Comparisons between singularity categories and relative stable categories of finite groups

TL;DR

This paper explores the relationship between the relative stable category and the singularity category of group algebras over various coefficient rings, revealing structural connections and decompositions.

Contribution

It establishes links between these categories for different types of coefficient rings, including self-injective and finite global dimension cases, and analyzes their monoidal and spectral properties.

Findings

Shared Verdier quotient when the ring is self-injective

Semi-orthogonal decomposition for rings with finite global dimension

Partial compatibility of the decomposition with monoidal structure

Abstract

We consider the relationship between the relative stable category of Benson, Iyengar, and Krause and the usual singularity category for group algebras with coefficients in a commutative noetherian ring. When the coefficient ring is self-injective we show that these categories share a common, relatively large, Verdier quotient. At the other extreme, when the coefficient ring has finite global dimension, there is a semi-orthogonal decomposition, due to Poulton, relating the two categories. We prove that this decomposition is partially compatible with the monoidal structure and study the morphism it induces on spectra.