Stratifying integral representations of finite groups

Abstract

We classify the localizing tensor ideals of the integral stable module category for any finite group G. This results in a generic classification of Z[G]-lattices of finite and infinite rank and globalizes the modular case established in celebrated work of Benson, Iyengar, and Krause. Further consequences include a verification of the generalized telescope conjecture in this context, a tensor product formula for integral cohomological support, as well as a generalization of Quillen's stratification theorem for group cohomology. Our proof makes use of novel descent techniques for stratification in tensortriangular geometry that are of independent interest.

Stratifying integral representations via equivariant homotopy theory

“…This result generalizes the main theorem of Benson, Iyengar, and Krause [BIK11a] for R a field, as well as [Bar21] for R a ring of integers in a number field, without depending on these earlier results. The proof uses ideas from our previous paper [Bar21], but amplifies them through the use of tools from equivariant homotopy theory.…”

“…This result generalizes the main theorem of Benson, Iyengar, and Krause [BIK11a] for R a field, as well as [Bar21] for R a ring of integers in a number field, without depending on these earlier results. The proof uses ideas from our previous paper [Bar21], but amplifies them through the use of tools from equivariant homotopy theory. It can be seen as a blueprint of similar classification results in spectral representation theory: that is, for derived categories of representations with coefficients in ring spectra.…”

“…Consider the category Lat(G, R) of R-linear Grepresentations whose underlying R-module is projective. Combined with the techniques of [Bar21], our main theorem leads to a 'generic' classification of objects in Lat(G, R):…”

“…Since such a classification up to isomorphism is inaccessible in general, we instead turn to a coarser classification problem. This takes place in a derived context, more specifically the derived category of R-linear G-representations Rep(G, R) from [Bar21]. While constructed homotopically, it also encompasses the algebraic category of representations.…”

“…This paper takes place in an appropriate framework of derived (or spectral) representation theory. The key category Rep(G, R) and its accompanying stable module category StMod(G, R) were introduced in [Bar21] as a suitable generalization of their more familiar algebraic cousins K(Inj(kG)) and StMod(G, k) from fields k to general ring (spectra) coefficients. A previous candidate was constructed by Benson, Iyengar, and Krause [BIK13], but was there found to be unsuitable for a cohomological stratification.…”