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 , generalizing an early result of Grothendieck and (2) when 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.
Citation
Siddharth Mathur. "Experiments on the Brauer map in high codimension." Algebra Number Theory 16 (3) 747 - 775, 2022. https://doi.org/10.2140/ant.2022.16.747
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