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具有Hochschild上链局部作用的范畴层

Sheaves of categories with local actions of Hochschild cochains

Dario Beraldo

备注：
To appear in Compositio Mathematica
发表信息：
Compositio Mathematica (2019), 155(8), 1521-1567
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摘要：

Hochschild上链的概念导致从$Aff$，仿射DG计划，monoidal DG类别的分配。我们表明，这种分配延伸，在一些适当的有限性条件下，函子$\mathbb H: Aff \to AlgBimod(DGCat)$，后者表示的范畴monoidal DG范畴和双模。现在，任何函子$\mathbb A: Aff \to AlgBimod(DGCat)$通过取模而产生范畴层理论$ShvCat^{\mathbb A}$。本文对$ShvCat^{\mathbb H}$进行了研究。模糊地说，这个理论将D-模理论归类，就像Gaitsgory的原始$ShvCat$将准相干层理论归类一样。我们开发了$ShvCat^{\mathbb H}$的功能性，它的下降特性，最重要的是，$\mathbb H$ -affineness的概念。然后，我们证明了$\mathbb H$ -仿射代数栈：$Y$栈满足一些温和的条件，$\infty$范畴$ShvCat^{\mathbb H}(Y)$是等价的$\infty$ -范畴的模块$\mathbb H(Y)$，monoidal DG范畴中定义的arXiv：1709.07867。作为一个应用，考虑一个准光滑栈$Y$和一个DG范畴$C$，其作用为$\mathbb H(Y)$。然后$C$承认在$Sing(Y)$中的奇异支撑理论，其中$Sing(Y)$是$Y$的奇异空间。

英文摘要：

The notion of Hochschild cochains induces an assignment from $Aff$, affine DG schemes, to monoidal DG categories. We show that this assignment extends, under some appropriate finiteness conditions, to a functor $\mathbb H: Aff \to AlgBimod(DGCat)$, where the latter denotes the category of monoidal DG categories and bimodules. Now, any functor $\mathbb A: Aff \to AlgBimod(DGCat)$ gives rise, by taking modules, to a theory of sheaves of categories $ShvCat^{\mathbb A}$. In this paper, we study $ShvCat^{\mathbb H}$. Vaguely speaking, this theory categorifies the theory of D-modules, in the same way as Gaitsgory's original $ShvCat$ categorifies the theory of quasi-coherent sheaves. We develop the functoriality of $ShvCat^{\mathbb H}$, its descent properties and, most importantly, the notion of $\mathbb H$-affineness. We then prove the $\mathbb H$-affineness of algebraic stacks: for $Y$ a stack satisfying some mild conditions, the $\infty$-category $ShvCat^{\mathbb H}(Y)$ is equivalent to the $\infty$-category of modules for $\mathbb H(Y)$, the monoidal DG category defined in arXiv:1709.07867. As an application, consider a quasi-smooth stack $Y$ and a DG category $C$ with an action of $\mathbb H(Y)$. Then $C$ admits a theory of singular support in $Sing(Y)$, where $Sing(Y)$ is the space of singularities of $Y$.

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