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

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.