A moving lemma for cohomology with support

TL;DR

This paper proves a moving lemma for cohomology with support on certain smooth quasi-projective varieties, leading to significant generalizations of classical theorems and properties in algebraic geometry.

Contribution

It introduces a moving lemma applicable to a broad class of cohomology theories, extending key theorems like effacement, Gersten conjecture, and purity in characteristic zero.

Findings

Generalizes effacement theorem for cohomology with support

Provides a finite level version of the Gersten conjecture in characteristic zero

Shows that refined unramified cohomology groups are motivic

Abstract

For a natural class of cohomology theories with support (including \'etale or pro-\'etale cohomology with suitable coefficients), we prove a moving lemma for cohomology classes with support on smooth quasi-projective k-varieties that admit a smooth projective compactification (e.g. if char(k)=0). This has the following consequences for such k-varieties and cohomology theories: a local and global generalization of the effacement theorem of Quillen, Bloch--Ogus, and Gabber, a finite level version of the Gersten conjecture in characteristic zero, and a generalization of the injectivity property and the codimension 1 purity theorem for \'etale cohomology. Our results imply that the refined unramified cohomology groups from [Sch23] are motivic.