The stable Picard group of Hopf algebras via descent, and an application

TL;DR

This paper develops a descent spectral sequence approach to compute the Picard group of the stable category of modules over certain finite-dimensional cocommutative Hopf algebras, with applications to Steenrod algebra modules.

Contribution

It introduces a novel spectral sequence framework for calculating Picard groups via descent, applicable to Hopf algebras and their module categories.

Findings

Spectral sequence effectively computes Picard groups in specific cases.

Application to $ ext{A}(1)$-modules demonstrates practical utility.

Provides solutions to lifting problems in module categories.

Abstract

Let $A$ be a cocommutative finite dimensional Hopf algebra over the field with two elements, satisfying some mild hypothesis. We set up a descent spectral sequence which computes the Picard group of the stable category of modules over $A$. The starting point is the observation that the stable category of $A$-modules can be reconstructed, as an $\infty$-category, as the totalization of a cosimplicial $\infty$-category whose layers are related to the stable categories of modules over the quasi-elementary sub-Hopf-algebras of $A$. This leads to a spectral sequence computing the Picard group which, in some cases, is completely understood. This also leads to a spectral sequence answering a lifting problem in the category of $A$-modules. We then show how to apply this machinery to compute Picard groups and solve lifting problems in the case of $\mathcal{A}(1)$-modules, where $\mathcal{A}(1)$ is the subalgebra of the Steenrod algebra generated by the two first Steenrod squares.