Rational $p$-adic Hodge theory for $d$-de Rham-proper stacks

Abstract

In this follow-up paper we show that smooth Hodge-proper stacks over ${\mathcal{O}}_K$ are ${\mathbb{Q}}_p$-locally acyclic: namely the natural map between étale ${\mathbb{Q}}_p$-cohomology of the algebraic and Raynaud generic fibers is an equivalence. This establishes the ${\mathbb{Q}}_p$-case of general conjectures made in our previous work. As a corollary, we get that if a smooth Artin stack over $K$ has a smooth Hodge-proper model over ${\mathcal{O}}_K$, its ${\mathbb{Q}}_p$-étale cohomology is a crystalline Galois representation. We then also establish a truncated version of the above results in more general setting of smooth $d$-de Rham-proper stacks over ${\mathcal{O}}_K$: here we only require first $d$ de Rham cohomology groups be finitely-generated over ${\mathcal{O}}_K$. As an application, we deduce a certain purity-type statement for étale ${\mathbb{Q}}_p$-cohomology of Raynaud generic fiber, as well as crystallinity of a first several étale cohomology groups in the presence of a Cohen--Macauley model over ${\mathcal{O}}_K$ in the schematic setting.

Appendix by Haoyang Guo