Local study of stable module categories via tensor triangulated geometry

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.

23 pages, all comments are welcome, v2 fixed some typos

References in corpus (2)

• Derived Algebraic Geometry III: Commutative Algebra
• The spectrum of prime ideals in tensor triangulated categories