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.