On a Generalized Brauer Group in Mixed Characteristic Cases

J. Math. Sci. Univ. Tokyo
Vol. 27 (2020), No. 1, Page 29-64.

Sakagaito, Makoto
On a Generalized Brauer Group in Mixed Characteristic Cases
[Full Article (PDF)] [MathSciNet Review (HTML)] [MathSciNet Review (PDF)]


Abstract:
We define a generalization of the Brauer group $\mathrm{H}_{B}^{n} (X)$ 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, $\mathrm{H}_{B}^{n}(X)$ agrees with the ├ętale motivic cohomology $\mathrm{H}^{n+1}_{\mathrm{\acute{e}t}}\left(X, \mathbb{Z}(n-1)\right)$. 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 $\mathrm{H}_{B}^{n}\left(R \right) \simeq \mathrm{H}_{B}^{n}\left(k\right)$ for a henselian local ring $R$ of a smooth algebra over a mixed characteristic discrete valuation ring and the residue field $k$ of $R$. 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.

Keywords: Brauer group, Gersten's conjecture, local-global principle.

Mathematics Subject Classification (2020): Primary 14F30; Secondary 14F22.
Received: 2020-06-05