On a generalized Brauer group in mixed characteristic cases

Abstract

We define a generalization of the Brauer group $H_{\mathrm{B}}^n(X)$ 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, $H_{\mathrm{B}}^n(X)$ agrees with the étale motivic cohomology $H_{\acute{\mathrm{e}}\mathrm{t}}^{n+1}\left(X,\mathbb{Z}(n-1)\right)$. 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 $H_{\mathrm{B}}^n\left(R\right)\simeq H_{\mathrm{B}}^n\left(k\right)$ 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.

29 pages, Final version, Revised after the referee report