Bloch-Ogus theory for smooth and semi-stable schemes in mixed characteristic

TL;DR

This paper advances Bloch-Ogus theory and verifies the Gersten conjecture for certain homology theories on smooth and semi-stable schemes in mixed characteristic, with implications for Galois symbol maps.

Contribution

It proves the Gersten conjecture for smooth schemes and a special case for semi-stable schemes in mixed characteristic, extending Bloch-Ogus theory.

Findings

Gersten conjecture proven for smooth schemes

Special case of Gersten conjecture for semi-stable schemes

Surjectivity of Galois symbol map for local rings over discrete valuation rings

Abstract

We study Bloch-Ogus theory and the Gersten conjecture for homology theories with duality satisfying certain properties, in particular for \'etale cohomology with finite coefficients coprime to the residue characteristic of the base, for smooth and semi-stable schemes in mixed characteristic. We prove the Gersten conjecture in the smooth case and prove a special case in the semi-stable situation. As a corollary of the smooth case we obtain the surjectivity of the Galois symbol map for arbitrary local rings over an excellent discrete valuation ring.