Hodge-Witt decomposition of relative crystalline cohomology

TL;DR

This paper proves a Hodge-Witt decomposition for the relative crystalline cohomology of certain smooth, proper schemes over artinian local rings, extending understanding of their cohomological structures in algebraic geometry.

Contribution

It establishes the degeneration of the relative de Rham-Witt spectral sequence and the Hodge-Witt decomposition for a broad class of schemes, including abelian schemes and Calabi-Yau varieties.

Findings

Spectral sequence degenerates under general assumptions

Crystalline cohomology admits a Hodge-Witt decomposition

Includes examples like abelian schemes and K3 type varieties

Abstract

For a smooth and proper scheme over an artinian local ring with ordinary reduction over the perfect residue field we prove - under some general assumptions - that the relative de Rham-Witt spectral sequence degenerates and the relative crystalline cohomology, equipped with its display structure arising from the Nygaard complexes, has a Hodge-Witt decomposition into a direct sum of (suitably Tate-Twisted) multiplicative displays. As examples our main results include the cases of abelian schemes, complete intersections, varieties of K3 type and some Calabi-Yau $n$-folds.