The derived Brauer map via twisted sheaves

Abstract

Let $X$ be a quasicompact quasiseparated scheme. The collection of derived Azumaya algebras in the sense of Toën forms a group, which contains the classical Brauer group of $X$ and which we call $B{r}^{†}(X)$ following Lurie. Toën introduced a map $\varphi :B{r}^{†}(X)\to H_{et}^2(X,{\mathbb{G}}_m)$ which extends the classical Brauer map, but instead of being injective, it is surjective. In this paper we study the restriction of $\varphi$ to a subgroup $Br(X)\subset B{r}^{†}(X)$, which we call the "derived Brauer group", on which $\varphi$ becomes an isomorphism $Br(X)\simeq H_{et}^2(X,{\mathbb{G}}_m)$. 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.

23 pages