Torsion in the crystalline cohomology of singular varieties

TL;DR

This paper demonstrates that the crystalline cohomology of mildly singular projective varieties can be infinitely generated, especially in cases with ordinary double points, highlighting complex behaviors in singular cohomological structures.

Contribution

It provides new examples and insights into the size and structure of crystalline cohomology for singular varieties, emphasizing the role of derived de Rham comparisons.

Findings

Crystalline cohomology can be infinitely generated in singular projective varieties.

Singularities like ordinary double points significantly affect cohomology size.

Comparisons between crystalline and derived de Rham cohomology are crucial for these results.

Abstract

This note discusses some examples showing that the crystalline cohomology of even very mildly singular projective varieties tends to be quite large. In particular, any singular projective variety with at worst ordinary double points has infinitely generated crystalline cohomology in at least two cohomological degrees. These calculations rely critically on comparisons between crystalline and derived de Rham cohomology.