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On the Trace of Graded Automorphisms 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 View metadata, citation and similar papers at core.ac.uk brought to you by CORE and provided by Elsevier - Publisher Connector James J. Zhang² 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 s s Ž.s Ý Žs .d phism . The trace of is defined to be Tr , t s d G 0 tr ¬ Atd . We prove that TrŽ.s , 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. Q 1997 Academic Press 1. INTRODUCTION Throughout the paper k will denote a field. A Z-graded vector space ` M s [d g Z Mddis called locally finite if dim M - for all d, where dim is always the dimension of a k-vector space. An N-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]. ² Work supported in part by the NSF. E-mail: [email protected]. 353 0021-8693r97 $25.00 Copyright Q 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 Z d d HtMdŽ.sÝdim Mt. dgZ 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 g with respect to a grading of g, then the series HqM Ž.is also called the q-dimension or character in the representation theory of Kac]Moody Lie algebraswx 9 . If g is an affine Kac]Moody algebra, the Ž. 2pit series HqM is a modular function of t with q s e . Moreover there is a more general notion}the 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 . s s Ž. A linear map of M s [d g Z Mdddis called graded if M ; M for all d g Z. The trace of a graded map s is defined to be < d TrMŽ.s , t s ÝtrŽ.s Mdt . dgZ We will omit M from TrMMŽ.s , 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.s , 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 s is a graded algebra endomorphism of A. Let Ci denote the induced matrix of s 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.s , 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.s , t has nice properties. For example, Tr Ž.s , 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.s , t is deter- Ž < . mined by finitely many terms Tr s 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 Z 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 ??? ª M ª ??? ª M ª M ª 0Ž. 2-1 Ž. i is exact and that fi i G 0 are k-linear graded endomorphisms of M Ž.Ý Ž .i Ž.f 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 fi 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 ??? 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 fi ¬ i s 2,...,n j f1 ¬L and f1 ¬ L, f10, f 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.f¬i 0 for the exact sequence is0 y iMd s 2 1 0 0ªMdddªMªMª0 for every degree d g Z. 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 si be a graded endomorphism of i M for i s 1 and 2, respectively. Then 12ss 1 s 2 s 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- s morphism s of A and M g Gr-A we define the s-twisted module M such s that M s M as a graded vector space with the action m) a s ms Ž.a for all m g M, a g A. We define a functor on Gr-A associated to s by Ž. s Ž. Ž. FMs sMand Ffs sffor any f g HomGr- A M, N . It is easy to see that Fs is indeed a functor from the category Gr-A to itself. If furthermore s s is an automorphism, then Fs is an invertible functor and A is isomorphic to A as a graded right A-module.
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