##
**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.

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) |

### Keywords:

derived algebraic geometry; stacks; enhancements of triangulated categories; Hochschild cohomology### Citations:

Zbl 1118.18010
PDF
BibTeX
XML
Cite

\textit{D. Ben-Zvi} et al., J. Am. Math. Soc. 23, No. 4, 909--966 (2010; Zbl 1202.14015)

### References:

[1] | Vigleik Angeltveit, The cyclic bar construction on \?_{\infty } \?-spaces, Adv. Math. 222 (2009), no. 5, 1589 – 1610. · Zbl 1178.55005 |

[2] | Bojko Bakalov and Alexander Kirillov Jr., Lectures on tensor categories and modular functors, University Lecture Series, vol. 21, American Mathematical Society, Providence, RI, 2001. · Zbl 0965.18002 |

[3] | D. Ben-Zvi and D. Nadler, Loop Spaces and Connections. arXiv:1002.3636. · Zbl 1246.14027 |

[4] | D. Ben-Zvi and D. Nadler, The character theory of a complex group. arXiv:0904.1247. · Zbl 1325.22011 |

[5] | J. Bergner, A survey of \( (\infty, 1)\)-categories. arXiv:math/0610239. · Zbl 1200.18011 |

[6] | Roman Bezrukavnikov, Noncommutative counterparts of the Springer resolution, International Congress of Mathematicians. Vol. II, Eur. Math. Soc., Zürich, 2006, pp. 1119 – 1144. · Zbl 1135.17011 |

[7] | Roman Bezrukavnikov and Michael Finkelberg, Equivariant Satake category and Kostant-Whittaker reduction, Mosc. Math. J. 8 (2008), no. 1, 39 – 72, 183 (English, with English and Russian summaries). · Zbl 1205.19005 |

[8] | M. Bökstedt, Topological Hochschild Homology. Preprint, 1985. |

[9] | Marcel Bökstedt and Amnon Neeman, Homotopy limits in triangulated categories, Compositio Math. 86 (1993), no. 2, 209 – 234. · Zbl 0802.18008 |

[10] | Alexey I. Bondal, Michael Larsen, and Valery A. Lunts, Grothendieck ring of pretriangulated categories, Int. Math. Res. Not. 29 (2004), 1461 – 1495. · Zbl 1079.18008 |

[11] | 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 (English, with English and Russian summaries). · Zbl 1135.18302 |

[12] | Claude Chevalley, Théorie des groupes de Lie. Tome II. Groupes algébriques, Actualités Sci. Ind. no. 1152, Hermann & Cie., Paris, 1951 (French). · Zbl 0054.01303 |

[13] | Kevin Costello, Topological conformal field theories and Calabi-Yau categories, Adv. Math. 210 (2007), no. 1, 165 – 214. · Zbl 1171.14038 |

[14] | P. Deligne, Catégories tannakiennes, The Grothendieck Festschrift, Vol. II, Progr. Math., vol. 87, Birkhäuser Boston, Boston, MA, 1990, pp. 111 – 195 (French). · Zbl 0727.14010 |

[15] | Vladimir Drinfeld, DG quotients of DG categories, J. Algebra 272 (2004), no. 2, 643 – 691. · Zbl 1064.18009 |

[16] | A. D. Elmendorf, I. Kriz, M. A. Mandell, and J. P. May, Rings, modules, and algebras in stable homotopy theory, Mathematical Surveys and Monographs, vol. 47, American Mathematical Society, Providence, RI, 1997. With an appendix by M. Cole. · Zbl 0894.55001 |

[17] | J. Francis, Derived algebraic geometry over \( \mathcal E_n\)-rings. MIT Ph.D. thesis, 2008. |

[18] | J. Francis, The cotangent complex and Hochschild homology of \( \mathcal E_n\)-rings. In preparation. |

[19] | Daniel S. Freed, Higher algebraic structures and quantization, Comm. Math. Phys. 159 (1994), no. 2, 343 – 398. · Zbl 0790.58007 |

[20] | Vladimir Hinich, Drinfeld double for orbifolds, Quantum groups, Contemp. Math., vol. 433, Amer. Math. Soc., Providence, RI, 2007, pp. 251 – 265. · Zbl 1142.18004 |

[21] | A. Hirschowitz and C. Simpson, Descente pour les \( n\)-champs. arXiv:math/9807049. |

[22] | Mark Hovey, John H. Palmieri, and Neil P. Strickland, Axiomatic stable homotopy theory, Mem. Amer. Math. Soc. 128 (1997), no. 610, x+114. · Zbl 0881.55001 |

[23] | Mark Hovey, Brooke Shipley, and Jeff Smith, Symmetric spectra, J. Amer. Math. Soc. 13 (2000), no. 1, 149 – 208. · Zbl 0931.55006 |

[24] | Po Hu, Igor Kriz, and Alexander A. Voronov, On Kontsevich’s Hochschild cohomology conjecture, Compos. Math. 142 (2006), no. 1, 143 – 168. · Zbl 1109.18001 |

[25] | A. Joyal, Quasi-categories and Kan complexes, J. Pure Appl. Algebra 175 (2002), no. 1-3, 207 – 222. Special volume celebrating the 70th birthday of Professor Max Kelly. · Zbl 1015.18008 |

[26] | André Joyal and Ross Street, Tortile Yang-Baxter operators in tensor categories, J. Pure Appl. Algebra 71 (1991), no. 1, 43 – 51. · Zbl 0726.18004 |

[27] | M. Kapranov, Rozansky-Witten invariants via Atiyah classes, Compositio Math. 115 (1999), no. 1, 71 – 113. · Zbl 0993.53026 |

[28] | Anton Kapustin, Lev Rozansky, and Natalia Saulina, Three-dimensional topological field theory and symplectic algebraic geometry. I, Nuclear Phys. B 816 (2009), no. 3, 295 – 355. · Zbl 1194.81224 |

[29] | Ralph M. Kaufmann, A proof of a cyclic version of Deligne’s conjecture via cacti, Math. Res. Lett. 15 (2008), no. 5, 901 – 921. · Zbl 1161.55001 |

[30] | Bernhard Keller, On differential graded categories, International Congress of Mathematicians. Vol. II, Eur. Math. Soc., Zürich, 2006, pp. 151 – 190. · Zbl 1140.18008 |

[31] | Maxim Kontsevich, Operads and motives in deformation quantization, Lett. Math. Phys. 48 (1999), no. 1, 35 – 72. Moshé Flato (1937 – 1998). · Zbl 0945.18008 |

[32] | Maxim Kontsevich, Rozansky-Witten invariants via formal geometry, Compositio Math. 115 (1999), no. 1, 115 – 127. · Zbl 0924.57017 |

[33] | Maxim Kontsevich and Yan Soibelman, Deformations of algebras over operads and the Deligne conjecture, Conférence Moshé Flato 1999, Vol. I (Dijon), Math. Phys. Stud., vol. 21, Kluwer Acad. Publ., Dordrecht, 2000, pp. 255 – 307. · Zbl 0972.18005 |

[34] | M. Kontsevich and Y. Soibelman, Notes on A-infinity algebras, A-infinity categories and non-commutative geometry I. Preprint arXiv:math/0606241. · Zbl 1202.81120 |

[35] | L. G. Lewis Jr., J. P. May, M. Steinberger, and J. E. McClure, Equivariant stable homotopy theory, Lecture Notes in Mathematics, vol. 1213, Springer-Verlag, Berlin, 1986. With contributions by J. E. McClure. · Zbl 0611.55001 |

[36] | J. Lurie, Derived Algebraic Geometry, MIT Ph.D. Thesis, 2004. |

[37] | Jacob Lurie, Higher topos theory, Annals of Mathematics Studies, vol. 170, Princeton University Press, Princeton, NJ, 2009. · Zbl 1175.18001 |

[38] | J. Lurie, Derived Algebraic Geometry 1: Stable infinity categories. arXiv:math.CT/ 0608228. |

[39] | J. Lurie, Derived Algebraic Geometry 2: Noncommutative algebra. arXiv:math.CT/ 0702299. |

[40] | J. Lurie, Derived Algebraic Geometry 3: Commutative algebra. arXiv:math.CT/ 0703204. |

[41] | Jacob Lurie, On the classification of topological field theories, Current developments in mathematics, 2008, Int. Press, Somerville, MA, 2009, pp. 129 – 280. · Zbl 1180.81122 |

[42] | J. Lurie, Derived Algebraic Geometry 6: \( E_k\) Algebras. arXiv:0911.0018. |

[43] | James E. McClure and Jeffrey H. Smith, A solution of Deligne’s Hochschild cohomology conjecture, Recent progress in homotopy theory (Baltimore, MD, 2000) Contemp. Math., vol. 293, Amer. Math. Soc., Providence, RI, 2002, pp. 153 – 193. · Zbl 1009.18009 |

[44] | Michael 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. · Zbl 1033.18003 |

[45] | Amnon Neeman, The connection between the \?-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. · Zbl 0868.19001 |

[46] | Amnon Neeman, The Grothendieck duality theorem via Bousfield’s techniques and Brown representability, J. Amer. Math. Soc. 9 (1996), no. 1, 205 – 236. · Zbl 0864.14008 |

[47] | Amnon Neeman, Triangulated categories, Annals of Mathematics Studies, vol. 148, Princeton University Press, Princeton, NJ, 2001. · Zbl 0974.18008 |

[48] | D. O. Orlov, Equivalences of derived categories and \?3 surfaces, J. Math. Sci. (New York) 84 (1997), no. 5, 1361 – 1381. Algebraic geometry, 7. · Zbl 0938.14019 |

[49] | Viktor Ostrik, Module categories over the Drinfeld double of a finite group, Int. Math. Res. Not. 27 (2003), 1507 – 1520. · Zbl 1044.18005 |

[50] | J. Roberts and S. Willerton, On the Rozansky-Witten weight systems. math.AG/ 0602653. · Zbl 1268.57007 |

[51] | L. Rozansky and E. Witten, Hyper-Kähler geometry and invariants of three-manifolds, Selecta Math. (N.S.) 3 (1997), no. 3, 401 – 458. · Zbl 0908.53027 |

[52] | Stefan Schwede and Brooke Shipley, Stable model categories are categories of modules, Topology 42 (2003), no. 1, 103 – 153. · Zbl 1013.55005 |

[53] | Brooke Shipley, \?\Bbb Z-algebra spectra are differential graded algebras, Amer. J. Math. 129 (2007), no. 2, 351 – 379. · Zbl 1120.55007 |

[54] | D. Tamarkin, Another proof of M. Kontsevich formality theorem. arXiv:math/9803025. |

[55] | D. Tamarkin, The deformation complex of a \( d\)-algebra is a \( (d+1)\)-algebra. arXiv:math/ 0010072. |

[56] | R. W. Thomason and Thomas Trobaugh, Higher algebraic \?-theory of schemes and of derived categories, The Grothendieck Festschrift, Vol. III, Progr. Math., vol. 88, Birkhäuser Boston, Boston, MA, 1990, pp. 247 – 435. · Zbl 0731.14001 |

[57] | Bertrand Toën, The homotopy theory of \?\?-categories and derived Morita theory, Invent. Math. 167 (2007), no. 3, 615 – 667. · Zbl 1118.18010 |

[58] | Bertrand Toën, Higher and derived stacks: a global overview, Algebraic geometry — Seattle 2005. Part 1, Proc. Sympos. Pure Math., vol. 80, Amer. Math. Soc., Providence, RI, 2009, pp. 435 – 487. · Zbl 1183.14001 |

[59] | Bertrand Toën and Gabriele Vezzosi, Homotopical algebraic geometry. I. Topos theory, Adv. Math. 193 (2005), no. 2, 257 – 372. · Zbl 1120.14012 |

[60] | Bertrand Toën and Gabriele Vezzosi, Homotopical algebraic geometry. II. Geometric stacks and applications, Mem. Amer. Math. Soc. 193 (2008), no. 902, x+224. · Zbl 1145.14003 |

[61] | Thomas Tradler and Mahmoud Zeinalian, On the cyclic Deligne conjecture, J. Pure Appl. Algebra 204 (2006), no. 2, 280 – 299. · Zbl 1147.16012 |

This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.