GAGA problems for the Brauer group via derived geometry

TL;DR

This paper advances the understanding of derived Azumaya algebras and the derived Brauer group, establishing new properties and injectivity results, and extends Grothendieck's existence theorem to broader contexts in derived algebraic geometry.

Contribution

It introduces a Beauville-Laszlo-style property for derived Azumaya algebras and proves injectivity of the derived Brauer group under certain conditions, extending classical results.

Findings

Derived Azumaya algebras satisfy a Beauville-Laszlo-type property.

The derived Brauer group injects into the Henselization's Brauer group for proper schemes.

Grothendieck's existence theorem holds for twisted sheaves without the resolution property.

Abstract

This paper is dedicated to a further study of derived Azumaya algebras. The first result we obtain is a Beauville-Laszlo-style property for such objects (considered up to Morita equivalence), which is consequence of a more general Beauville-Laszlo kind of statement for quasi-coherent sheaves of categories. Next, we prove that given any (derived) scheme $X$, proper over the spectrum of a quasi-excellent Henselian ring, the derived Brauer group of $X$ injects into the one of the Henselization of $X$ along the base, generalizing a classical result of Grothendieck and a more recent theorem of Geisser-Morin. As a separate application, we deduce that Grothendieck's existence theorem holds for the stable $\infty$-categories of twisted sheaves even when the corresponding $\bbG_m$-gerbe does not satisfy the resolution property, offering an improvement of a result of Alper, Rydh and Hall.