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Residue maps, Azumaya algebras, and buildings

Igor A. Rapinchuk

Abstract

The goal of this note is to give an explicit formula for the residues of twists of the matrix algebra in terms of the twisting cocycle. Combined with the Fixed Point Theorem for actions of finite groups on affine buildings, this leads to a quick proof of the well-known characterization of unramified algebras in terms of Azumaya algebras.

Residue maps, Azumaya algebras, and buildings

Abstract

The goal of this note is to give an explicit formula for the residues of twists of the matrix algebra in terms of the twisting cocycle. Combined with the Fixed Point Theorem for actions of finite groups on affine buildings, this leads to a quick proof of the well-known characterization of unramified algebras in terms of Azumaya algebras.
Paper Structure (4 sections, 3 theorems, 35 equations)

This paper contains 4 sections, 3 theorems, 35 equations.

Table of Contents

  1. Introduction
  2. Coboundary map and factor sets
  3. Residue maps: proof of Theorem \ref{['T:1']}
  4. Azumaya algebras and buildings: proof of Theorem \ref{['T:2']}

Key Result

Theorem 1

$r([A(c)])$ is represented by $b = (b_{\sigma}) \in Z^1(\mathcal{L}/\mathcal{K} , \mathbb{Q}/\mathbb{Z})$ with

Theorems & Definitions (4)

  • Theorem 1
  • Theorem 2
  • Proposition 1
  • proof