论文标题
通过扭曲的滑轮衍生的brauer地图
The derived Brauer map via twisted sheaves
论文作者
论文摘要
令$ x $为准化的准驱动方案。从托恩（Toën）的意义上讲，派生的阿祖玛亚（Azumaya）代数的集合，其中包含$ x $的古典brauer组，我们称之为$ br^\ dagger（x）$之后。 toën引入了一个地图$ ϕ：br^\匕首（x）\ to h^2_ {et}（x，x，\ mathbb g_m）$，它扩展了经典的brauer地图，但不是注射性，它是露天的。在本文中，我们研究了$ ϕ $对亚组$ br（x）\ subset br^\ dagger（x）$的限制，我们称之为“衍生的brauer group”，$ ϕ $在其上变成同构$ br（x）\ simeq h^2_ e et}（x，x，x，x，x，x，x，\ nathbb g_mmbb g_m）$。该地图可以解释为经典Brauer地图的派生版本，该版本提供了一种在经典的Brauer组和共同的Brauer组之间“填补空白”的方法。 Lurie利用了可观的$ \ infty $类别的理论来介绍了$ br（x）$的组。在那儿，提到的阿贝尔群体的同构是从相当于可逆的可抗可廉价$ \ mathcal $ \ mathcal o_x $ linear类别的“ brauer空间”之间的$ \ infty $分类中得出的。我们提供了$ \ infty $ - 类别的替代证明，它通过结缔组织的凝结物以及其通过结缔组织扭曲的滑轮从左到右表征函子。我们还证明，这种等价具有对称的单体结构，从而证明了binda a porta的猜想。
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 $Br^\dagger(X)$ following Lurie. Toën introduced a map $ϕ:Br^\dagger(X)\to H^2_{et}(X,\mathbb G_m)$ which extends the classical Brauer map, but instead of being injective, it is surjective. In this paper we study the restriction of $ϕ$ to a subgroup $Br(X)\subset Br^\dagger(X)$, which we call the "derived Brauer group", on which $ϕ$ 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.
