Local study of stable module categories via tensor triangulated geometry

TL;DR

This paper explores the stable module categories of certain Hopf algebras using tensor-triangulated geometry, revealing new structural insights and applications to Picard groups and localizations.

Contribution

It introduces a framework for analyzing stable categories of Hopf algebras via spectrum covers and $ extit{ extbf{infinity}}$-stacks, extending Margolis' work.

Findings

Identification of spectrum covers related to Margolis' work

Construction of $ extit{ extbf{infinity}}$-stacks from localizations

Applications to Picard groups and stable category localizations

Abstract

We investigate the particular properties of the stable category of modules over a finite dimensional cocommutative graded connected Hopf algebra $A$, via tensor-triangulated geometry. This study requires some mild conditions on the Hopf algebra $A$ under consideration (satisfied for example by all finite sub-Hopf-algebras of the modulo $2$ Steenrod algebra). In particular, we study some particular covers of its spectrum of prime ideals $\mathrm{Spc}(A)$, which are related to Margolis' Work. We then exploit the existence of Margolis' Postnikov towers in this situation to show that the localization at an open subset $U$ of $\mathrm{Spc}(A)$, for various $U$, assembles in an $\infty$-stack. Finally, we turn to applications in the study of Picard groups of Hopf algebras and localizations in the stable categories of modules.