On a generalized Brauer group in mixed characteristic cases

TL;DR

This paper introduces a generalized Brauer group for schemes, proves a Gersten-type conjecture in mixed characteristic cases, and applies these results to establish local-global principles for Galois cohomology over function fields.

Contribution

It defines a new generalized Brauer group in mixed characteristic, proves a key conjecture for this group, and applies findings to Galois cohomology in algebraic geometry.

Findings

Generalized Brauer group agrees with étale motivic cohomology in certain cases

Proves Gersten-type conjecture for the generalized Brauer group in mixed characteristic

Establishes local-global principles for Galois cohomology over function fields

Abstract

We define a generalization of the Brauer group $\operatorname{H}_\mathrm{B}^{n}(X)$ for an equi-dimensional scheme $X$ and $n>0$. In the case where $X$ is the spectrum of a local ring of a smooth algebra over a discrete valuation ring, $\operatorname{H}_\mathrm{B}^{n}(X)$ agrees with the \'{e}tale motivic cohomology $\operatorname{H}^{n+1}_{\mathrm{\acute{e}t}}\left(X, \mathbb{Z}(n-1)\right)$. We prove (a part of) the Gersten-type conjecture for the generalized Brauer group for a local ring of a smooth algebra over a mixed characteristic discrete valuation ring and an isomorphism $ \operatorname{H}_\mathrm{B}^{n}\left( R \right) \simeq \operatorname{H}_\mathrm{B}^{n}\left( k \right) $ for a henselian local ring $R$ of a smooth algebra over a mixed characteristic discrete valuation ring and the residue field $k$. As an application, we show local-global principles for Galois cohomology groups over function fields of smooth curves over a mixed characteristic excellent henselian discrete valuation ring.