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Integral transforms and Drinfeld centers in derived algebraic geometry


Authors: David Ben-Zvi, John Francis and David Nadler
Journal: J. Amer. Math. Soc. 23 (2010), 909-966
MSC (2010): Primary 14-XX; Secondary 55-XX
DOI: https://doi.org/10.1090/S0894-0347-10-00669-7
Published electronically: April 1, 2010
MathSciNet review: 2669705
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Abstract: We study the interaction between geometric operations on stacks and algebraic operations on their categories of sheaves. We work in the general setting of derived algebraic geometry: our basic objects are derived stacks $ X$ and their $ \infty$-categories $ {QC}(X)$ of quasi-coherent sheaves. (When $ X$ is a familiar scheme or stack, $ {QC}(X)$ is an enriched version of the usual quasi-coherent derived category $ D_{qc}(X)$.) We show that for a broad class of derived stacks, called perfect stacks, algebraic and geometric operations on their categories of sheaves are compatible. We identify the category of sheaves on a fiber product with the tensor product of the categories of sheaves on the factors. We also identify the category of sheaves on a fiber product with functors between the categories of sheaves on the factors (thus realizing functors as integral transforms, generalizing a theorem of Toën for ordinary schemes). As a first application, for a perfect stack $ X$, consider $ {QC}(X)$ with its usual monoidal tensor product. Then our main results imply the equivalence of the Drinfeld center (or Hochschild cohomology category) of $ {QC}(X)$, the trace (or Hochschild homology category) of $ {QC}(X)$ and the category of sheaves on the loop space of $ X$. More generally, we show that the $ \mathcal{E}_n$-center and the $ \mathcal{E}_n$-trace (or $ \mathcal{E}_n$-Hochschild cohomology and homology categories, respectively) of $ {QC}(X)$ are equivalent to the category of sheaves on the space of maps from the $ n$-sphere into $ X$. This directly verifies geometric instances of the categorified Deligne and Kontsevich conjectures on the structure of Hochschild cohomology. As a second application, we use our main results to calculate the Drinfeld center of categories of linear endofunctors of categories of sheaves. This provides concrete applications to the structure of Hecke algebras in geometric representation theory. Finally, we explain how the above results can be interpreted in the context of topological field theory.


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Additional Information

David Ben-Zvi
Affiliation: Department of Mathematics, University of Texas, Austin, Texas 78712-0257
Email: benzvi@math.utexas.edu

John Francis
Affiliation: Department of Mathematics, Northwestern University, Evanston, Illinois 60208-2370
Email: jnkf@math.northwestern.edu

David Nadler
Affiliation: Department of Mathematics, Northwestern University, Evanston, Illinois 60208-2370
Email: nadler@math.northwestern.edu

DOI: https://doi.org/10.1090/S0894-0347-10-00669-7
Received by editor(s): October 23, 2008
Received by editor(s) in revised form: March 4, 2010
Published electronically: April 1, 2010
Article copyright: © Copyright 2010 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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