Cohomologically proper stacks over Zp̳ : algebra, geometry and representation theory

Abstract

Abstract In this thesis, we study a class of so-called cohomologically proper stacks from various perspectives, mainly concentrating on the p-adic context. Cohomological properness is a relaxed properness condition on a stack which roughly asks the cohomology of any coherent sheaf to be finitely generated over the base. We extend some of the techniques available for smooth proper schemes to smooth cohomologically proper stacks, featuring in particular recently developed theory of prismatic co-homology and the classical Deligne-Illusie method for the Hodge-to-de Rham degeneration. As main applications we prove the Totaro's conjectural inequality between the dimensions of the de Rham and the singular F[subscript p]-cohomology of the classifying stack of a reductive group, compute the ring of prismatic characteristic classes at non-torsion primes and give some new examples of the Hodge-to-de Rham degeneration for stacks in characteristic 0. We also study some descent properties of certain Brauer group classes on conical resolutions, a question having some applications to the theory of Fedosov quantizations in characteristic p. Some surprising results about the G[subscript m]-weights of differential 1-forms that are obtained along the way, originally motivated the attempt to generalize the integral p-adic Hodge theory to the setting of cohomologically proper stacks.