JOURNAL OF ALGEBRA 189, 353᎐376Ž. 1997 ARTICLE NO. JA966896
On the Trace of Graded Automorphisms
Naihuan Jing*
Department of Mathematics, North Carolina State Uni¨ersity, Raleigh, North Carolina 27695
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James J. Zhang†
Department of Mathematics, Uni¨ersity of Washington, Seattle, Washington 98195
Communicated by J. T. Stafford
Received January 9, 1996
Let A A be a connected algebra with a graded algebra endomor- s [d G 0 d Ž. Ý Ž .d phism . The trace of is defined to be Tr , t s d G 0 tr ¬ Atd . We prove that TrŽ. , t is a rational function if A is either finitely generated commutative or right noetherian with finite global dimension or regular. A version of Molien’s theorem follows in these three cases. If A is a regular algebra or a Frobenius algebra we prove a reciprocity for the trace. We also partially generalize a theorem of Watanabe on the Gorenstein property to the noncommutative case. ᮊ 1997 Academic Press
1. INTRODUCTION
Throughout the paper k will denote a field. A ޚ-graded vector space ϱ M s [d g ޚ Mddis called locally finite if dim M - for all d, where dim is always the dimension of a k-vector space. An ގ-graded algebra A s Ž. [d G 0 Ad is called connected if A0 s k. The tri¨ial A-module Ar [i) 0 Ai is denoted by kA or simply by k if no confusion occurs. The Hilbert series
* Work supported in part by the NSA. E-mail: [email protected]. † Work supported in part by the NSF. E-mail: [email protected].
353
0021-8693r97 $25.00 Copyright ᮊ 1997 by Academic Press All rights of reproduction in any form reserved. 354 JING AND ZHANG of M M is defined to be s [d g ޚ d
d HtMdŽ.sÝdim Mt. dgޚ The Hilbert series is an efficient tool to study commutative and noncom- mutative graded algebrasŽ seewx 4, 10, 21, 22. . For example, the Hilbert series provides a measurement of the size of a graded module by giving its Gelfand᎐Kirillov dimensionwx 10 . If M is a graded module over an infinite dimensional Kac᎐Moody Lie algebra ᒄ with respect to a grading of ᒄ, then the series HqM Ž.is also called the q-dimension or character in the representation theory of Kac᎐Moody Lie algebraswx 9 . If ᒄ is an affine Kac᎐Moody algebra, the Ž. 2i series HqM is a modular function of with q s e . Moreover there is a more general notionᎏthe trade of endomorphisms used widely in mathematical physics, for instance, the traces of certain vertex operators are n-point functions in the statistical mechanics models in connection with affine Lie algebraswx 3, 8 . Ž. A linear map of M s [d g ޚ Mdddis called graded if M ; M for all d g ޚ. The trace of a graded map is defined to be < d TrMŽ. , t s ÝtrŽ. Mdt . dgޚ
We will omit M from TrMMŽ. , t if no confusion occurs. Clearly TrŽ. id , t Ž. is the Hilbert series HtMd. We say M is left bounded if M s 0 for all d<0. If M is left bounded and locally finite, then TrŽ. , t belongs to the ring ktww ,ty1 xx of Laurent power series. The trace has been used to study fixed subrings of commutative polyno- mial algebras since the last century. Suppose that A is a graded polyno- u mial algebra kVwx, where V is a positively graded vector space [is1 Vi and that is a graded algebra endomorphism of A. Let Ci denote the induced matrix of on Viilet n be the dimension of Vi. Using a special ordered monomial basis of A Žsee the proof ofwx 4, 2.5.1. , it is not hard to prove that 1 TrŽ. , t s .Ž. 1-1 Łuidet I Ct is1Ž.niiy
Molien’s theoremwx 4, 2.5.2 follows easily fromŽ. 1-1 . Benson’s bookwx 4 gives a good account of invariant theory for commutative rings. Recently in ring theory there have been increased activities to study noncommutative graded rings coming from quantum groupswx 11, 13 and noncommutative projective geometrywx 2, 17 . One important class is that of THE TRACE OF GRADED AUTOMORPHISMS 355 graded regular algebras defined by Artin and Schelterwx 1 that are viewed as noncommutative analogues of commutative polynomial algebras. A connected algebra A is called Gorenstein if it has left and right injective iŽ. iŽ. dimension n and if Ext k, AAAs Ext k, A s 0 for all i / n and nŽ. nŽ. Ext k, AAA( Ext k, A ( k. If, moreover, A has finite global dimen- sion, then A is called regular. The definition of regularity here is the definition given by Artin and Schelter without the condition that A has polynomial growth. There are two subtle points in studying fixed subrings in the noncommutative case. First a finitely presented regular algebra need not be noetherianwx 22, 3.4 . Second the automorphism group of a noncom- mutative regular algebra is not easy to determine, even for the Sklyanin algebrawx 18 . Examining the trace of the graded automorphisms provides a way to study the automorphism group and fixed subrings. The purpose of the paper is to extend some standard results in commu- tative invariant theory to noncommutative algebras. In particular, we partially generalize Molien’s theorem and a thoerem of Watanabewx 4, 5.3.2 to the noncommutative caseŽ. Sections 5 and 6 . In order to do so we prove that Tr Ž. , t has nice properties. For example, Tr Ž. , t is rational if A is either finitely generated commutative or right noetherian with finite global dimension or regularŽ. Section 2 . As a consequence, Tr Ž. , t is deter- Ž < . mined by finitely many terms Tr A d . For regular algebras and Frobe- nius algebras, we prove that there is a reciprocity formula for the trace Ž.Section 3 . We also list the traces for all graded algebra automorphisms of the Sklyanin algebraŽ. Section 4 . In this paper e¨ery graded ¨ector space is assumed to be locally finite and an endomorphism or automorphism of an algebra refers to the algebraic structure.
2. PROPERTIES
Similar to Euler characteristic Tr is additive with respect to exact sequences.
i i EMMA L 2.1. Let M s [d g ޚ Md be graded ¨ector spaces for i G 0 with i the minimal degree of M going to the positi¨e infinity when i goes to infinity. Suppose that the complex
n 1 0 иии ª M ª иии ª M ª M ª 0Ž. 2-1 Ž. i is exact and that i i G 0 are k-linear graded endomorphisms of M Ž.Ý Ž .i Ž. commuting with the boundary maps of 2-1 . Then iG 0 y1Tr i,ts0. Proof. First we assume that the complexŽ. 2-1 has finite length, i.e., i there is an n such that M s 0 for all i ) n. Using the commutativity between i s and boundary maps ofŽ. 2-1 we can break Ž. 2-1 into two exact 356 JING AND ZHANG
sequences:
n 2 0 ª M ª иии ª M ª L ª 0Ž. 2-2 and
1 0 0 ª L ª M ª M ª 0Ž. 2-3 Ä4Ä4Ä4 such that i ¬ i s 2,...,n j 1 ¬L and 1 ¬ L, 10, commute with the boundary maps ofŽ. 2-2 and Ž. 2-3 , respectively. By induction we only need to consider the case when n s 2. By the definition of Tr it suffices to 2 i prove Ý Ž.1trŽ.¬i 0 for the exact sequence is0 y iMd s
2 1 0 0ªMdddªMªMª0
for every degree d g ޚ. This follows immediately from linear algebra. In general we consider the subcomplex of degree less than m. Since the minimal degree of M i goes to the positive infinity, such a subcomplex has finite length and hence the formula holds. The general case follows by letting m go to the positive infinity.
1 2 The tensor product over k is denoted by m. Let M and M be two 1 2 graded vector spaces. The graded tensor product M m M is left bounded i and locally finite as long as M are. Let i be a graded endomorphism of i M for i s 1 and 2, respectively. Then
12 1 2 TrMmMŽ.Ž.Ž.Ž.12m,tsTrM1, t TrM2, t . 2-4 Let Gr-A be the category of graded right A-modules with morphisms being graded A-homomorphisms of degree 0. For a graded algebra endo- morphism of A and M g Gr-A we define the -twisted module M such that M s M as a graded vector space with the action
m) a s m Ž.a
for all m g M, a g A. We define a functor on Gr-A associated to by Ž. Ž. Ž. FM sMand Ff sffor any f g HomGr- A M, N . It is easy to see that F is indeed a functor from the category Gr-A to itself. If furthermore is an automorphism, then F is an invertible functor and A is isomorphic to A as a graded right A-module. Let M and N be graded A-modules. A graded k-linear map from M to N is called -linear if
Ž.ma s Ž.Ž.m a
for all m g M, a g A. For example, itself is a -linear endomorphism of A. A graded map : M ª N is -linear if and only if it is a graded A-homomorphism from M to N . THE TRACE OF GRADED AUTOMORPHISMS 357
ޚ For a graded A-module M s [d g ޚ Md and an integer s g we define Ž. Ž. Ž. the shift module M s s [d g ޚ Msddsby Ms sMqd. By a noncommu- tative graded version of Nakayama’s lemma, every left bounded and locally finite graded projective right A-module P is free, namely, it has a form of Ž. [iiAys. Thus P s V m A, where V is a left bounded and locally finite graded vector space [iikx with deg xis s i. It is clear that V ( P mAAk , where mA is the tensor product over A. Given a graded endomorphism 00: V ª V, m is a -linear endomorphism of V m A for a graded endomorphism of A. Given a -linear endomorphism : V m A ª V m A, we can define Ј to be mAkid . Then Ј is a graded endomorphism of V. In general is not equal to Ј m . The next lemma shows that we still have a formula similar toŽ. 2-4 .
LEMMA 2.2. Let A be a connected algebra with a graded endomorphism and let P s V m A be a left bounded and locally finite free A-module. If is a -linear graded endomorphism of P, then
TrPVAŽ., t s Tr ŽЈ, t .Ž.Tr , t ,
where Ј s mAkid . Proof. First we assume that dim V is finite and use induction on dim V. If dim V s 1, then P s AŽ.ys and every -linear endomorphism of P is of the form
Ž.a s Ž.1 Ž.a ,
where Ž.1 g k. Hence
s TrŽ., t s Ž.Ž.1 t Tr , t s Tr ŽЈ, t .Ž.Tr , t . If dim V ) 1, let s be the lowest degree in V. Since A is connected, Psss V and has to preserve the summand Vsm A. Extending the field does not change the trace, so we may assume that k is algebraically closed. Ž.Ž. Let x g Vs be an eigenvector of and hence, of Ј and x s x for some g k. Then the map restricted to kx m A is a -linear endomor- phism of kx m A and the map induced on Ž.Vrkx m A, still denoted by , is a -linear endomorphism of Ž.Vrkx m A. Therefore we have a short exact sequence
0 ª kx m A ª V m A ª Ž.Vrkx m A ª 0 Ä 4 such that < TrVmAkxŽ., t s Tr mA Ž., t q TrŽVrkx.mA Ž., t . 358 JING AND ZHANG Similarly, we have Ј Ј Ј TrVkŽ., t s Tr xV Ž., t q Tr rkx Ž., t . Hence the statement follows by induction. Next we consider the general case. Let VF nis [ FniV . Then VFnm A is a-stable submodule of V m A. Since V is left bounded and locally finite, VFn is finite dimensional and hence the formula holds for VFnm A.Itis Ž.Ž . easy to see that V m A F n s VF n m A F n. Therefore TrŽ., t lim Tr <<, t lim Tr Ј, t TrŽ. , t s Ž.VFnmAVs Ž. Ž.FnmA nªϱnªϱ sTrŽ.Ž.Ј, t Tr , t . Let A be a connected algebra and M be a graded right A-module. Since A is connected, M has a minimal free resolutionŽ. MFR . We say that M has a finitely generated MFR if every free module in the MFR of M is finitely generated. We say that M has a finite MFR if M has a finitely generated MFR of finite length. THEOREM 2.3. Let A be a connected algebra with a graded endomor- phism . 1. Let M be a graded right A-module with a finite MFR. If is a -linear graded endomorphism of M, then TrMAŽ., t s ft Ž.Ž.Tr , t 1 for some fŽ. t g ktw,ty x. 2. If B s ArI is a graded factor algebra of A, preser¨es the ideal I, and BA has a finite MFR, then TrBAŽ. , t s pt Ž.Ž.Tr , t for some pŽ. t g kwx t with pŽ.0 s 1. Ž. Ž.y1 Ž. 3. If kAA has a finite MFR, then Tr , t s p t for some p t g kwx t with pŽ.0 s 1. 4. If A is regular or right noetherian with finite global dimension, then Ž. Ž.y1 Ž. Ž. TrA , t s p t for some p t g kwx t with p 0 s 1. Ž. Ýli Remark. Consider pt s1q is1 cti as an element in the power Ž.y1 Ý Ž.jŽÝ i.jŽ series ring ktww xx. Then pt s1q jG1 y1 iG1 ctiiwhere c s 0 .Ž.y1Ý i for i G l . Writing pt s1q iG1 dtii, we see that d is an integral polynomial of degree i in c1,...,cii. Conversely, every c is an inte- gral polynomial of degree i in d1,...,dii. Thus d for i ) l is an THE TRACE OF GRADED AUTOMORPHISMS 359 Ž. Ž.y1 integral polynomial in d1,...,dl. In our case of Tr , t s pt , every Ž < .Ž<.Ž<. tr AAiis uniquely determined by tr 1, . . . ,tr Al, where l s deg ptŽ.. Proof. 1. Suppose that the finite MFR of M is ⅷ n n 1 0 P [ 0 ª P ª P y ª иии ª P ª M ª 0, i i i where P s V m A for some finite dimensional graded vector spaces V Ž. iiy1 is0, 1, . . . , n . The boundary map from P to P is denoted by ␦i. ؒ ؒ Applying F to P , we see that Ž P . is a resolution of graded right A-module M . By definition can be viewed as an A-homomorphism from M to M . By the universal property of projectiveŽ. free resolutionw 7, ii Theorem 6.3x , there exists a sequence of A-homomorphisms i: P ¬ P ␦ ␦ such that iy1 iiis for i G 0, where y1s . Consider ias an i endomorphism of P , i is a -linear endomorphism commuting with the boundary maps. By Lemmas 2.1 and 2.2, we obtain nn iiX TrMiVŽ, t .syÝÝ Ž.Ž1Tr,t .sy Ž.1Tri Ži,t .ŽTrA , t . is0is0 sftŽ.TrA Ž , t .,Ž. 2-5 n i X y1 Ž. Ý Ž. iŽ . where fts is0 y1TrVi,tgktw,t x. From this proof we see that if M has a finitely generated MFR, part 1 still holds true with 1 ftŽ.gktww ,ty xx. 2. Let M s B and be the induced map COROLLARY 2.4. Let B be a finitely generated connected commutati¨e algebra and be a graded endomorphism of B. Then TrŽ. , t is a rational function. As an illustration in using Theorem 2.3 to compute the trace we give a simple and well-known result about free algebras. EXAMPLE 2.5. The trace of graded endomorphisms of the free algebra ²: ²: kx1,..., xn . Let A s kx1,..., xn be the free algebra generated by x1,..., xniwith deg x s 1. Then the trivial module k has a finite MFR, 0ªVmAªAªkª0, n where V s [is1 kxi. Every graded endomorphism of A is determined Ž. by its action on the linear space V or is given by a matrix C s cij n=n, Ž. Ž . Ž.y1 Ž. where xijijjs Ý cx. By Theorem 2.3, Tr , t s pt and pt s ŽX.Ž. 1yTrV 1, t s 1 y rt for some r g k. However, Tr , t s 1 q 1 tr Ctqиии . Hence r s tr C and TrŽ.Ž , t s 1 y tr Ct .y. 3. RECIPROCITY One elegant result of Stanleywx 21, 12.7 states that if A is a connected commutative Cohen᎐Macaulay ring, then A is Gorenstein if and only if Ž y1 . l Ž. HtAAs"tH t for some l g ޚ. In the noncommutative case it was proved inwx 22, 2.4 and 3.1 that if A is connected and either regular or Ž y1 . l Ž. right noetherian with finite global dimension, then HtAAs"tH t for some l. In this section we will show that such a reciprocity formula also holds for the trace on regular and Frobenius algebras. THEOREM 3.1. Let A be a connected regular algebra of global dimension n Ž. Ž.y1 and let be a graded automorphism of A. Suppose that Tr , t s pt Žy1. y1yl Ž. as in Theorem 2.3.4. Then py1 t s atpt orequi¨alently Ž y1 y1. l Ž. Ž. Tr , t s at Tr , t , where l s deg p t and a is the leading coeffi- cient of p Ž. t . Proof. Bywx 22, 3.1 , k has a finite MFR, ⅷ ny1 1 P [ 0ªAŽ.yl ª AŽ.ysiiªиииª A Ž.ys ªAªkª0 [[ii j where l ) si ) 0. By Lemma 3.2.1Ž. to be proved below , there is a -linear ؒ ؒ automorphism [ Ä4j of the complex P . Then the argument of Theo- Ž. Ž.y1 Ž. Ýn Ž.j ŽX. rem 2.3 implies that Tr , t s pt and pt s js0y1Tr j,t, X ؒk where jjAs m k. Since A is regular, the dual complex P [ ]HomŽP ؒ, A.Žis an MFR of the left A-module kl.. Applying Ž᎐.k y1 k HomŽ.᎐, A to Ä4jj, we obtain a -linear automorphism Ä 4of the THE TRACE OF GRADED AUTOMORPHISMS 361 exact sequence of left A-modules, ny1 1 0 ¤ klŽ.¤Al Ž.¤ AsŽ.ii¤иии ¤ AsŽ.¤A¤0. [[ii By Lemmas 2.1 and 2.2, Ž. kknyj TrŽ.dj Ž. nyj ky1 syÝŽ.1Trž/Ž.jЈ,tTrŽ. , t . j Ž. Ž k . yl Since kl is one dimensional, Tr d isomorphism 0 such that ␦0 6 6 PM0 0 6 6 0 y1 ␦06 6 PM0 0Ž. 3-1 0 0 commutes. Since P is projective and ␦0: P ª M is surjective, there is 362 JING AND ZHANG a morphism 0 such thatŽ. 3-1 commutes. Tensoring with the trivial A-module k we see that y1 maps a generating set of M to a generating 0 set of M . Since P is a projective cover, which is also free, ␦0 maps a 0 generating set of P to a generating set of M. Hence 0 mA k is an isomorphism. By Nakayama’s lemma for graded modules, 0 is an isomor- phism. ⅷ Part 2 is obtained by applying F s to and part 3 follows from parts 1 and 2. If A is a connected regular algebra with finite GK-dimension, then Ž. Ž.y1 Ž. Ž HtA spt and every root of pt is a root of 1 combiningwx 22, 2.2 andwx 22, 3.1. . Next we will show a similar property for the trace. PROPOSITION 3.3. Let A be a connected regular algebra with finite GK-di- mension and suppose that is a graded automorphism of A of finite order. Ž. Ž.y1 Ž. Then Tr , t s p t according to Theorem 2.3.4 and e¨ery root of p t has absolute ¨alue 1. Moreo¨er, if p Ž. t is a polynomial with integer coeffi- cients, then e¨ery root of p Ž. t is a root of 1. Proof. As in the proof of Theorem 3.1, we consider an MFR of k, ny1 1 ؒ P [ 0ªAŽ.yl ª AŽ.ysiiªиииª A Ž.ys ªAªkª0, [[ii j Ž. where l ) si) 0. By the proof of Theorem 2.3 or 3.1, pt s Ýn Ž.i ŽX.Ž. is0 y1Tr i ,t. Hence the leading coefficient of pt is equal to Ž.n Ž. Ž. Ž. Ä p4Ž y1 bif d x s b x for all x g A yl . By Lemma 3.2.3, i where Ž. Ž.y1 ؒ . pdenotes the order of is a morphism of P . Then HtA spt , Ž. Ýn Ž.i ŽŽp..Ј Ž. where pt s is0 y1Tr i ,t. Thus the leading coefficient of pt np np is Ž.y1 b . Bywx 22, 3.1.4 , Ž.y1 b is either 1 or y1. As a consequence, b is a root of 1. By definition, TrŽ. , t Ý tr Ž <.t d. Since has finite order, s d G 0 A d < Ž < . every eigenvalue of AAddis a root of 1 and tr has absolute value less than or equal to dim A . Thus lim< trŽ <.<1rdlimŽ. dim A 1rd1 ddªϱA dF dF Ž.Ž.ŁlŽ. because A has finite GK-dimension . Write pt s is11yrti .By Ž.lqn wx22, 2.1 , < root of pt Ž.is a root of 1wx 5, Lemma 2, p. 72 . Next we consider the trace on Frobenius algebras. Recall that an algebra A is called Frobenius if it is finite dimensional and Hom kŽ.A, k is isomorphic to A as a left and as a right A-module. Bywx 16, 3.2 , a finite THE TRACE OF GRADED AUTOMORPHISMS 363 n dimensional connected algebra A s [is0 Ainwith A / 0 is Frobenius Ä4 if and only if dim Anis 1, and for every x g A y 0 and 0 F i F n, xAnyins AxyinsA. Suppose that is a graded automorphism of A. Pick a basis Äx ii,..., x 4of A Ž.Žfor i 1,...,n and write x i. 1 wii s js ii Ži. ÝjЈ cxjЈjjЈand Cijjs c Ј. Then i TrŽ. , t s Ýtr Cti. i inyin If A is Frobenius, we can assume that xxjjЈ s␦ijx1 wx16, 3.2 . Thus Ž. nnŽ.y1 Ž. CCinyiscI11 and hence cC11 insCyi. This implies the following result. n HEOREM T 3.4. Let A s [is0 Ai be a connected Frobenius algebra with Ž. An skx / 0 and a graded automorphism of A. Suppose that x s cx = n 1 1 for some c g k [ k y Ä40.Then TrŽ. , t s ct TrŽy, ty.. The following immediate consequence will be used in the next section. COROLLARY 3.5. Let A and be as in Theorem 3.4. Then, for all 1in1, trŽy1 < .Ž0 if and only if tr <.0. FFyAAins yis 4. INDUCED AUTOMORPHISMS ON EXT In this section we will make the methods in Section 2 more precise and then reprove Theorem 3.1 by using Theorem 3.4. At the end we will list the traces for all graded automorphisms of the Sklyanin algebra. d Given graded right A-modules M and N, let Hom AŽ.M, N denote the Ž. set of all A-module homomorphisms h: M ª N such that hMii:Nqd ޚ Ž. dŽ. for all i g . We set Hom A M, N s [d g ޚ Hom A M, N and we denote i Ž. the corresponding derived functors by Ext A M, N .If M has a finitely i Ž. iŽ. generated MFR, then Ext A M, N s Ext A M, N . This is not true in general. If M is a graded right A-module and N is a graded left A A-module, the usual Tor-groups Tori Ž.M, N have graded structures, AŽ. which we denote by Tori M, N .If Ais a connected regular or noethe- rian Gorenstein algebra of injective dimension n, then it follows from i definition that Ext Ž.k, A s 0 for all i / n, and bywxw 22, 3.1.2 and 23, nŽ. nŽ.Ž. 0.3.1x , there is an integer l such that Ext k, AAA( Ext k, A ( kl. Let be a graded endomorphism of A. Suppose that a graded right A-module M has a finitely generated MFR n 1 0 ؒ P [ иии ª P ª иии ª P ª P ª M ª 0. 364 JING AND ZHANG Let y1 be a -linear endomorphism of M. By the universal property of Ä4 ؒ projective resolution, there is a -linear endomorphism s i of the complex P ؒ. Note that ؒ is not unique, but two different ؒs are always i ؒ homotopic. Hence induces a k-linear endomorphism i on Ext Ž.M, k Ž i .Ži. sHom P , k by sending f g Hom P , k to fi. It is easy to see that ؒؒ . Ž i fi gHom P , k and it is independent of the choices of . Similarly Ž.i induces a k-linear endomorphism iAon Tor iM, k s P m k. We define the following double series: i i i ⅷ TrExt Ž.M, k Ž.Ž.y1E, t, u syÝ1TrxtŽ.M,kiŽ.,tu iG0 and i i Tr Ž.Ž. , t, u 1Tr Ž.Ž.,tu. TorⅷŽ M, k. y1TorsyÝ iM,ki iG0 ii Ži.Ži.U i We may write P as V m A and hence Hom P , k s V and P mA i i U i k s V . By the k-linear graded duality between ŽV . and V , we obtain 1 that Tr i Ž. , ty Tr Ž. , t . Hence Ext Ž M, k. i s Tor i Ž M, k. i y1 Tr ⅷŽ.Ž. , t , u Tr Ž.Ž. , t, u . 4-1 Ext M, k y1Tors ⅷŽM,k.y1 y1 Note that Tr Ž. , t, u and hence Tr ⅷŽ M, k.Ž , t , u.is in TorⅷŽ M, k. y1Extw y1x y1 ⅷ Ž. kuwxww t,t xx,butingeneralTrExt Ž M , k . y1,t,u is not in kuww ,uty1 xxww ,ty1 xx. By the proof of Theorem 2.3.1,Ž. 2-5 , and Ž. 4-1 , we have i X TrMŽ. 1, t TrAV Ž.Ž.Ž. , t Ý1Tr i,t y s ½5y i iG0 i TrAŽ. , t Ý Ž.1TrTor Ž.M, kiŽ.,t s½5y i iG0 Tr Ž. , t Tr Ž , t,1 . sA TorⅷŽ M, k. y1 y1 ⅷ s TrAŽ., t TrExt Ž.M, k Ž.y1, t ,1 . A special case is when M s k and y1 s idk . Then y1 is a -linear automorphism of k for any . Rewrite y1 as y1. It follows from Ž. Trk y1, t s 1 that 1 y1 y TrAŽ., t s TrExtⅷŽ.k, k Ž.y1, t , 1 .Ž. 4-2 iŽ. Ž. ޚ iŽ. Since both Ext k, k and Tori k, k have -grading, [iG 0 Ext k, k and Ž. ޚ2 [iG 0 Tori k, k are -graded vector spaces. By definition, for every THE TRACE OF GRADED AUTOMORPHISMS 365 iŽ. Ž. i)0, nonzero homogeneous elements in Ext k, k wTori k, k , respec- tivelyx have negative degreesŽ. positive degrees, respectively . For every ޚ2 endomorphism the -linear map y1 of k induces -graded endomor- Ä4 iŽ. Ä4 Ž. phisms [ iion [ G0Ext k, k and [ iion [ G0Torik, k . LEMMA 4.1. Let A be a connected algebra, and be graded endomor- phisms of A, and , , , and be defined as abo¨e. Suppose that k has a finitely generated MFR. Then 1. s . As a consequence if is an automorphismn of A, then iŽ. Ž.op is an automorphism of [iG 0grExt k, k . Further, Aut A acts naturally ޚ2 iŽ. on the -graded ¨ector space [iG 0 Ext k, k . 2. s . As a consequence if is an automorphism of A, then is Ž. Ž. an automorphism of [iG 0grTorik, k . Further, Aut A acts naturally on ޚ2 Ž. the -graded ¨ector space [iG 0 Tori k, k . Proof. 1. By the universal property of projective resolution, there exist ؒ Ä4 ؒ Ä4 ؒ endomorphisms s iiand s of the MFR P ª k ª 0, where ؒ ؒ y1 sidk , 0 s and y1 s idk , 0 s . The composition is a ؒؒؒؒ -linear endomorphism of P . Hence sŽ. . Applying HomŽ.᎐, k , we obtain that ؒؒ ؒ HomŽ.Ž. , k s Hom Ž , k .Hom Ž , k .. Thus Ž. is iifor all i G 0 and s .If is an automorphism, we have y1 y1 y1 id id iŽ. . s s As [iG 0Ext k , k s Hence is an automorphism. The last assertion is clear. Part 2 can be proved similarly. iŽ. ޚ2 By using the Yoneda product, [iG 0 Ext k, k is a -graded algebra, denoted by A࠻. The next lemma shows that is also an algebra automor- phism. LEMMA 4.2. Let A be a connected algebra such that k has a finitely Ž.op generated MFR. Then e¨ery element in the group Autgr A acts naturally as a ޚ2-graded algebra automorphism of A࠻. Ä4ؒؒ Proof. Let P ª k ª 0 be an MFR of k and s i be a -linear . Ž ؒ automorphism of P with y1 s idk and 0 s Lemma 3.2.1 . By Ž.i definition, iisends f to f for all f g Hom P , k . For every f g i j HomŽP , k.Žand every g g Hom P , k., we lift g to a complex morphism jqss Ä4gss, where g is an A-homomorphism from P to P for all s as inw 16, Ž.2-2xwx . By the discussion of 16, Sect. 2 , the Yoneda product of f and g is 366 JING AND ZHANG иŽ iqj .Ž . Äy1 4 fgsfgijg Hom P , k . We can lift g s g to sgssqjby Äy1 4 checking that sssg qjare A-homomorphisms commuting with the boundary maps. Hence и и y1 Ž.f Ž.g s f iiiig qjis fg qjs Ž.fg . Thus is an algebra homomorphism and the statement follows from Lemma 4.1.1. By definition A࠻ is ޚ2-graded. The trace of any ޚ2-graded automor- phism is defined by nm ࠻ < ࠻ TrAŽ., t, u s ÝtrŽ. An,mtu. 2 Ž.n,mgޚ Hence ࠻ и TrAŽ., t, u s TrExt Ž.k, k Žy1, t, yu ., and byŽ. 4-2 , we have 1 1 y1 y y1 y ࠻ TrAŽ., t s TrExtⅷŽ.k, k Ž.y1, t ,1 sTrA Ž., t , y1. THEOREM 4.3. Let A be a connected algebra such that k has a finitely generated MFR. Let be a graded automorphism of A and let denote the ࠻ Ž. Žy1 .y1 induced automorphism on A . Then TrAA , t s Tr ࠻ , t , y1. If A is regular, then bywx 19 , A࠻is Frobenius. Similar to Theorem 3.4, we can prove that, for every ޚ2-graded automorphism of A࠻, l n y1 y1 y1 TrAA࠻ Ž., t, u s bt u Tr ࠻ Ž. , t , u = for some b g k and l, n g ގ Žthe details are left to the interested reader. . In particular, we have n l y1 y1 TrAA࠻࠻Ž.Ž. , t, y1 sy1 bt Tr Ž. , t , y1, which recovers Proposition 3.1 by Theorem 4.3. Ž.op Ž ࠻. In general Autgr A \ Autgr A . However if A is Koszul, then Ž.op Ž ࠻. Autgr A ( Autgr A . A connected algebra A is called quadratic if all defining relations of A generate a subspace RA in A11m A . Its dual !ŽU.² H: H algebra is then defined as A [ TA1 rRAA, where R is the orthogo- U U nal subspace of RA in A11m A .If Ais quadratic, then the graded ࠻࠻ !Ä4 subalgebra on the diagonal [n Ayn, ni; A is isomorphic to A. Let x THE TRACE OF GRADED AUTOMORPHISMS 367 Ä i4 U ! be a basis for A11; A and let x be the dual basis for A ; A. Define i eAi[ÝŽ.x*mx. 2 Then eA s 0. This special element gives rise to the so-called Koszul complex of right A-modules, ! ! иии ª Ž.An* m A ª иии ª Ž.A1* m A ª k m A ª k ª 0Ž. 4-3 with left multiplication by eA as the boundary mapsŽ. or called differentials . We say that A is Koszul if the Koszul complexŽ. 4-3 is exact. One standard ࠻ ! fact is that a connected algebra A is Koszul if and only if A s A Žin this ࠻ Ä4. case we set deg x s n for all x g Ayn , n y 0 . For more information on Koszul algebras, seewx 16, Sect. 5 . < Let be a graded endomorphism of a quadratic algebra A. Then A1 < is essentially an n = n matrix. The transpose of A1 gives rise to the dual graded endomorphism for the dual algebra A!. In the matrix format if Ž. Žj. i xjiijiijsÝcxfor c g k, then x s Ýijicx. It is routine to check that is a graded endomorphism of A if and only if is a graded endomorphism of A!. The construction shows that there is a one-to-one correspondence between graded endomorphisms of A and those of A!.In Ž.op Ž !. ! ࠻ particular, Autgr A ( Autgr A . Identifying A with A we obtain that Ž. Ž.Ž.y1 swuse 4-3x and that TrAA࠻ , t , y1 s Tr ! , yt . The follow- ing corollary is a consequence of Theorem 4.3. COROLLARY 4.4. Let A be a Koszul algebra. Suppose that is a graded algebra automorphism of A and that is the dual automorphism of A!. Then Ž. Ž .y1 TrAA , t s Tr ! , yt . Remark. Corollary 4.4 holds for any graded algebra endomorphism . One can directly prove it by checking that gives rise to a complex ÄŽ < ! . 4 Ž. morphism A n * m ¬ n G 0 on the Koszul complex 4-3 and by invoking Lemmas 2.1 and 2.2. As an application of Corollary 4.4 we calculate the trace for graded automorphisms of the Sklyanin algebra. For simplicity we assume that k is the complex number field ރ in this case. Let ␣123, ␣ , ␣ g ރ satisfy ␣123123q␣q␣q␣␣␣ s0 Ž.Ž.Ž.Ž. and ␣123, ␣ , ␣ / y1, 1, ␣3, ␣ 1, y1, 1 , 1, ␣2, y1 . The Sklyanin alge- bra S is the connected ރ-algebra with four generators x0123, x , x , x and six relations xx0iiyxx0s␣ijkŽ.xx qxx kj, xx0iiqxx0sxxjkyxx kj, 368 JING AND ZHANG where Ž.i, j, k is a cyclic permutation of Ž. 1, 2, 3 . The Sklyanin algebra S is a Koszul algebra with its Koszul dual being Frobeniuswx 17, 4.7 and 4.9 . The graded automorphism group of the Sklyanin algebra S was studied inwx 18 . It is given by an exact sequence = 1 ª ރ ª Autgr Ž.S ª E4 ª 1, = where the automorphisms corresponding to ރ and E4 are described as = n follows. If r g ރ , the graded automorphism grris defined by g : x ª r x Ä4 = Ž. for all x g Sn y 0 . It is easy to see that ރ lies in the center of Autgr S . Ž. Ž. As inwx 18 , E4 is isomorphic to the abelian group ޚr 4 [ ޚr 4 and is expressed as ÄÄ4a q b ¬ a, b g 0, 1r4, 1r2, 3r44 , where is a complex number with ImŽ. ) 0. The corresponding graded automorphism is de- Ž. noted by ⌽ for g E4. For detailed information, seewx 17 and wx 18 . Ž. = Every graded automorphism of S has a form gr⌽ for r g ރ and Ž. ŽŽ.. gE4. It is obvious that Tr , t s Tr ⌽ , rt and hence it suffices to compute all TrŽŽ..⌽ , t . By using Corollary 3.5, we manage to compute the traces on the Koszul dual S!, and by Corollary 4.4, we obtain 1 TrŽ.⌽Ž.0,t s 4, Ž.1yt 1 1 1 Tr ⌽ , t s Tr ⌽ , t s Tr ⌽ q , t s , ž/ž/ 22 ž/ž/222 žž2 / /Ž.1yt 1 1 Tr ⌽ , t s Tr ⌽ , t s 4 , ž/ž/44ž/ž/ 1yt 11 Tr ⌽ q , t s 4 . ž/ž/44 1qt ŽŽ.. For other g E4 ,Tr⌽ ,t also has a simple form. It is routine to check that TrŽ.Ž.Ž.⌽Ž. 12q , t [ Tr ⌽ Ž.Ž.Ž. 1⌽ 2, t s Tr ⌽ 1, t Ä4Ä4 for all 12g 1r4, r4, 1r4 q r4 and g 1r2, r2, 1r2 q r2 . De- tails are left to the interested reader. 5. FIXED SUBRINGS A group action of G on aŽ. graded ring A is a group homomorphism f: Ž. Ž. GªAut A wor F: G ª Autgr A in the graded casex . We will simply write g for the image fgŽ.. For example, we will write TrŽ.g, t instead of TrŽŽ.fg,t .. The fixed subring is G A [ Ä4a g A ¬ gaŽ.sa,᭙ggG. THE TRACE OF GRADED AUTOMORPHISMS 369 A great deal has been done for fixed subrings. For the commutative case, seewx 4 , and for the noncommutative case, w 12 x . In this section we will discuss some basic properties of fixed subrings for noncommutative graded algebras. In the rest of this section we assume that G is a finite subgroup of AutŽ.A 1 and that<< G yg k. As noticed more than 100 years ago, there is a projection of A associated to G: y1 G:aª< LEMMA 5.1. Let A be an algebra and let G be defined as in Ž.5-1 . G G G 1. G is an A -bimodule projection from A to A , namely, A s A [ G G C, where C s ker GG, as A -bimodule and y1 y1 HtGŽ.Tr Ž , t . Tr < y1 TrAAGŽ.h, t s < THEOREM 5.3. Let A be a connected algebra such that kA has a finite Ž. Ä MFR. Suppose that G is a finite subgroup of Autgr A and H s h g Ž. ŽG. G4 Autgr A ¬ hA sA ., Then for e¨ery h g H ¨iewed as an element in G Autgr Ž A .Ž.,TrAGh,t is a rational function. Ž. Proof. By Theorem 2.3.3, TrA g, t is a rational function for all g g Autgr Ž.A . By Lemma 5.2, TrAGŽ.h, t is a rational function. Using Lemma 5.2, we can derive a formula for the Hilbert series of Veronese subrings. EXAMPLE 5.4. Veronese subrings of a connected algebra. Let A be a connected algebra and n be a positive integer. The nth Veronese subring Žn. A is [iinA . There are two obvious gradings: one is deg x s i and the Ä4 Žn. other is deg x s in for all x g Ain y 0 . Here we consider A as a graded subring of A and use the second grading. The Veronese subrings = can be viewed as fixed subrings as follows. Let r g k [ k y Ä40 and n Ä4 define grn: x ¬ rxfor all x g A y 0 . Then gris a graded automor- = phism of A. In fact, r ¬ gr is a group homomorphism from k to Autgr Ž.A with image lying in the center of AutgrŽ.A .If kcontains an nth primitive root n of 1, then the fixed subring of A under the group action G Ä g i ¬ i 0,...,n 14 is the Veronese subring AŽn.. Therefore s n s y 11ny1 ny1 ii HtAAŽn.Ž.sÝÝTr Ž.g, t s HAnŽ. t . nnn is0 is0 Note that the above formulaŽ. the left end equals the right end can be proved easily for any field k, without using group action and the trace function, where n is taken to be a nth root of 1 in ރ. Let h be any graded automorphism of A. Then h is also a graded automorphism of AŽ n..By Lemma 5.2, we have 11ny1 ny1 ii TrAAŽn.Ž.h, t s ÝÝTr Ž.hg, t s TrAnŽ.h, t . nnn is0 is0 Next we consider Frobenius algebras. PROPOSITION 5.5. Let A be a connected finite dimensional Frobenius algebra with the highest degree n and let G be a finite subgroup of Autgr Ž.A . If g < id for all g G, then AG is Frobenius. A n s A n g G n Proof. Let A s [is0 Binn. By assumption, B s A ( k. For every nonzero element x g Bin, xA yins AxyinsAwx16, 3.2 . Applying the G Ž. A-bimodule map GnLemma 5.1.1 , we have xB yins BxyinsB. Hence AGis Frobeniuswx 16, 3.2 THE TRACE OF GRADED AUTOMORPHISMS 371 Without the condition g < id , AG may not be Frobenius. The first A n s A n such counterexample was given inwx 14 . Here is an easy way to produce more counterexamples. By Example 5.4, the mth Veronese subring AŽ m. is a fixed subring if m g k. Let A be a finite dimensional Frobenius algebra with the highest degree n. Let m be an integer such that n s qm q i with Žm. 0-i-m, and dim Aqm ) 1. Then bywx 16, 3.2 , A is not Frobenius. If s Ais not isomorphic to kxwxrŽx., then such integers m always exist. Also Žm. note that if m ) n, then A s k is Frobenius. 6. GORENSTEIN PROPERTY If A is a connected noetherian PI algebra, then A having finite left and right injective dimension implies that A is Gorensteinwx 20, 1.1 . Hence the definition of Gorenstein propertyŽ. see Section 1 is equivalent to the classical one when A is connected and PIŽ. including commutative . A variation of Watanabe’s theoremwx 4, 5.3.2 states that if kV wxis the polynomial algebra over a finite dimensional vector space V and ⌳Ž.V *is 1 the exterior algebra, and if G is a finite subgroup of GLŽ.V with < PROPOSITION 6.1. Let A be a regular Koszul algebra of global dimension n and let be a graded automorphism of A. Then Ž y1 y1. n Ž. = 1. TrAA , t s at Tr , t for some a g k . !!Ž .Ž .Ž! . 2. TrAm AAAm , t s Tr , t rTr , yt . y1 y1 n !!ŽŽ . .Ž. Ž . 3. TrAmAAm , t sy1TrmAm ,t. 1n !!Žy.Ž. Ž. 4. HtAmAAsy1 HtmA. Ž. y1 5. Let G be a finite subgroup of Autgr A with<< G g k acting ! ! G naturally on A m A and letŽ A m A. be the fixed subring. Then 1 n ! GŽ y .Ž. ! GŽ. HtŽAmA.Žsy1 HtAmA.. Proof. 1. By Theorem 4.3, it is equivalent to show that ŽŽ .y1 y1. yn Ž . TrAA!! , t s bt Tr , t , which follows from Theorem 3.4 and the fact that A!is Frobeniuswx 16, 5.10 . Part 2 follows fromŽ. 2-4 and Theorem 4.3, part 3 follows from part 2 and Theorem 3.4, and part 4 is a special case of part 3. 372 JING AND ZHANG 5. ByŽ. 5-2 and part 3, we have y1 y1 y1 !G << 1 HtŽAmA.Ž.sGÝTrAmAŽ.m , t gG y1 y1y1 << ! sG ÝTrAmAŽ.Ž.m , t gG ny1 << ! syŽ.1 G ÝTrAmAŽ.m , t gG n !G syŽ.1 HtŽAmA. Ž.. As in the commutative case, part 5 is supporting evidence for a noncom- mutative version of Watanabe’s theorem. Naturally we have the following conjecture. Conjecture.IfAis a noncommutative noetherian Koszul regular alge- Ž. y1 Ž !.G bra and G is a finite subgroup of Autgr A with < LEMMA 6.2. Let A be a connected algebra and B be a connected Frobe- nius algebra. 1. TheŽ. right injecti¨e dimensions of A m B and A are the same. 2. A m B is Gorenstein if and only if A is. Proof. 1. Since k has global dimension zero and B is a finite dimen- sional algebra of injective dimension zero, by the Kunneth¨ formula for double complexes we have i i Ext AmBABŽ.Ž.Ž.M m N, A m B ( Ext M, A m Hom N, B Ž6-1. for every rightŽ. or left A-module M and every finite dimensional rightŽ or .Ž.Ž. left B-module N. Since Hom B N, B / 0 6-1 implies that injdim A F injdim A m B. It remains to show that i Ext AmBŽ.L, A m B s 0Ž. 6-2 for all A m B-modules L, and for all i ) injdim A.If LsMmN,it follows fromŽ. 6-1 . In general we consider a chain of submodules, l l 1 L > LJ > иии > LJ > LJ q s 0, Ž. where J s A m [iG1 Bi and l is the highest degree of B. Hence it sss1 s suffices to showŽ. 6-2 for all L [ LJ rLJ q. By definition, LJs0, and THE TRACE OF GRADED AUTOMORPHISMS 373 ss Ž. Ž. it implies that LAmBAB( L m k . Hence 6-2 follows from 6-1 . Thus injdim A s injdim A m B. Ž. 2. Letting M s kABand N s k in 6-1 , we obtain i i Ext AmBAŽ.Ž.Ž.k mBAABB, A m B ( Ext k , A m Hom k , B i (Ext AAŽ.k , A m k.Ž. 6-3 Hence A is Gorenstein if and only if A m B is. Let A be a regular Koszul algebra of global dimension n. Then k has a finite MFR of length n such that the ith free module in the free resolution n n Ž.mi Ž. Ž. Ž. is A yi , where mds 1. In particular, Ext k, A ( Ext k, k ( kn. The Koszul dual A!is Frobeniuswx 16, 5.10 with the highest degree n, and !Ž !.Ž. hence HomA kA!, A ( k yn . Therefore, by Lemma 6.2 and a graded version ofŽ. 6-3 , we obtain the following. COROLLARY 6.3. Let A be a regular Koszul algebra of global dimension n. Then ! 1. A m A is a Gorenstein algebra of injecti¨e dimension n. n ! 2. Ext !Žk ! , A A . k. Am A Am A m s Next we consider the Veronese subrings AŽm. of a Gorenstein ring A. We will give a necessary and sufficient condition for AŽm. being Goren- steinŽ. Proposition 6.5 . LEMMA 6.4. Let A be a noetherian connected algebra generated in degree 1 and let m be a positi¨e integer. If I is a graded injecti¨e module of A Žm. containing no finite dimensional submodules, then I is a graded injecti¨e Žm. module o¨er A . Proof. A graded module M is called torsion if it is a union of finite dimensional submodules and M is called torsion-free if it contains no nontrivial finite dimensional submodules. We denote by Tors the subcate- gory of Gr-A of all torsion modules and QGr-A the quotient category Gr-ArTors. Let : Gr-A ª QGr-A be the canonical functor and let : QGr-A ª Gr-A be a right adjoint to Žseew 2,Ž. 2.2.2x. . Similarly we have an adjoint pair of Žm. and Žm. between Gr-AŽm. and QGr-AŽ m.. Con- Žm.Žm. sider the fuhnctor Ž.᎐ : Gr-A ª Gr-A defined by taking the mth Veronese of graded modules. It induces a functor, also denoted by Ž.᎐ Žm.: Žm. Žm. Žm. QGr-A ª QGr-A by sending Ž.M to ŽM .. Hence we have a 374 JING AND ZHANG commutative diagram Žm. Ž. 6 Gr-A y Gr-AŽm. Žm. 6 6 Žm. Ž. 6 QGr-A y QGr-AŽm. Žm. Žm. Byw 2, 5.10Ž. 3x the bottom functor Ž.M ª ŽM .is a category equivalence; thus it sends injectives to injectives. By definition, it sends the shift ŽŽ..Mim to Žm.ŽMiŽm.Ž... Now let I be a graded injective A-module containing no finite dimen- sional submodules. Then I is torsion-free and byw 2, 7.1Ž. 3x , I s ŽŽ..I . Bywx 5, 4.5.4 , Ž.I is injective and hence Žm.ŽI Žm..is injective in QGr-AŽm.. Thus, byw 5, 4.5.3Ž. 2xw and 2, 7.1Ž. 3x , Žm.Žm.ŽIŽm..is an injective AŽm.- module. Byw 2,Ž. 2.2.2x , we have Žm. Žm. Žm. Žm. Žm. Žm. Žm. Ž.I s HomŽ.A , Ž.I Žm. Žm. Žm. Žm. sHomŽ.A , Ž.Ž.I i [igޚ Žm.Žm.Žm.Žm. s HomŽ. Ž.A , Ž.Ž.Ž.Ii2, 2.2.2 [igޚ sHomŽ. Ž.A , Ž.Ž.Iim [igޚ Ž.m sž/HomŽ.A, IiŽ.2, Ž 2.2.2 . [igޚ Žm. sI. Therefore I Žm.Žis an injective A m.-module. PROPOSITION 6.5. Let A be a noetherian connected Gorenstein algebra generated in degree 1. Suppose that n is the injecti¨e dimension of A and that n Ext Ž.Ž.k, A ( k l for some integer l. Let m be a positi¨e integer. 1. If m< l, then the Veronese subring AŽm. is a Gorenstein algebra of injecti¨e dimension n. Ž m. 2. If m ¦ l and if dim Ad ) 1 for all d ) 0, then A has infinite injecti¨e dimension. Proof. Consider a minimal graded injective resolution of AA: 0 1 ny1 n 0 ª AA ª I ª I ª иии ª I ª I ª 0. THE TRACE OF GRADED AUTOMORPHISMS 375 i i Since Ext Ž.k, A s 0 for each i - n, I Ž.for i - n contains no finite n dimensional submodules. Bywx 23, 0.3.3 , I ( A*Ž.l . Taking the mth Veronese submodules, we have Žm. 01Ž.mm Ž. n1Ž.m Ž.m 0ªAªŽ.Iª Ž.Iªиии ª ŽI y .ª A* Ž.l ª 0. Ž.6-4 By Lemma 6.4, every ŽI i.Žm.Ž.for i - n is an injective AŽm.-module and obviously it contains no finite dimensional submodules. In particular,Ž. 6-4 i Ž Ž m.. is a part of an injective resolution and hence ExtAŽ m. k, A s 0 for all i - n. Žm. Žm. Žm. 1. If ml< , then A*Ž.l s ŽA .Ž* lrm ., which is an injective A - module. HenceŽ. 6-4 is an injective resolution of AŽm.and the injective Žm. dimension of A is no more than n. Applying Hom AAŽm.ŽŽ.Ž.k m., ᎐ to 6-4 , i Žm. n Žm. Ž.Ž . Ž.Ž . we obtain that Ext AAmk, A s 0 for all i / n and Ext mk, A s Žm. klŽ.rm. Therefore A is a Gorenstein algebra of injective dimension n. 2. Suppose that injdim AŽm. - ϱ and that m does not divide l.We i Žm. Ž. Ž.Ž . will prove that dim AsAs 1 for some s ) 0. By 6-4 , Ext m k, A s 0 iŽm.iynŽm. Ž.ŽŽ..ŽŽ.. for all i - n and Ext AAmk, A ( Ext mk, A* l for all i G n. Since AlŽ.Žm. is locally finite and left bounded, by the k-linear graded duality, iy n Žm. iyn Žm. Ž .ŽŽŽ.. .ŽŽ . . Ext AAm k, A* l ( Ext mA yl , k for all i G n.Hence iŽm. Žm. Ext AŽm.ŽŽ.k, A* l .is finite dimensional for all i and bywx 23, 0.3.1 , A is n Žm. Ž.ŽŽ.. Gorenstein. By letting i s n we also obtain Ext A mk, A* l Žm. Žm. Ž .ŽŽ. . (Hom A m Al ,k /0. Hence the injective dimension of A is n Žm. 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