Blow-up invariance of cohomology theories with modulus

TL;DR

This paper establishes conditions under which cohomology theories with modulus are invariant under blow-ups, providing new proofs and defining blow-up invariant Witt vector cohomology with modulus.

Contribution

It introduces a sufficient condition for blow-up invariance of cohomology theories with modulus and defines a new blow-up invariant Witt vector cohomology with modulus.

Findings

Proved blow-up invariance of Hodge cohomology with modulus.

Defined Witt vector cohomology with modulus and proved its invariance.

Provided a short proof of Kelly-Miyazaki's blow-up invariance result.

Abstract

In this paper, we study cohomology theories of $\mathbb{Q}$-modulus pairs, which are pairs $(X, D)$ consisting of a scheme $X$ and a $\mathbb{Q}$-divisor $D$. Our main theorem provides a sufficient condition for such a cohomology theory to be invariant under blow-ups with centers contained in the divisor. This yields a short proof of the blow-up invariance of the Hodge cohomology with modulus proved by Kelly-Miyazaki. We also define the Witt vector cohomology with modulus using the Brylinski-Kato filtration and prove its blow-up invariance.