The derived Brauer map via twisted sheaves

TL;DR

This paper explores a derived version of the Brauer map for schemes, establishing an isomorphism between the derived Brauer group and the cohomological Brauer group, and provides an alternative proof of a key equivalence involving prestable categories.

Contribution

It introduces a derived Brauer map that extends the classical one, and offers a new proof of the equivalence between the Brauer space and certain mapping spaces, confirming a conjecture.

Findings

The derived Brauer map is an isomorphism on a specific subgroup.

An alternative proof of the equivalence of $oldsymbol{ ext{Brauer space}}$ and mapping spaces is provided.

The equivalence preserves a symmetric monoidal structure.

Abstract

Let $X$ be a quasicompact quasiseparated scheme. The collection of derived Azumaya algebras in the sense of To\"en forms a group, which contains the classical Brauer group of $X$ and which we call $Br^\dagger(X)$ following Lurie. To\"en introduced a map $\phi:Br^\dagger(X)\to H^2_{et}(X,\mathbb G_m)$ which extends the classical Brauer map, but instead of being injective, it is surjective. In this paper we study the restriction of $\phi$ to a subgroup $Br(X)\subset Br^\dagger(X)$, which we call the "derived Brauer group", on which $\phi$ becomes an isomorphism $Br(X)\simeq H^2_{et}(X,\mathbb G_m)$. This map may be interpreted as a derived version of the classical Brauer map which offers a way to "fill the gap" between the classical Brauer group and the cohomogical Brauer group. The group $Br(X)$ was introduced by Lurie by making use of the theory of prestable $\infty$-categories. There, the mentioned isomorphism of abelian groups was deduced from an equivalence of $\infty$-categories between the "Brauer space" of invertible presentable prestable $\mathcal O_X$-linear categories, and the space $Map(X,K(\mathbb G_m,2))$. We offer an alternative proof of this equivalence of $\infty$-categories, characterizing the functor from the left to the right via gerbes of connective trivializations, and its inverse via connective twisted sheaves. We also prove that this equivalence carries a symmetric monoidal structure, thus proving a conjecture of Binda an Porta.