$p$-adic Hodge theory for Artin stacks

TL;DR

This paper extends integral p-adic Hodge theory to Artin stacks using prismatic cohomology, establishing new links between étale cohomology, Galois representations, and stacky structures, with applications to inequalities and characteristic classes.

Contribution

It develops a p-adic Hodge theory framework for Artin stacks, including new results on cohomology equivalences and applications to conjectural inequalities.

Findings

Galois representations are crystalline for Hodge-proper stacks.

The natural map between étale cohomology of algebraic and Raynaud fibers is often an equivalence.

Established a theory of A_inf-characteristic classes.

Abstract

This work is devoted to the study of integral $p$-adic Hodge theory in the context of Artin stacks. For a Hodge-proper stack, using the formalism of prismatic cohomology, we establish a version of $p$-adic Hodge theory with the \'etale cohomology of the Raynaud generic fiber as an input. In particular, we show that the corresponding Galois representation is crystalline and that the associated Breuil-Kisin module is given by the prismatic cohomology. An interesting new feature of the stacky setting is that the natural map between \'etale cohomology of the algebraic and the Raynaud generic fibers is often an equivalence even outside of the proper case. In particular, we show that this holds for global quotients $[X/G]$ where $X$ is a smooth proper scheme and $G$ is a reductive group. As applications we deduce Totaro's conjectural inequality and also set up a theory of $A_{\mathrm{inf}}$-characteristic classes.