2022/11/30 by Guo, Haoyang, Kubrak, Dmitry, Prikhodko, Artem
#Algebraic Geometry (math.AG) #FOS: Mathematics #Number Theory (math.NT)
paper · doi:10.48550/arxiv.2211.17227
In this follow-up paper we show that smooth Hodge-proper stacks over \mathcal OK are \mathbb Qp-locally acyclic: namely the natural map between étale \mathbb Qp-cohomology of the algebraic and Raynaud generic fibers is an equivalence. This establishes the \mathbb Qp-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 OK, its \mathbb Qp-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 OK: here we only require first d de Rham cohomology groups be finitely-generated over \mathcal OK. As an application, we deduce a certain purity-type statement for étale \mathbb Qp-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 OK in the schematic setting.