Residue maps, Azumaya algebras, and buildings (Q6544507)

Residue due maps are a fundamental tool in studying Galois cohomology and Brauer groups. In the paper under review, the author studies the residue map \N\[\Nr: \mathrm{Br}\left( \mathcal{L}/\mathcal{K}\right) \longrightarrow \mathrm{H}^1\left( \mathcal{L}/\mathcal{K}, \mathbb{Q}/\mathbb{Z}\right),\N\]\Nwhere \(\mathcal{K}\) is a field complete with respect to a complete valuation, \(\mathcal{L}/\mathcal{K}\) is a nontrivial finite unramified Galois extension, and \(\mathrm{Br}\left( \mathcal{L}/\mathcal{K}\right)\) denotes the relative Brauer group of \(\mathcal{L}\) over \(\mathcal{K}\).\NAny element in $\mathrm{Br}\left(\mathcal{L}/\mathcal{K}\right)$ may be represented by the twist $A(c)$ of the matrix algebra by some $1$-cocycle $c \in {Z}^1\left( \mathcal{L}/\mathcal{K}, \mathrm{PGL}_n\right)$ for some $n$. In this paper, an explicit description of the residue of the Brauer class associated to $A(c)$ is computed. The proof relies on calculations in Galois cohomology.\N\NUsing this result and the Fixed Point Theorem for actions of finite groups on affine buildings, a new proof of the characterization of unramified algebras in terms of Azumaya algebras is given.