(t) : r + 2t + sp + f.*

r+t!-tz-+ts

Now let 1 be the ideal of S that is generated by {. Then /t : (0) and /2: infinite dihedral : klV) where

I

I dim

t=4

i'

T

I

ri

il

li

ii

ii Therefore,

T, < l+2+3 +4(n-2):4n-2. (2.2)

SLIDE 33

60-

Let R : klo,bl r <, ) be the skew group ring of the cyclic group 1r) = C2 of order 2

ver the commutative polynomial ring kla,b],

with e acting by intercha"nging the variables c a¡rd ó. Then R is a finitely generated module over its center kla+b,cü], and hence B is a Noetherian Pl-algebra. The subspace V::{x,a}¡

f R generates

R, and an explicit finitc presentation of R is given by

R = K {X,A}I(*_ I, XAXA- AXAX).

(Use the "universal properties"

f polynomial

and skew group rings.) We claim that

S:: crv(R) = r{X,A\l(*,XA"XA-AXA"X; n : l) ::7.

Putting {:: x-lk and a:: a*k in S, we have €2: ,2+l{tl

a¡ld

€ao€o : boa*fn+z) : ab"+fn+z): a€on€. So ,9 is a homomorphic image of ?. The n- th homogeneous component To

f

T is spanned by bhe images

f

A¡,Ai:ú4t-i-t (0 < t< n-t), md Y4iYAn-i-z (l <, < n-z). Therefore, dim T, 1ln 1 (" ) 2), a¡rd so

i d;* T; < t+2t+Dtrr-r): n'+2.

i{ i:2 The corresponding elements

f

R' , namely ai (0 < i 1 n),

i roi- ''-

1 :

i6i-;-1,

(0 < t l=i-l < n-r) , and xa;rai-;-2: ai-;-2bi (t < i < j-2{n-2), are linearly independent and belong to Il') . As in (Z.f ) we conclude that

Ddims;:dimÁ"\-n2+2

t'={

a¡rd that T = S. Further, dim T,: dim So: 2n- I (n > 2), and so the PoincarJ series

f

^S is given by

Pv(t) : r+2t+ D(2n- r)t' : l+Ú3 n)2 Gl;F

The given explicit presentation of S = ? shows that S is not finitely presented: no finite subset

f {*, XA" XA- AXA" X ; n > I } generates

all relations.

_ 6l _ For completeness, we remark that ^9 has infinite uniform dimension as (^9,^9)-bimodule. For, if I denotes the ideal of S that is generated by (, then It: (0) and t': r.9, /¡, where each 1¡ ': €ot€'^S: €ot€.f [o] : ]["] €ot€: ,9 .goiq is an ideal of S. However, since S is a finitely generated module over its center, Lemma (11.2.4) guarantees the existence

f a finie-dimensional

generating subspace W oI R so that grw(R ) is well-behaved. Here we can take W :: {a,b,r )¡. As above, one shows that

gTW(R ) = ,t {X,A,B)I(* , AB- BA , XA_ BX , B- A-X).

So gr¡y(.R) is finitely presented and Noetherian, being a finite normalizing extension

f the

polynomial algebra k lA,Bl.

(Z.e) An affine PI-alS.b." B hil

neratin

subs esI/CWsuch that Pv( f ) is rational but Pr¡¿( t) is not ( J.T. Stafford lStl). Let (*+(t) Éle.u.el)

t: I t'l ri',;,'i) e Mz(kfx'Y'z])'

where klr,y,z] is the commutative polynomial ring in r ,y,z over ¡t ""td (...), denotes the ideal

f /c[a,y,z] lhat is generated

by the indicated elements. The ring fi will be an appropriate factor of ,5, and hence r? will be affine PI, as ,9 is. Pick any sequence {ór I d € [J ] with ói € {0,1} *d such that the sequence e¡ : 6;+r,

ó; is not recursive (thus De;ti is not

rational). Put

I : J{6} : : < zri6;,zy'(r-ór) | i € lN)* C klx,y,z) and let

,,:[(2.2,^zxa):, 1r'"rll c ^s.

' ' - [ (22, ny) (r,ry))

It is readily checked that 1 is an ideal of S, so we can define R ::

S lI. We write e;¡ for the images in .R of the matrix units e,7 in S a¡rd xe;¡, !e¡¡, ze;¡ fot the images of xe;¡ e1rc' For our two generating spaces

f R we take

SLIDE 34

62-

|'.

(0tt,

e12, 2A21, e22, tC22, UeZZ) *

and l'l/ i: {V, zte21}¡ It is convenient to write

symbolically, md similarly for all l'). t^2) : and Now e¡¡zz'-2 : O if zr"-, e t t¡ 6,_z: I

f "I, and so exactly one of the two terms in

dim V") : d&n IA"- 1) + Z (" > 2) , whence

Pv(t) : D dim(V") lÁn- r)¡.¿n

¿)0 . : I +bú + 7t' l-t' Now consider the same calculalions for

* :f{,1,',}, *,:lrJ

In this case,

wtü :

The number

f nonzero

+ lrr (" > 2).

'q ¡ if er1zy"-, * o, by definirion ,l)-position above is nonzero. Thus

:l+5t+7Dt,"

¡)2

+ W("- ') (n > 2). is ( t- ó "_ ,) + ( l- ( 1- 6,_r)) : r- eo_2.

': [l]] ':l,J

Thus

(u!),',',,, ,,,1' ,'i ,1,',,"r) , í1,')

iff zy"- the (r

[u,'; ),' ;i

,,"'

J íi,' )

berms in the (f.t)-position Thus dizn tyY(t) : ttim Wb-t) + 7 - implies that P¡at( t) is not rational.

63-

€n-2 (" > 2), and so our choice of the sequence {c;} (2.4) Growth series

f some

. We refer to (III.2.3) for the norations Eb), pÁú) etc.

(u) The free abelian group

Pe(t) : (r+r)'l(t- t), :

with E:{rfr, ..,e,+r} gives

Put Gr-t :: (x1,.. ttr_t> g G

r. Cleally,

,E'(') : U 1A(") n G,_tx!)

ll lSo ar¡d E-t:: {r¡+t,. , md E."l n G,_tzl

. ,t,!l ) and use induction on a-1:-t Ej-'¡ tr D . T.herefore,

#rr"): É #E,glttt)

l:-n

: 2'D #89, - *nJ:) ¿{

So, putbing Qo(t), : "lo #El"l.úo and similarly for E,_,, we have Qs(t):2Qq_,(,) *

Qe,_,(t)

: Q, l¿)'r+ú l-t The assertion now follows by induction, since ee(t): pe(t)1 1_ ú) and similarly for es,_r(t). (b) The Heisenberg group G-lz,ylrty-rry central) with E:{rtt,y*t}: The coefficients d, .. : #E(")- #Eb- t) of p¿,(r) satisfy the recursion

O: do+a- \do+z + 4dn+s- ldn+s + 6do+{ - Sdo+s+ 4dn+2_ Sdo+t + do (r, > l). A proof of this fact has been given by Bernd Weber (Bonn, Ph.D. thesis in prepa.ration). By Lemma (III.1.2) , Pp(t) is rational. In fact, P¿(ü) has a pole of order 4 at t: t; rhe orher poles are * f and L e2ni/3, each of order l. From the pole at t :1, we deduce that d(G) : 4 which agrees with Theorem (I.Z.l). (.) The group G: {a,z lz;oz-ia

- aziaz-t;

, € Z) with E: {o*,,r*t}, Elementary

SLIDE 35

but sqmewhat tedious calculations show that

Po(t) : (t- t)1(tt¿)3(t+¿2)= ( l- r- t'- tt)(r- zt- P) As r[2-1 < 1 is a pole

f PB(ú),

we conclude from Lemma (III.1.a) that the numbers #El"l grow exponentially. This agrees with Theorems (III.2.3)(a) and (1.2.4), N G is solvable butnot nilpotent'by-finite .

The above group G is the universal case of all semidirect products of the form

A x<z> ,

where A is abelian a¡¡d is a cyclic Zlz)-module, A: a'Z<z). The Heisenberg group also has this form:

(t r)

c=(7,@n) xto rJ

It would be interesting to compute Pe(t), with E: {o*t,t*t\ z€GL"(T,),A>T'".

for various other REFERENCES tA]

D. Anick, On monomial algebras
f finite global dimension, Trans. Amer. Math. Soc.

291 (1985), 291-310. [At M] M.F. Atiyah and I.G. Macdonald,'Introduction to Commutative Algebra, Addison- Wesley Publ. Co., Reading, 1969. tB]

H. Bass, The degree of polynomial growth of finitely generated nilpobent groups, Proc.

London Math. Soc. 25 ( 1972), 603-614. [8"]

M. Benson, Growth series
f finite extensions of Z"

a¡e rational, Invent. Math.73 ( 1983) ,251-269. IBer]

A. Berele,

Homogeneous polynomial identities, Israel J. Math. 42 (1982), 258'272. fBergl] G.M. Bergman, A note on growth functions of algebras and semigroups, mimeographed notes, University of California, Berkeley 1978. IBerg2] G.M. Bergman, Gelfa¡rd-Kirillov dimension of factor rings, preprint (UC-Berkeley, le87). IBern] I.N. Bernstein, Modules over a ring of differential operators. Study of the fundamental solutions

f equations

with constant, coefficients,

Funct. Anal. Appl. 5 (1971), 89-101.

[Bo-Kr]W. Borho a¡¡d

H. Kraft, Über die Gelfa¡rd-Kirillov

Dimension, Math. Ann.220 (1976), t-24. [Bour] N. Bourbaki, Groupes et Algébres de Lie, chap. 4-6, Hermann, Paris, 1968. [Bou2] N. Bourbaki, A]gébre commutative, chap. 3-4, Hermann, Pa.ris' 1967' tB.l

A. Braun, The nilpobency of the radical in a finitely generated Pl-ring, J. Algebra 89

( 1984), 375-396. lC] J.W. Cannon, The growth of the closed surface groups and the compact hyperbolic coxet€r SrouPS' PrePrint. [Ch-H] A.W. Chatters and C.R. Hajarnavis, Rings with Chain Conditions, Pitman, London' 1980.

J. Dixmier, Algébres enveloppantes,

Gauthier-Villars, Paris, 1974.

E. Formanek, Invariants a¡rd the ring of generic matrices, J. Algebra 89 (1984), 178-

223.

tDl tFl

SLIDE 36

66-

[G-Ki] I.M. Gelfa¡rd and A.A. Kirillov, Sur les corps lies aux algebres enveloppantes des algebres de Lie, Publ. Math. IHES 31 (1966), 5-19. [Got]

R. Gordon, Primary decomposition in right noetheria¡r

rings, Comm. Alg. 2 (1974), 49t-524. IGo2] R. Gordon, Artinian quotient rings of FBN rings, J. Algebra 35 (1975), 304-307. IGov] V.E. Govorov, Graded algebras,

Matem. Zametki 12 (1972), lg7-204.

[G.] R.I. Grigorchuk, On Milnor's problem of group growth, Soviet Math. Dokl. 28 ( 1983) , 23-26. [Gro] M. Gromov, Groups of polynomial growth and expanding maps, Publ. Math. IHES 53 (1981) ,52-TB. II-R]

K. Ireland and M. Rosen, A Classical Introduction to Modern Number Theory,

Graduate Texts in Mathematics 84, Springer, New York, 1982. lI J.C. Jantzen, Einhüllende Algebren halbeinfacher Lie-Algebren, Springer, Berlin, 1983. [K-L] G.R. Krause and T.H. Lenagan, Growth of Algebras and Gelfand-Kirillov Dimension, Pitman, London, 1985. tLl T.H. Lenaga.n, Gelfand-I{irillov dimension is exact for noetherian PI-algebras, Canadian

Math. Bull. 27 (1984)

,247-25o. [Lot]

M. Lorenz, Group rings and division rings, in: Methods in Ring Theory (F. Van

Oystaeyen ed.) , Reidel, Dordrecht, 1984. lLo2l

M. Lorenz, On the tra¡rscendence

degree oI group algebras of nilpotent grou¡rs, Glasgow

Math. J. 25 (1984), 167-174.

[Lo3] M. Lorenz, On Gelfand-Kirillov dimension a¡rd related topics, to appear in J. Algebra. [Lo-Sml M. Lorenz a¡rd L.W. Small, On the Gelfand-Kirillov dimension of noetherian PI-

algebras, Contemp. Math. 13, pp.199-205, Amer. Math. Soc., Providence, 1982.

L. Makar-Limanov,

The skew field of fractions of the Weyl algebra contains a free subalgebra, Comm. Algebra 11 (1983), 2003-2006.

lMc-Rl J. McConnell a¡rd J. Robson, Noncommutative Noetheria¡r Rings, Wiley-Interscience (to appear). lMill

J. Milnor, A note on curvature

and fundamental group, J. Diff. Geom. 2 (1968), l-7.

tMl

_67 _ [Mi2] J' Milnor, Growth of finitely generated solvable troups, J. Diff.

Geom. 2 (1968),

447

449.

[N-V.O.] C. Nastasescu and F. Van Amsterdam, 1982. [P-Sz] G. Polya and G. Szegd, Aufgaben Heidelberg, 1971. Oystaeyen, Graded Ring Theory, North-Holla¡rd, und Lehrsátze aus der Analysis II, 4. Aufl., Springer, ta]

D. Quillen, Higher algebraic

K-theory I, in: Lecture Notes in Math., Vol.34t, pp.Sb- 147, Springer, Heidelberg, 1973. tRl J.-E. Roos, A mathematical introduction, in: Lect. Notes in Math., Vol. 1183, Springer, Heidelberg, 1986. [R"] J.M. Rosenblatt, Invariant measures and growth conditions, Trans. Amer. Math. Soc. 193 (1e74), 33-53. [Row] L.H. Rowen, Polynomial ldentities in Ring Theory, Academic Press, New York, lgg0. tS]

P. Schapira,

Microdifferential Systems in the Complex Domain, Springer, HeidelSr.rg, I 985. [Sh] J.B. Shea¡er, A graded algebra with non-rational Hilbert series, J. Algebra 69 (lf)lro). 228-23r. ISm-St Wa] L.W. Small, J.T. Stafford, a¡rd R.B. Warfield, Affine algebras

f Gelf;ur<l-11

¡¡¡ll,,r dimension one are PI, Math. Proc. cambridge philos. Soc.92 (lgg5) ,4o7-41,t IS"]

L. Solomon, The orders
f finite Chevalley

groups, J. Algebra g (1966), 37(i-:rl:r ISt] J.T. Stafford, handwritüen notes (Leeds, 1984). tT]

P. Tauvel, Sur la dimension de Gelfa¡rd-Kirillov, Comm. Algebra lQ ( trlx'.1)

u tt¡ u,i l ITi]

J. Tits, Free subgroups

in linear groups, J. Algebra 20 (1572), 2,.r0-170 [W]

P. Wagreich, The growth function of a discrete

group, in: l,r,ct N¡rr¡.r rn \lr¡f.... ...

Vol. 959, Springer,

Heidelberg,l982. Group Actions, Vancorrvr.r, lilx,' [W"] R.B. Wa¡field, The Gelfa¡rd-Kirillov dimension of a modrrlr., l,r"l¡rrlr,n,r ,i ,. 1 e85) . [Wo]

J. Wolf, Growth of finitely generated

solvable gr()lr¡)s ¡g¡rl r rnr ¡r'r¡ l . . manifolds, J. Diff. Geom. 2 (1968) ,42L-446.

SLIDE 37

68-

tzl D.B. Zagier, Zetafunktionen und quadratische Kiirper, Springer, Heidelberg, 1981.

CUADERNOS DE ALGEBRA

1- M. L. TEPLY - F in i teness Cond i *ir¡rrs

r,

Tor s iort Thec.¡r ies - (Noviembre 1- 984). 2- A. IIARTINEZ SEVILLA. Teor las de Extensión ,/ Cohctmolr.rqJa de seni grupes. (Novienbre 1.985). 3- A- VERSCHOREN. Hecke Actions artd ReJat¡ve Irtvariarrt-<- (Noviembre 1.986)

4-

M- BULLEJOS LORENZO- Cohonttlogla he Abe I jarra ¡ La Sucesiórr Exacta Larga- (Marzo 1.987). 5- F- Van OYSTAEYEN. Graded Rirrgs Applied tr.r the :itudy

f

Orders and their Invariants, 6. tt- P- CARRASCO CÁERASCO. Hiperc()nPIeir.>s Cruzados: Cnhr.rnr.¡ Iogla y E.xterr siones. (Novienbre

1. 987).

7. M- LORENZ - Gelfarrd-firrillctv Llimertsion and P<,tincaré Serie-<- (Marzo 1- 988).

l I

{

I I

SLIDE 38

CUADERNOS DE ALGEBRA

MARTIN LORENZ GE LFAND-KIRIL,L O V DI ME N S I ON AND POINCARE SEKIES

GRANADA, MARZO 1988

Departamento de Algebra y Fundamentos. Universidad de Granada.