Fiberwise building and stratification in tensor triangular geometry

TL;DR

This paper develops a fiberwise criterion for stratification in tensor triangular geometry, enabling the analysis of localizing tensor ideals via coproduct-preserving tt-functors, with applications to derived categories of group modules.

Contribution

It introduces conditions on tt-functors that determine localizing tensor ideals fiberwise, extending stratification results to new categories like permutation modules and group schemes.

Findings

Established a fiberwise criterion for stratification in tt-categories.

Proved the big derived category of permutation modules is stratified.

Extended methods to representations of finite group schemes over Noetherian bases.

Abstract

We establish conditions on a family of coproduct-preserving tt-functors $f_i\colon \mathcal{T}\to \mathcal{T}_i$ between tt-categories with small coproducts, ensuring that the localizing tensor ideal generated by an object $x \in \mathcal{T}$ is determined by those objects whose image under each $f_i$ lies in the localizing tensor ideal generated by $f_i(x)$ for all $i$. This leads to a fiberwise criterion for stratification in the setting of rigidly-compactly generated tt-categories. As an application, we prove that the big derived category of permutation modules for a finite group over an arbitrary Noetherian base is stratified. Moreover, our methods extend to the category of representations of a finite group scheme over a Noetherian base, thereby recovering a recent result from the literature.