Fiberwise building and stratification in tensor triangular geometry

2025/01/22 by Gómez, Juan Omar · 1 citation
#18F99 #18G65 #18G80 #20C10 #20C20 #Category Theory (math.CT) #FOS: Mathematics #Representation Theory (math.RT)

paper · doi:10.48550/arxiv.2501.12711

Abstract

We give conditions on a family of coproduct-preserving tt-functors f_i\colon T→ T_i between tt-categories with small coproducts, ensuring that the localizing tensor-ideal generated by an object x in T is built from those objects whose image under f_i lies in the localizing tensor-ideal generated by f_i(x) for all i. This allows us to provide a criterion for stratification on a fiberwise level when restricted to big tt-categories. As an application, we show that the big derived category of permutation modules for a finite group over an arbitrary Noetherian base is indeed stratified. Furthermore, our approach extends to the category of representations of a finite group scheme over a Noetherian base, thereby recovering a recent result from the literature.

Cited by

• The Balmer spectrum of integral permutation modules

Related

• Costratification and actions of tensor-triangulated categories
• Homological stratification and descent
• Lifting (co)stratifications between tensor triangulated categories
• Stratifying integral representations of finite groups
• Descent in tensor triangular geometry