2016/01/12 by Nicolas Ricka, Ricka, Nicolas
Mathematics · #19L41 #55P42 #55S10 #Advanced Topics in Algebra #Algebraic Topology (math.AT) #Algebraic structures and combinatorial models #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology #math.AT #msc:19L41 #msc:55P42 #msc:55S10
paper · pdf · doi:10.48550/arxiv.1601.03098
27 pages, comments are welcome
openalex publication_date 2016/01/12 · arxiv created 2016/12/08 · arxiv updated 2016/12/09 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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.