The Avrunin and Scott theorem and a truncated polynomial algebra.
Let \(G\) be a finite group and \(k\) be an algebraically closed field of characteristic \(p>0\). Over the past 30 years, the cohomology and representation theory of \(kG\)-modules has been greatly influenced by the idea of cohomological support varieties which provide a geometric way to study cohomology. The success of this tool in the modular representation theory of finite groups has led to its use in many other areas. One key to this success has been the ability to provide non-cohomological descriptions of support varieties. The first such description, in the case of finite groups, was conjectured by J. Carlson. More precisely, for an elementary Abelian \(p\)-group, Carlson introduced the notion of a rank variety for a module which is defined representation-theoretically without using cohomology. For the trivial module \(k\), the support variety and rank variety are naturally isomorphic. For any finite-dimensional module, the support (rank) variety is a subvariety of the support (rank) variety for the trivial module. Carlson conjectured that under this isomorphism, for any finite-dimensional module, the support variety and rank varieties should correspond. This conjecture was first proven by G. Avrunin and L. Scott and is the theorem referred to in the title of the paper. The goal of this paper is to investigate the analogous question in the context of modules for a truncated polynomial algebra \(\Lambda_m=k[X_1,X_2,\dots,X_m]/(X_i^2)\) over an algebraically closed field \(k\) of arbitrary characteristic. For \(\Lambda_m\), support varieties are defined using Hochschild cohomology. This is based on work of \textit{N. Snashall} and \textit{Ø. Solberg} [Proc. Lond. Math. Soc. (3) 88, No. 3, 705--732 (2004; Zbl 1067.16010)] and of the authors with \textit{N. Snashall, Ø. Solberg} and \textit{R. Taillefer} [\(K\)-Theory 33, No. 1, 67--87 (2004; Zbl 1116.16007)]. Rank varieties were introduced by the authors [in Math. Z. 247, No. 3, 441--460 (2004; Zbl 1078.16008)]. The main result of this paper is to show that for any finite-dimensional \(\Lambda_m\)-module the support variety and rank variety again correspond under an isomorphism identifying the two varieties for the trivial module. The authors use an interesting new approach of identifying both the support and rank varieties in terms of stable maps. With this description, they show that the two varieties correspond. The authors also show that their methods provide a new proof of the Avrunin and Scott theorem for elementary Abelian groups.
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