On Problems about a generalization of the Brauer group

TL;DR

This paper introduces a generalized Brauer group using Bloch's cycle complex, proves the Gersten conjecture in certain cases, and extends related theorems to broader contexts in algebraic geometry.

Contribution

It defines a new generalized Brauer group, proves the Gersten conjecture for it in specific cases, and generalizes Artin's theorem on Brauer groups.

Findings

Proved Gersten conjecture for generalized Brauer groups in some cases

Established Gersten conjecture for logarithmic Hodge-Witt cohomology in certain rings

Extended Artin's theorem to a broader class of Brauer groups

Abstract

In this paper, we define a generalization of the Brauer groups by using Bloch's cycle complex on etale site. We prove the Gersten conjecture of generalized Brauer group on some cases. As an application we prove the Gersten conjecture of the logarithmic Hodge-Witt cohomology for a two dimensional regular local ring which is smooth over a spectrum of some discrete valuation ring of characteristic $p>0$. Moreover we consider a generalization of Artin's theorem on Brauer groups.