Dimensional vanishing of the saturated de Rham-Witt complex

TL;DR

This paper demonstrates that the saturated de Rham-Witt complex exhibits a dimensional vanishing property even for singular schemes, surpassing classical cohomology theories and aligning more with étale cohomology.

Contribution

It provides partial evidence that the saturated de Rham-Witt complex behaves better for singular schemes by establishing a strong dimensional vanishing property.

Findings

Saturated de Rham-Witt complex satisfies a dimensional vanishing property.

This vanishing is stronger than that of classical de Rham-Witt and crystalline cohomology.

The property aligns more closely with étale cohomology.

Abstract

The saturated de Rham-Witt complex, introduced by Bhatt-Lurie-Mathew in arXiv:1805.05501, is a variant of the classical de Rham-Witt complex which is expected to behave better for singular schemes. We provide partial justification for this expectation by showing that the saturated de Rham-Witt complex satisfies a dimensional vanishing property even in the presence of singularities. This is stronger than the vanishing properties of the classical de Rham-Witt complex, crystalline cohomology, or de Rham cohomology, and is instead comparable to \'etale cohomology.