Pinching Azumaya algebras

TL;DR

The paper proves that under certain finite modifications of schemes, Azumaya algebra representations of Brauer classes are preserved, extending Ferrand's pinching results to Azumaya algebras.

Contribution

It extends Ferrand's pinching technique to Azumaya algebras, showing the preservation of Azumaya algebra representations under finite scheme modifications.

Findings

Azumaya algebra representations are preserved under finite modifications.

Extension of Ferrand's pinching result to Azumaya algebras.

Provides conditions for cohomological Brauer classes to be represented by Azumaya algebras.

Abstract

We show, that for a morphism of schemes from X to Y, that is a finite modification in finitely many closed points, a cohomological Brauer class on Y is represented by an Azumaya algebra if its pullback to X is represented by an Azumaya algebra. Part of the proof uses an extension of a result by Ferrand, on pinching of finite locally free sheaves, to Azumaya algebras.