Hodge-to-de Rham degeneration for stacks

Abstract

We introduce a notion of a Hodge-proper stack and extend the method of Deligne-Illusie to prove the Hodgeto-de Rham degeneration in this setting. In order to reduce the statement in characteristic 0 to characteristic p, we need to find a good integral model of a stack (a so-called spreading), which, unlike in the case of schemes, need not to exist in general. To address this problem we investigate the property of spreadability in more detail by generalizing standard spreading out results for schemes to higher Artin stacks and showing that all proper and some global quotient stacks are Hodge-properly spreadable. As a corollary we deduce a (non-canonical) Hodge decomposition of the equivariant cohomology for certain classes of varieties with an algebraic group action.

On endomorphisms of the de Rham cohomology functor

“…In particular the Deligne-Illusie splitting in [KP21a], the Drinfeld splitting in [BL21] as well as the splitting induced by Theorem 1.1 must all agree.…”

$p$-adic Hodge theory for Artin stacks

“…One can also show that if X is Hodge-proper over O K the Hodge-to-de Rham spectral sequence for X K degenerates (Theorem 4.3.33). This gives a partial strengthening of results of [KP19] on the degeneration of Hodge-to-de Rham spectral sequence: namely, to show that it degenerates for a given smooth Artin stack over K it is enough to find a Hodge-proper model over O K . Comparing with the results of [KP19], it shows that it's enough to "Hodgeproperly spread out" a Hodge-proper stack to a single prime, rather than all but finite set of them.…”

“…This gives a partial strengthening of results of [KP19] on the degeneration of Hodge-to-de Rham spectral sequence: namely, to show that it degenerates for a given smooth Artin stack over K it is enough to find a Hodge-proper model over O K . Comparing with the results of [KP19], it shows that it's enough to "Hodgeproperly spread out" a Hodge-proper stack to a single prime, rather than all but finite set of them. Finally, it is also possible to describe the Breuil-Kisin module associated to…”

“…In [KP19] we used the notion of Hodge-properness to prove a version of the Hodge-to-de Rham degeneration in the context of Artin stacks in characteristic 0 via reduction to a large enough characteristic p (analogous to the work of Deligne-Illusie [DI87]). While that can be seen as an extension of a result related to the classical Hodge theory over C, in this paper we attempt to extend the results of p-adic Hodge theory to smooth Hodge-proper stacks over O K .…”

“…We start by recalling the notion of Hodge-proper stacks (introduced by the authors in [KP19]) in Section 1. The first two subsections (1.1, 1.2) contain various technical results about bounded below coherent modules and basic properties of Hodge-proper stacks respectfully.…”