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 \(\textsf{Br}^\dagger (X)\) following Lurie. Toën introduced a map \(\phi :\textsf{Br}^\dagger (X)\rightarrow H ^2_{\acute{e }t }(X,{\mathbb {G}}_{\textrm{m}})\) 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 \(\textsf{Br}(X)\subset \textsf{Br}^\dagger (X)\), which we call the derived Brauer group, on which \(\phi \) becomes an isomorphism \(\textsf{Br}(X)\simeq H ^2_{\acute{e }t }(X,{\mathbb {G}}_{\textrm{m}})\). 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 \(\textsf{Br}(X)\) 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 \(Map (X,K ({\mathbb {G}}_{\textrm{m}},2))\). 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.