Mackey Algebras which are Gorenstein

Abstract

We complete the picture available in the literature by showing that the integral Mackey algebra is Gorenstein if and only if the group order is square-free, in which case it must have Gorenstein dimension one. We illustrate this result by looking in details at the examples of the cyclic group of order four and the Klein four group.Let G be a finite group and R a commutative ring of coefficients. The Mackey algebra µ R (G), introduced in [TW95], is a finite-rank free R-algebra whose representations form precisely the category of Mackey functors for G with R-coefficients. It can be defined as the endomorphism algebra of H≤G G/H in the category of finite G-sets and R-linear combinations of spans of G-maps. In this note we will say that a (unital, associative, non-necessarily commutative) ring S is Gorenstein 1 if its injective dimension over itself as a left module is finite. The latter number n is the Gorenstein dimension of S, and we will sometimes say for short that S is n-Gorenstein. Famously, Gorenstein rings are ubiquitous in mathematics: see [Bas63] for the classical case of local commutative rings, and e.g. [DSŠ17] for a wide-ranging look at non-commutative rings and small categories.The goal of this note is to clarify the literature on the following point:Theorem 1. The integral Mackey algebra µ Z (G) is Gorenstein if and only if the order of G is square-free, in which case it has Gorenstein dimension one.Note that this contradicts [Gre92, Lemma 2.2], which states that projective Mackey functors over any finite group have injective dimension one. The lemma in question was used to prove that the the finitistic dimension of µ Z (G) equals one, and a different proof of the latter theorem was provided (together with several other similar results) by Bouc, Stancu and Webb; see [BSW17, Theorem 1.2]. Still, the question whether µ Z (G) is always Gorenstein was left open. As it turns out, it is Gorenstein precisely for groups G with square-free order. These are also the only groups for which µ Z (G) can be made into a symmetric Z-algebra, by a theorem of Rognerud [Rog15] (we will use this result, so more on this below); and they are also precisely the groups for which the Burnside ring B(G) is (1-)Gorenstein or even a symmetric Z-algebra, by theorems of Krämer [Krä74] and Gustafson [Gus77].For one direction of our theorem we actually know somewhat more:Date

Thick tensor ideals of right bounded derived categories

“…It is thus a main subject in tensor triangular geometry to determine/describe the Balmer spectrum of a given tensor triangulated category. Such studies have been done for these thirty years considerably widely; one can find ones at least in stable homotopy theory [7,13,16], commutative algebra [15,19,23], algebraic geometry [2,22,24], modular representation theory [6,8,9,14] and motivic theory [12,21].…”