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

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 ∞-category, as the totalization of a cosimplicial ∞-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 A(1)-modules, where A(1) is the subalgebra of the Steenrod algebra generated by the two first Steenrod squares.2010 Mathematics Subject Classification. 55S10,55P42,19L41. Key words and phrases. Stable category of modules, Steenrod algebra, Homotopical descent. The author is indebted to Akhil Mathew for suggesting such an approach to the study of Picard groups of Hopf algebras.