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

TL;DR

This paper proves that smooth Hodge-proper stacks over $\\mathcal O_K$ are $\mathbb Q_p$-locally acyclic, establishing conjectures about their étale cohomology and Galois representations, with extensions to $d$-de Rham-proper stacks.

Contribution

It demonstrates the $\mathbb Q_p$-local acyclicity of smooth Hodge-proper stacks and extends results to $d$-de Rham-proper stacks, confirming conjectures and providing new cohomological insights.

Findings

Étale $\mathbb Q_p$-cohomology of Hodge-proper stacks matches Raynaud generic fibers.

Smooth Artin stacks with Hodge-proper models have crystalline Galois representations.

Partial results on cohomology purity and crystallinity in more general settings.

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 \'etale $\mathbb Q_p$-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 $\mathbb Q_p$-\'etale 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 \'etale $\mathbb Q_p$-cohomology of Raynaud generic fiber, as well as crystallinity of a first several \'etale cohomology groups in the presence of a Cohen--Macauley model over $\mathcal O_K$ in the schematic setting.