Annihilating the cohomology of group schemes

TL;DR

This paper demonstrates that cohomology classes in finite flat group schemes can be eliminated via finite covers, and applies this to provide a conceptual proof of a key theorem on Cohen-Macaulay algebras in positive characteristic.

Contribution

It introduces a new approach to annihilating cohomology classes using covers, offering a more conceptual proof of Hochster-Huneke's theorem.

Findings

Cohomology classes can be killed by finite covers of the base scheme.

A new proof of Hochster-Huneke's theorem is provided, avoiding previous equational methods.

The approach applies to both finite flat and abelian schemes with different types of covers.

Abstract

Our goal in this note is to show that cohomology classes with coefficients in finite flat group schemes can be killed by finite covers of the base scheme, and similarly for abelian schemes with "finite covers" replaced by "proper covers." We apply this result to commutative algebra by giving a new and more conceptual proof of Hochster-Huneke's theorem on the existence of big Cohen-Macaulay algebras in positive characteristic; all previous proofs of this result were equational or cocycle-theoretic in nature.