On the trace of graded automorphisms (Q1355589)

Let \(M =\bigoplus_{d\in\mathbb{Z}}M_d\) be a graded vector space over a commutative field \(k\) with \(\dim_k(M_d)<\infty\) for every integer \(d\). For a graded endomorphism \(\sigma\), define its trace as \(\text{Tr}_M(\sigma,t)=\sum_{d\in\mathbb{Z}}\text{tr}(\sigma|M_d)t^d\). For \(\sigma=\text{Id}_M\), the trace is the Hilbert-Poincaré series. The authors prove that the trace \(\text{Tr}_A(\sigma,t)\) is a rational function if \(A\) is either a finitely generated commutative or right noetherian with finite global dimension or regular connected positively graded \(k\)-algebra (Theorem 2.3 and Corollary 2.4). This result is later extended to \(\text{Tr}_{A^G}(\sigma,t)\), where \(A^G\) is the fixed subring under a finite group \(G\) of graded automorphisms and \(\sigma(A^G)=A^G\) (Theorem 5.3). This gives a version of Molien's classical theorem. Here, regular means that \(A\) has finite global dimension \(n\) and \(\text{Ext}^i(k,A_A)=\text{Ext}^i(k,{_AA})=0\) for \(i\neq n\) and \(\text{Ext}^n(k,A_A)\cong\text{Ext}^n(k,{_AA})\cong k\), where \({_Ak_A}\cong A/\bigoplus_{n>0}A_d\). This notion stems from \textit{M. Artin} and \textit{W. Schelter} [Adv. Math. 66, 171-216 (1987; Zbl 0633.16001)]. If global dimension is replaced by injective dimension, then \(A\) is said to be Gorenstein. A reciprocity formula of the form \(\text{Tr}(\sigma^{-1},t^{-1})=at^l\text{Tr}(\sigma,t)\), for \(a\) a nonzero scalar in \(k\), is proved for a graded automorphism \(\sigma\) of a regular or Frobenius connected algebra \(A\) (Theorems 3.1 and 3.4). If \(A\) is a connected Koszul algebra with Koszul dual \(A^!\), then the formula \(\text{Tr}(\sigma,t)=\text{Tr}_{A^!}(\sigma^{\tau},-t)^{-1}\) is used to compute the trace for graded automorphisms of the Sklyanin algebra (Section 4). For the notion of Koszul dual, see the book by \textit{Yu. Manin} [Quantum groups and non-commutative geometry (1988; Zbl 0724.17006)]. The paper also contains the following conjecture: If \(A\) is a noncommutative noetherian Koszul regular algebra and \(G\) is a finite group of graded automorphisms of \(A\) with \(|G|^{-1}\in k\), then \((A\otimes A^!)^G\) is Gorenstein. This conjecture is proved for some special groups (Section 6).