Monoidal model structures over infinite groups

TL;DR

The paper extends monoidal model structures to module categories over group algebras $kG$ for infinite groups by combining Hovey’s cotorsion-pair framework with Gorenstein homological algebra. It leverages Kropholler’s ${\bf LH}$ and Talelli’s $\Phi$, $\Phi_{flat}$ operations to enlarge the class of groups for which a Gorenstein projective model structure exists and yields a compactly generated tensor triangulated homotopy category. Key contributions include establishing tensor-closure properties for Gorenstein projectives over suitable $k$, constructing compact generators via induction from subgroups, and proving FP$_{\infty}$-type equivalences for finitely generated Gorenstein projective modules in broad group classes. The results provide a systematic approach to derive and analyze monoidal, compactly generated homotopy theories for derived categories of $kG$-modules across large infinite group classes, with concrete consequences for representation theory and homological algebra.

Abstract

The stable category of modules over the algebra of a finite group with coefficients in a field is a compactly generated tensor triangulated category, that has been studied extensively in representation theory. In this paper, we provide a plethora of infinite groups G, for which the category of kG-modules (where k is a commutative coherent ring of finite global dimension) admits a monoidal model structure, in the sense of Hovey, whose associated homotopy category is a compactly generated tensor triangulated category. To that end, we use a technique recently introduced by the authors, which is based on Kropholler's operation LH and the second author's operation Φ.

Theorems & Definitions (21)

• Proposition 2.1
• Lemma 2.2
• Theorem 2.3
• Corollary 2.4
• Corollary 2.5
• Lemma 3.1
• Lemma 3.2
• Lemma 3.3
• Lemma 3.4
• Lemma 3.5