Higher displays arising from filtered de Rham-Witt complexes

TL;DR

This paper establishes a new structure called higher displays on crystalline cohomology for certain schemes, linking it to the Nygaard filtration and providing explicit descriptions in relative settings, with implications for deformation theory.

Contribution

It introduces higher displays as a relative analogue of Fontaine's strongly divisible lattices on crystalline cohomology, and describes their explicit structure via the Nygaard filtration.

Findings

Crystalline cohomology admits a higher display structure under certain conditions.

Explicit description of the relative display using the Nygaard filtration.

Existence of a crystal of relative displays when the mod p reduction has a smooth, versal deformation space.

Abstract

For a smooth projective scheme $X$ over a ring $R$ on which $p$ is nilpotent that meets some general assumptions we prove that the crystalline cohomology is equipped with the structure of a higher display which is a relative version of Fontaine's strongly divisible lattices. Frobenius-divisibilty is induced by the Nygaard filtration on the relative de Rham-Witt complex. For a nilpotent PD-thickening $S/R$ we also consider the associated relative display and can describe it explicitly by a relative version of the Nygaard filtration on the de Rham-Witt complex associated to a lifting of $X$ over $S$. We prove that there is a crystal of relative displays if moreover the mod $p$ reduction of $X$ has a smooth and versal deformation space.