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Integral transforms and Drinfeld centers in derived algebraic geometry. (English) Zbl 1202.14015

Derived categories of (quasi-)coherent sheaves on a scheme \(X\) encode geometry in algebraic data. Any such category is triangulated and it has been known for a long time that certain operations cannot be performed in this setting: the category has to be enriched. Over a ring of characteristic zero one could work with pretriangulated dg-categories or \(A_\infty\)-categories. A more general solution, which in characteristic zero is equivalent to the former two, is provided by \(\infty\)-categories.
In the paper under review, the authors study the interaction between geometric operations on stacks and algebraic operations on their categories of sheaves. They work in the setting of derived algebraic geometry: the basic objects are so called perfect derived stacks and their \(\infty\)-categories of quasi-coherent sheaves. Derived stacks arise naturally if one takes fibre products or more complicated limits on schemes and stacks, similar to how stacks occur from performing quotients on schemes. The class of perfect derived stacks the authors work with is a fairly broad one: it includes, for example, quasi-compact derived schemes with affine diagonal and, in characteristic zero, quotients of a quasi-projective derived scheme by a linear action of an affine group. One useful property of this class of objects is the following. If \(X\) is a perfect stack and \(\text{QC}(X)\) is its \(\infty\)-category of quasi-coherent sheaves, then compact objects and perfect complexes in \(\text{QC}(X)\) coincide.
One of the main results of the paper is the identification of the category of sheaves on a fibre product of two perfect stacks wich the tensor product of the categories of sheaves on the factors. The authors furthermore show that the category of sheaves on a fibre product can be identified with functors between the categories of sheaves on the factors (so functors can be realized as integral transforms; this generalises a result of B. Toën for ordinary schemes in [Invent. Math. 167, No. 3, 615–667 (2007; Zbl 1118.18010)]).
The above results are applied in several instances. For example, it is shown that the Drinfeld center (or Hochschild cohomology category) of \(\text{QC}(X)\), the trace (or Hochschild homology category) of \(\text{QC}(X)\) and the category of sheaves on the loop space of \(X\) are equivalent. The authors also use their results to provide concrete applications to the structure of Hecke algebras in geometric representation theory. Furthermore, they explain how the above results can be interpreted in the context of topological field theory.
Reviewer: Pawel Sosna (Bonn)

MSC:

14F05 Sheaves, derived categories of sheaves, etc. (MSC2010)
18E30 Derived categories, triangulated categories (MSC2010)
14A20 Generalizations (algebraic spaces, stacks)

Citations:

Zbl 1118.18010
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