#### Vol. 16, No. 3, 2022

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Experiments on the Brauer map in high codimension

### Siddharth Mathur

Vol. 16 (2022), No. 3, 747–775
##### Abstract

Using twisted sheaves, formal-local methods, and elementary transformations we show that separated algebraic spaces which are constructed as pushouts or contractions (of curves) have enough Azumaya algebras. This implies (1) Under mild hypothesis, every cohomological Brauer class is representable by an Azumaya algebra away from a closed subset of codimension $\ge 3$, generalizing an early result of Grothendieck and (2) $\mathrm{Br}\left(X\right)={\mathrm{Br}}^{\prime }\left(X\right)$ when $X$ is an algebraic space obtained from a quasiprojective scheme by contracting a curve. This result is valid for all dimensions but if we specialize to surfaces, it solves the question entirely: there are always enough Azumaya algebras on separated surfaces.

##### Keywords
Brauer group, twisted sheaves, Brauer map, algebraic stacks, gerbes
##### Mathematical Subject Classification
Primary: 14A20, 14F22
##### Milestones
Revised: 22 June 2021
Accepted: 30 July 2021
Published: 9 July 2022
##### Authors
 Siddharth Mathur Mathematisches Institut Heinrich-Heine Universität Düsseldorf Germany