Dieudonné Theory via Cohomology of Classifying Stacks

Mondal, Shubhodip

doi:10.48550/arxiv.2002.05413

Cited by 1 publication

(2 citation statements)

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“…Example 2.5.6. In [Mon20] it is shown that for a finite group G over a perfect field k of characteristic p the corresponding Dieudonné module M (G) can be expressed as H 2 crys (BG/k) with the natural Frobenius action on it.…”

Section: Thenmentioning

confidence: 99%

“…At the same time the idea of using classifying stacks to construct examples of schemes with a certain pathological behavior of cohomology (which goes back at least to [Ser58]) has bore some fresh fruit with the construction of counterexamples to degeneration of the HKR spectral sequence in char p and Hodge-de Rham degeneration over ramified mixed characteristic local rings in [ABM19] and [Li20] respectfully. Besides that in [Mon20] the classical notion of a Dieudonné module (as well as the more recent prismatic Dieudonné module of [AB19]) of a p-divisible group was given a natural interpretation in terms of crystalline (resp. prismatic) cohomology of the corresponding classifying stack.…”

Section: Introductionmentioning

confidence: 99%

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$p$-adic Hodge theory for Artin stacks

Kubrak¹,

Prikhodko²

2021

Preprint

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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 étale 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 étale 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 inf -characteristic classes.

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Section: Thenmentioning

confidence: 99%