The tt-geometry of permutation modules. Part I: Stratification

Abstract

We consider the derived category of permutation modules over a finite group, in positive characteristic. We stratify this tensor triangulated category using Brauer quotients. We describe the set underlying the tt-spectrum of compact objects, and discuss several examples.Permutation modules. Among the easiest representations to construct, permutation modules are simply the k-linearizations k(X) of G-sets X. And yet they play an important role in subjects as varied as derived equivalences [Ric96], Mackey functors [Yos83], or equivariant homotopy theory [MNN17], to name a few. The authors' original interest stems from yet another connection, namely the one with Voevodsky's theory of motives [Voe00], specifically Artin motives. For a gentle introduction to these ideas, we refer the reader to [BG21].We consider a 'small' tensor triangulated category, the homotopy category (1.2) K(G) := K b perm(G; k)

Quillen stratification in equivariant homotopy theory

“…In general, this is a problem that poses substantial difficulties, as witnessed by the case of Sp G for which we can still only resolve this question for abelian groups [BHN + 19]. To illustrate this point further, note that Theorem C, (a) gives a short and independent proof of a recent result of Balmer-Gallauer [BG22b] for cyclic p-groups, the case relevant for the motivic tt-geometry of Artin motives over finite fields. However, it requires additional techniques to determine the remaining specialization relations in Figure 3, and this corresponds precisely to the gluing data between the strata.…”