Finite generation of Tate cohomology

📝 Abstract

Let G be a finite group and let k be a field of characteristic p. Given a finitely generated indecomposable non-projective kG-module M, we conjecture that if the Tate cohomology $\HHHH^*(G, M)$ of G with coefficients in M is finitely generated over the Tate cohomology ring $\HHHH^*(G, k) $, then the support variety V_G(M) of M is equal to the entire maximal ideal spectrum V_G(k). We prove various results which support this conjecture. The converse of this conjecture is established for modules in the connected component of k in the stable Auslander-Reiten quiver for kG, but it is shown to be false in general. It is also shown that all finitely generated kG-modules over a group G have finitely generated Tate cohomology if and only if G has periodic cohomology.

💡 Analysis

Let G be a finite group and let k be a field of characteristic p. Given a finitely generated indecomposable non-projective kG-module M, we conjecture that if the Tate cohomology $\HHHH^*(G, M)$ of G with coefficients in M is finitely generated over the Tate cohomology ring $\HHHH^*(G, k) $, then the support variety V_G(M) of M is equal to the entire maximal ideal spectrum V_G(k). We prove various results which support this conjecture. The converse of this conjecture is established for modules in the connected component of k in the stable Auslander-Reiten quiver for kG, but it is shown to be false in general. It is also shown that all finitely generated kG-modules over a group G have finitely generated Tate cohomology if and only if G has periodic cohomology.

📄 Content

arXiv:0804.4246v5 [math.RT] 13 Jun 2010 FINITE GENERATION OF TATE COHOMOLOGY JON F. CARLSON, SUNIL K. CHEBOLU, AND J´AN MIN´AˇC Dedicated to Professor Luchezar Avramov on his sixtieth birthday. Abstract. Let G be a finite group and let k be a field of characteristic p. Given a finitely generated indecomposable non-projective kG-module M, we conjecture that if the Tate cohomology ˆH ∗(G, M) of G with coefficients in M is finitely generated over the Tate cohomology ring ˆH ∗(G, k), then the support variety VG(M) of M is equal to the entire maximal ideal spectrum VG(k). We prove various results which support this conjecture. The converse of this conjecture is established for modules in the connected component of k in the stable Auslander-Reiten quiver for kG, but it is shown to be false in general. It is also shown that all finitely generated kG- modules over a group G have finitely generated Tate cohomology if and only if G has periodic cohomology.

1. Introduction Tate cohomology was introduced by Tate in his celebrated paper [14] where he proved the main theorem of class field theory in a remarkably simple way using Tate cohomology. After Cartan and Eilenberg’s treatment [9] of Tate cohomology and Swan’s basic results on free group actions on spheres [13], Tate cohomology became one of the basic tools in current mathematics. Our aim in this paper is to address a fundamental question: when is the Tate cohomology with coefficients in a module finitely generated over the Tate cohomology ring of the group. Suppose G be a finite group and let k be a field of characteristic p. If M is a finitely generated kG-module, then a well-known result in group cohomology due to Golod, Evens and Venkov says that H∗(G, M) is finitely generated as a graded module over H∗(G, k). Our goal is to investigate a similar finite-generation result for Tate cohomology. More precisely, if M is a finitely generated kG-module, then we want to know whether the Tate cohomology ˆH ∗(G, M) of G with coefficients in M is finitely generated as a graded module over the Tate cohomology ring ˆH ∗(G, k). In Section 2 we explain one reason for being interested in this problem. In general, it seems that the Tate cohomology of a module is seldom finitely generated, which is a striking contrast to the situation with ordinary cohomology. However, there are some notable exceptions. Our investigations have led us to a conjecture which we state as follows. Date: August 18, 2021. 2000 Mathematics Subject Classification. Primary 20C20, 20J06; Secondary 55P42. Key words and phrases. Tate cohomology, finite generation, periodic modules, support varieties, stable module category, almost split sequence. The first author is partially supported by a grant from NSF and the third author is supported from NSERC. 2 JON F. CARLSON, SUNIL K. CHEBOLU, AND J´AN MIN´AˇC Conjecture 1.1. Let G be a finite group and let M be an indecomposable finitely gen- erated kG-module such that H∗(G, M) ̸= {0}. If ˆH ∗(G, M) is finitely generated over ˆH ∗(G, k), then the support variety VG(M) of M is equal to the entire maximal ideal spectrum VG(k) of the group cohomology ring. The condition that H∗(G, M) ̸= 0 is certainly necessary since there are many modules with proper support varieties and vanishing cohomology [5]. Perhaps it is necessary to require that M lies in the thick subcategory of the stable category generated by k. We have evidence for the conjecture from two directions. First, the results of [3] indicate that products in negative Tate cohomology are often zero and we can use this to develop boundedness conditions on finitely generated modules over Tate cohomology. Under the right circumstances, these conditions imply infinite generation of the Tate cohomology. Secondly, for groups having p-rank at least two, we can show that many periodic modules fail to have finitely generated Tate cohomology. Indeed, we prove that for any such group there is at least one module whose Tate cohomology is not finitely generated. Hence, the only groups having the property that every finitely generated kG-module has finitely generated Tate cohomology have p-rank one or zero. On the other hand, in general, there are numerous modules which have finitely gener- ated Tate cohomology. In the last section we show some ways in which these modules can be constructed. It turns out that the constructions are consistent with the Auslander- Reiten quiver for kG-modules. That is, if a nonprojective module in a connected com- ponent of the Auslander-Reiten quiver has finitely generated Tate cohomology, then so does every module in that component. The paper is organized as follows. We begin in Section 2 by explaining how we had naturally arrived at the problem of finite generation of Tate cohomology. Sections 3 and 4 deal with modules whose Tate cohomology is not finitely generated and contain proofs in the two directions mentioned above. In Section 5 we prove affirmative results which provides a good source of modules whose Tate cohomology is fi

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