The prime spectra of relative stable module categories

For a finite group G and an arbitrary commutative ring R, Broué has placed a Frobenius exact structure on the category of finitely generated RG-modules by taking the exact sequences to be those that split upon restriction to the trivial subgroup. The corresponding stable category is then tensor triangulated. In this paper we examine the case R = S/tn, where S is a discrete valuation ring having uniformising parameter t. We prove that the prime ideal spectrum (in the sense of Balmer) of this ‘relative’ version of the stable module category of RG is a disjoint union of n copies of that for kG, where k is the residue field of S

Comparisons between singularity categories and relative stable categories of finite groups

We consider the relationship between the relative stable category of 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