Crystals and Chern classes

TL;DR

This paper investigates the properties of Chern classes of sheaves with crystal structures within crystalline and prismatic cohomology, revealing vanishing and torsion phenomena with bounds related to Bernoulli numbers.

Contribution

It establishes vanishing of Chern classes in crystalline cohomology and torsion bounds in prismatic cohomology for sheaves with crystal structures, advancing understanding of their cohomological behavior.

Findings

Chern classes vanish in crystalline cohomology.

Chern classes are torsion with bounded exponents in prismatic cohomology.

Formulation of questions on syntomic Chern classes.

Abstract

The goal of this paper is to study the Chern classes of coherent sheaves (and more generally perfect complexes) that admit crystal structures in the setting of crystalline cohomology and more generally relative prismatic cohomology. In the former theory, we show that the Chern classes vanish on the nose; in the latter theory, we show the classes are torsion with uniformly bounded exponents determined by suitable Bernoulli numbers. We also formulate some questions about syntomic Chern classes of such sheaves.