卷积类别的Morita等效性：arXiv附录：0805.0157
Morita equivalence for convolution categories: Appendix to arXiv:0805.0157
摘要：
在这个简短的后记我们的论文“积分变换和Drinfeld中心在派生代数几何”，我们描述了森田等价派生，归类矩阵代数所隐含的理论发展，因为它的外观。我们在拟相干层的完美栈X和它们的稳定无穷范畴Q（X）的背景下工作。完美栈包括特征为零的所有簇和公共栈，它们的层的稳定无穷范畴是它们的拟相干导出范畴的良好行为的加细，满足函数空间的公共性质的自然模拟。对于完美堆叠pi：X-->Y的态射，我们将层的范畴化矩阵代数Q（X x_Y X）与其具有么半群卷积积的导出纤维积相关联。我们表明，对于π忠实平坦（作为一个推论的1-仿射定理的Gaisgory）或π适当和满射和X，Y光滑（作为一个应用程序的正常下降，比照。Gaitsgory和Preygel），Q（X x_Y X）与Q（Y）之间存在Morita等价，即它们的稳定模无穷范畴的无穷范畴的等价。特别是，这直接意味着他们的德林费尔德中心的识别（如先前在0805.0157中建立的），更一般地说，其相关的拓扑域论的识别。另一个结果是，对于一个特征为零的仿射代数群G和一个代数子群K，传递到K-不变量导致了一个等价，从具有代数G-作用的稳定无穷范畴到Hecke范畴Q的模（K \G /K）。
英文摘要：
In this brief postscript to our paper "Integral transforms and Drinfeld centers in derived algebraic geometry", we describe a Morita equivalence for derived, categorified matrix algebras implied by theory developed since its appearance. We work in the setting of perfect stacks X and their stable infinity-categories Q(X) of quasicoherent sheaves. Perfect stacks include all varieties and common stacks in characteristic zero, and their stable infinity-categories of sheaves are well behaved refinements of their quasicoherent derived categories, satisfying natural analogues of common properties of function spaces. To a morphism of perfect stacks pi:X-->Y, we associate the categorified matrix algebra Q(X x_Y X) of sheaves on the derived fiber product equipped with its monoidal convolution product. We show that for pi faithfully flat (as a corollary of the 1-affineness theorem of Gaisgory) or for pi proper and surjective and X,Y smooth (as an application of proper descent, cf. Gaitsgory and Preygel), there is a Morita equivalence between Q(X x_Y X) and Q(Y), that is, an equivalence of their infinity-categories of stable module infinity-categories. In particular, this immediately implies an identification of their Drinfeld centers (as previously established in 0805.0157), and more generally, an identification of their associated topological field theories. Another consequence is that for an affine algebraic group G in characteristic zero and an algebraic subgroup K, passing to K-invariants induces an equivalence from stable infinity-categories with algebraic G-action to modules for the Hecke category Q(K\G/K).
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