Remote Access Journal of the American Mathematical Society
Green Open Access

Journal of the American Mathematical Society

ISSN 1088-6834(online) ISSN 0894-0347(print)



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
Published electronically: April 1, 2010
MathSciNet review: 2669705
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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.

References [Enhancements On Off] (What's this?)

  • [A] V. Angeltveit, The cyclic bar construction on $ A_\infty$ $ H$-spaces. Adv. Math. 222 (2009), 1589-1610. arXiv:math/0612165. MR 2555906
  • [BK] B. Bakalov and A. Kirillov, Jr., Lectures on tensor categories and modular functors. University Lecture Series, 21. American Mathematical Society, Providence, RI, 2001. MR 1797619 (2002d:18003)
  • [BN1] D. Ben-Zvi and D. Nadler, Loop Spaces and Connections. arXiv:1002.3636.
  • [BN2] D. Ben-Zvi and D. Nadler, The character theory of a complex group. arXiv:0904.1247.
  • [Ber] J. Bergner, A survey of $ (\infty, 1)$-categories. arXiv:math/0610239.
  • [Be] R. Bezrukavnikov, Noncommutative counterparts of the Springer resolution. International Congress of Mathematicians. Vol. II, 1119-1144, Eur. Math. Soc., Zürich, 2006. arXiv:math.RT/0604445. MR 2275638 (2009d:17026)
  • [BeF] R. Bezrukavnikov and M. Finkelberg, Equivariant Satake category and Kostant-Whittaker reduction. Moscow Math J. 8 (2008), no. 1, 39-72, 183. arXiv:0707.3799. MR 2422266 (2009d:19008)
  • [Bo] M. Bökstedt, Topological Hochschild Homology. Preprint, 1985.
  • [BoN] M. Bökstedt and A. Neeman, Homotopy limits in triangulated categories. Compositio Math. 86 (1993), no. 2, 209-234. MR 1214458 (94f:18008)
  • [BLL] A. Bondal, M. Larsen and V. Lunts, Grothendieck ring of pretriangulated categories. Int. Math. Res. Not. 2004, no. 29, 1461-1495. arXiv:math/0401009. MR 2051435 (2005d:18014)
  • [BV] A. Bondal and M. Van den Bergh, Generators and representability of functors in commutative and noncommutative geometry. Mosc. Math. J. 3 (2003), no. 1, 1-36, 258. MR 1996800 (2004h:18009)
  • [Ch] C. Chevalley, Théorie des groupes de Lie. Tome II. Groupes algébriques. Actualités Sci. Ind. no. 1152. Hermann & Cie., Paris, 1951. MR 0051242 (14:448d)
  • [C] K. Costello, Topological conformal field theories and Calabi-Yau categories. Adv. Math. 210 (2007), no. 1, 165-214. arXiv:math/0412149. MR 2298823 (2008f:14071)
  • [De] P. Deligne, Catégories tannakiennes. The Grothendieck Festschrift, Vol. II, 111-195, Progr. Math., 87, Birkhäuser Boston, Boston, MA, 1990. MR 1106898 (92d:14002)
  • [D] V. Drinfeld, DG quotients of DG categories. J. Algebra 272 (2004), no. 2, 643-691. MR 2028075 (2006e:18018)
  • [EKMM] A. Elmendorf, I. Kriz, M. Mandell and J.P. May, Rings, modules, and algebras in stable homotopy theory. With an appendix by M. Cole. Mathematical Surveys and Monographs, 47. American Mathematical Society, Providence, RI, 1997. MR 1417719 (97h:55006)
  • [F1] J. Francis, Derived algebraic geometry over $ \mathcal E_n$-rings. MIT Ph.D. thesis, 2008.
  • [F2] J. Francis, The cotangent complex and Hochschild homology of $ \mathcal E_n$-rings. In preparation.
  • [Fr] D. Freed, Higher algebraic structures and quantization. Comm. Math. Phys. 159 (1994) 343-398. MR 1256993 (95c:58034)
  • [H] V. Hinich, Drinfeld double for orbifolds. Quantum groups, 251-265, Contemp. Math., 433, Amer. Math. Soc., Providence, RI, 2007. arXiv:math.QA/0511476. MR 2349625 (2009a:14002)
  • [HS] A. Hirschowitz and C. Simpson, Descente pour les $ n$-champs. arXiv:math/9807049.
  • [HPS] M. Hovey, J. Palmieri and N. Strickland, Axiomatic stable homotopy theory. Mem. Amer. Math. Soc. 128 (1997), no. 610. MR 1388895 (98a:55017)
  • [HSS] M. Hovey, B. Shipley and S. Schwede, Symmetric Spectra. J. Amer. Math. Soc. 13 (2000), no. 1, 149-208. arXiv:math/9801077. MR 1695653 (2000h:55016)
  • [HKV] P. Hu, I. Kriz and A. Voronov, On Kontsevich's Hochschild cohomology conjecture. Compos. Math. 142 (2006), no. 1, 143-168. MR 2197407 (2006j:18003)
  • [Jo] A. Joyal, Quasi-categories and Kan complexes. J. Pure Appl. Algebra 175 (2002) 207-222. MR 1935979 (2003h:55026)
  • [JS] A. Joyal and R. Street, Tortile Yang-Baxter operators in tensor categories. J. Pure Appl. Algebra 71 (1991), no. 1, 43-51. MR 1107651 (92e:18006)
  • [Kap] M. Kapranov, Rozansky-Witten invariants via Atiyah classes. Compositio Math. 115 (1999), no. 1, 71-113. math.AG/9704009. MR 1671737 (2000h:57056)
  • [KRS] A. Kapustin, L. Rozansky and N. Saulina, Three-dimensional topological field theory and symplectic algebraic geometry I. Nuclear Phys. B 816 (2009), no. 3, 295-355. arXiv:0810.5415. MR 2522724
  • [Kau] R. Kaufmann, A proof of a cyclic version of Deligne's conjecture via cacti. Math. Res. Lett. 15 (2008), no. 5, 901-921. math.QA/0403340. MR 2443991 (2009i:18006)
  • [Ke] B. Keller, On differential graded categories. International Congress of Mathematicians. Vol. II, 151-190, Eur. Math. Soc., Zürich, 2006. arXiv:math.AG/0601185. MR 2275593 (2008g:18015)
  • [K1] M. Kontsevich, Operads and motives in deformation quantization. Moshé Flato (1937-1998). Lett. Math. Phys. 48 (1999), no. 1, 35-72. MR 1718044 (2000j:53119)
  • [K2] M. Kontsevich, Rozansky-Witten theory via formal geometry. Compositio Math. 115 (1999), no. 1, 115-127. arXiv:dg-ga/9704009. MR 1671725 (2000h:57057)
  • [KS1] M. Kontsevich and Y. Soibelman, Deformations of algebras over operads and the Deligne conjecture.Conference Moshé Flato 1999, Vol. I (Dijon), 255-307, Math. Phys. Stud., 21, Kluwer Acad. Publ., Dordrecht, 2000. MR 1805894 (2002e:18012)
  • [KS2] M. Kontsevich and Y. Soibelman, Notes on A-infinity algebras, A-infinity categories and non-commutative geometry I. Preprint arXiv:math/0606241.
  • [LMS] L.G. Lewis, Jr., J.P. May, and M. Steinberger, Equivariant stable homotopy theory. With contributions by J. McClure. Lecture Notes in Mathematics, 1213. Springer-Verlag, Berlin, 1986. MR 866482 (88e:55002)
  • [L1] J. Lurie, Derived Algebraic Geometry, MIT Ph.D. Thesis, 2004.
  • [L2] J. Lurie, Higher topos theory. Annals of Mathematics Studies, 170. Princeton University Press, 2009. arXiv:math.CT/0608040. MR 2522659
  • [L3] J. Lurie, Derived Algebraic Geometry 1: Stable infinity categories. arXiv:math.CT/ 0608228.
  • [L4] J. Lurie, Derived Algebraic Geometry 2: Noncommutative algebra. arXiv:math.CT/ 0702299.
  • [L5] J. Lurie, Derived Algebraic Geometry 3: Commutative algebra. arXiv:math.CT/ 0703204.
  • [L6] J. Lurie, On the Classification of Topological Field Theories. Current developments in mathematics, 2008, pp. 129-280, Int. Press, Somerville, MA, 2009. arXiv:0905.0465. MR 2555928
  • [L7] J. Lurie, Derived Algebraic Geometry 6: $ E_k$ Algebras. arXiv:0911.0018.
  • [MS] J. McClure and J. Smith, A solution of Deligne's Hochschild cohomology conjecture. Recent progress in homotopy theory (Baltimore, MD, 2000), 153-193, Contemp. Math., 293, Amer. Math. Soc., Providence, RI, 2002. MR 1890736 (2003f:55013)
  • [M] M. Müger, From subfactors to categories and topology II. The quantum double of tensor categories and subfactors. J. Pure Appl. Algebra 180 (2003), no. 1-2, 159-219. arXiv:math.CT/0111205. MR 1966525 (2004f:18014)
  • [N1] A. Neeman, The connection between the $ K$-theory localization theorem of Thomason, Trobaugh and Yao and the smashing subcategories of Bousfield and Ravenel. Ann. Sci. École Norm. Sup. (4) 25 (1992), no. 5, 547-566. MR 1191736 (93k:18015)
  • [N2] A. Neeman, The Grothendieck duality theorem via Bousfield's techniques and Brown representability. J. Amer. Math. Soc. 9 (1996), no. 1, 205-236. MR 1308405 (96c:18006)
  • [N3] A. Neeman, Triangulated categories. Annals of Mathematics Studies, 148. Princeton University Press, Princeton, NJ, 2001. MR 1812507 (2001k:18010)
  • [O] D. Orlov, Equivalences of derived categories and $ K3$ surfaces. Algebraic geometry, 7. J. Math. Sci. (New York) 84 (1997), no.5, 1361-1381. arXiv:math.AG/9606006. MR 1465519 (99a:14054)
  • [Os] V. Ostrik, Module categories over the Drinfeld double of a finite group. Int. Math. Res. Not. 2003, no. 27, 1507-1520. arXiv:math.QA/0202130. MR 1976233 (2004h:18005)
  • [RobW] J. Roberts and S. Willerton, On the Rozansky-Witten weight systems. math.AG/ 0602653.
  • [RW] L. Rozansky and E. Witten, Hyper-Kähler geometry and invariants of three-manifolds. Selecta Math. (N.S.) 3 (1997), no. 3, 401-458. MR 1481135 (98m:57041)
  • [SSh] S. Schwede and B. Shipley, Stable model categories are categories of modules. Topology 42 (2003), no. 1, 103-153. MR 1928647 (2003g:55034)
  • [Sh] B. Shipley, $ H{\mathbb{Z}}$-algebra spectra are differential graded algebras. Amer. J. Math. 129 (2007), no. 2, 351-379. MR 2306038 (2008b:55015)
  • [T1] D. Tamarkin, Another proof of M. Kontsevich formality theorem. arXiv:math/9803025.
  • [T2] D. Tamarkin, The deformation complex of a $ d$-algebra is a $ (d+1)$-algebra. arXiv:math/ 0010072.
  • [TT] R. Thomason and T. Trobaugh, Higher algebraic $ K$-theory of schemes and of derived categories. The Grothendieck Festschrift, Vol. III, 247-435, Progr. Math., 88, Birkhäuser Boston, Boston, MA, 1990. MR 1106918 (92f:19001)
  • [To1] B. Toën, The homotopy theory of dg categories and derived Morita theory. Invent. Math. 167 (2007), no. 3, 615-667. arXiv:math.AG/0408337. MR 2276263 (2008a:18006)
  • [To2] B. Toën, Higher and Derived Stacks: a global overview. Proceedings 2005 AMS Summer School in Algebraic Geometry. Proc. Sympos. Pure Math., Vol. 80, Part 1, Amer. Math. Soc., Providence, RI, 2009. arXiv:math.AG/0604504. MR 2483943
  • [ToVe1] B. Toën and G. Vezzosi, Homotopical Algebraic Geometry I: Topos theory. Adv. Math. 193 (2005), no. 2, 257-372. arXiv:math.AG/0404373. MR 2137288 (2007b:14038)
  • [ToVe2] B. Toën and G. Vezzosi, Homotopical Algebraic Geometry II: Geometric stacks and applications. Memoirs of the AMS 193 (2008). arXiv:math.AG/0404373. MR 2394633 (2009h:14004)
  • [TZ] T. Tradler and M. Zeinalian, On the cyclic Deligne conjecture. J. Pure Appl. Algebra 204 (2006), no. 2, 280-299. math.QA:0404218. MR 2184812 (2007j:16015)

Similar Articles

Retrieve articles in Journal of the American Mathematical Society with MSC (2010): 14-XX, 55-XX

Retrieve articles in all journals with MSC (2010): 14-XX, 55-XX

Additional Information

David Ben-Zvi
Affiliation: Department of Mathematics, University of Texas, Austin, Texas 78712-0257

John Francis
Affiliation: Department of Mathematics, Northwestern University, Evanston, Illinois 60208-2370

David Nadler
Affiliation: Department of Mathematics, Northwestern University, Evanston, Illinois 60208-2370

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.

American Mathematical Society