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Six operations on dg enhancements of derived categories of sheaves

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Abstract

We lift Grothendieck–Verdier–Spaltenstein’s six functor formalism for derived categories of sheaves on ringed spaces over a field to differential graded enhancements. Our main tools come from enriched model category theory.

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References

  1. Beke, T.: Sheafifiable homotopy model categories. Math. Proc. Camb. Philos. Soc. 129(3), 447–475 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  2. Bondal, A.I., Kapranov, M.M.: Enhanced triangulated categories. Mat. Sb. 181(5), 669–683 (1990)

    MATH  Google Scholar 

  3. Bondal, A.I., Larsen, M., Lunts, V.A.: Grothendieck ring of pretriangulated categories. Int. Math. Res. Not. 29, 1461–1495 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  4. Borceux, F.: Handbook of Categorical Algebra 1, Volume 50 of Encyclopedia of Mathematics and Its Applications. Cambridge University Press, Cambridge (1994)

    Google Scholar 

  5. Cisinski, D.-C., Déglise, F.: Local and stable homological algebra in Grothendieck abelian categories. Homol. Homot. Appl. 11(1), 219–260 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  6. Cirone, E.R.: A strictly-functorial and small dg-enhancement of the derived category of perfect complexes (2017). arXiv:1502.06573v2

  7. Cisinski, D.-C.: Localizing an arbitrary additive category. MathOverflow (2010). http://mathoverflow.net/q/44155 (version: 2010-10-29)

  8. Dugger, D., Hollander, S., Isaksen, D.C.: Hypercovers and simplicial presheaves. Math. Proc. Camb. Philos. Soc. 136(1), 9–51 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  9. Drinfeld, V.: DG quotients of DG categories. Revised Publication (2008). arXiv:math/0210114v7

  10. Gillespie, J.: The flat model structure on complexes of sheaves. Trans. Am. Math. Soc. 358(7), 2855–2874 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  11. Gillespie, J.: Kaplansky classes and derived categories. Math. Z. 257(4), 811–843 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  12. Ganter, N., Kapranov, M.: Representation and character theory in 2-categories. Adv. Math. 217(5), 2268–2300 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  13. Gabber, O., Ramero, L.: Foundations for almost ring theory. Preprint, Release 7 (2017). arXiv:math/0409584v12

  14. Groth, M.: Monoidal derivators and additive derivators (2012). arXiv:1203.5071v1

  15. Guillermou, S.: dg-methods for Microlocalization. Publ. Res. Inst. Math. Sci. 47(1), 99–140 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  16. Hirschhorn, P.S.: Model Categories and Their Localizations, Volume 99 of Mathematical Surveys and Monographs. American Mathematical Society, Providence (2003)

    Google Scholar 

  17. Hörmann, F.: Six functor formalisms and fibered multiderivators (2017). arXiv:1603.02146v2

  18. Hovey, M.: Model Categories, Volume 63 of Mathematical Surveys and Monographs. American Mathematical Society, Providence (1999)

    Google Scholar 

  19. Kelly, G.M.: Basic concepts of enriched category theory. Repr. Theory Appl. Categ. 10, vi+137 (2005)

    MathSciNet  MATH  Google Scholar 

  20. Keller, B.: On differential graded categories. In International Congress of Mathematicians. Vol. II, pp. 151–190. European Mathematical Society, Zürich (2006)

  21. Kashiwara, M., Schapira, P.: Sheaves on Manifolds, Volume 292 of Grundlehren der Mathematischen Wissenschaften. Springer, Berlin (1994)

    Google Scholar 

  22. Kashiwara, M., Schapira, P.: Categories and Sheaves, Volume 332 of Grundlehren der Mathematischen Wissenschaften. Springer, Berlin (2006)

    Google Scholar 

  23. Kuznetsov, A.: Height of exceptional collections and Hochschild cohomology of quasiphantom categories. J. Reine Angew. Math. 708, 213–243 (2015)

    MathSciNet  MATH  Google Scholar 

  24. Lawson, T.: Localization of enriched categories and cubical sets (2016). arXiv:1602.05313v1

  25. Leinster, T.: Higher Operads, Higher Categories, Volume 298 of London Mathematical Society Lecture Note Series. Cambridge University Press, Cambridge (2004)

    Book  Google Scholar 

  26. Lipman, J.: Notes on derived functors and Grothendieck duality. In Foundations of Grothendieck Duality for Diagrams of Schemes, Volume 1960 of Lecture Notes in Mathematics, pp. 1–259. Springer, Berlin (2009)

  27. Lunts, V.A., Orlov, D.O.: Uniqueness of enhancement for triangulated categories. J. Am. Math. Soc. 23(3), 853–908 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  28. Lunts, V.A., Schnürer, O.M.: Smoothness of equivariant derived categories. Proc. Lond. Math. Soc. 108(5), 1226–1276 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  29. Lunts, V.A., Schnürer, O.M.: New enhancements of derived categories of coherent sheaves and applications. J. Algebra 446, 203–274 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  30. Lunts, V.A.: Categorical resolution of singularities. J. Algebra 323(10), 2977–3003 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  31. Lurie, J.: Higher Topos Theory, Volume 170 of Annals of Mathematics Studies. Princeton University Press, Princeton (2009)

    Google Scholar 

  32. Liu, Y., Zheng, W.: Enhanced six operations and base change theorem for Artin stacks (2017). arXiv:1211.5948v3

  33. Lane, S.M.: Categories for the Working Mathematician, Volume 5 of Graduate Texts in Mathematics, 2nd edn. Springer, New York (1998)

    Google Scholar 

  34. May, J.P., Ponto, K.: More Concise Algebraic Topology, Localization, Completion, and Model Categories. Chicago Lectures in Mathematics. University of Chicago Press, Chicago (2012)

    MATH  Google Scholar 

  35. Nadler, D.: Microlocal branes are constructible sheaves. Sel. Math. (N.S.) 15(4), 563–619 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  36. Polishchuk, A., van den Bergh, M.: Semiorthogonal decompositions of the categories of equivariant coherent sheaves for some reflection groups (2017). arXiv:1503.04160v4

  37. Riehl, E.: Categorical Homotopy Theory, Volume 24 of New Mathematical Monographs. Cambridge University Press, Cambridge (2014)

    Book  Google Scholar 

  38. Schnürer, O.M.: Six operations on dg enhancements of derived categories of sheaves and applications. In: Opening Perspectives in Algebra, Representations, and Topology (Barcelona, 2015), Trends in Mathematics. Birkhäuser, Basel (2016)

  39. Shulman, M.: Homotopy limits and colimits and enriched homotopy theory (2009). arXiv:math/0610194v3

  40. Spaltenstein, N.: Resolutions of unbounded complexes. Compos. Math. 65(2), 121–154 (1988)

    MathSciNet  MATH  Google Scholar 

  41. Schnürer, O.M., Soergel, W.: Proper base change for separated locally proper maps. Rend. Semin. Mat. Univ. Padova 135, 223–250 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  42. Théorie des topos et cohomologie étale des schémas. Tome 1: Théorie des topos. Lecture Notes in Mathematics, Vol. 269. Springer, Berlin (1972). Séminaire de Géométrie Algébrique du Bois-Marie 1963–1964 (SGA 4), Dirigé par M. Artin, A. Grothendieck, et J. L. Verdier. Avec la collaboration de N. Bourbaki, P. Deligne et B. Saint-Donat

  43. Théorie des topos et cohomologie étale des schémas. Tome 2. Lecture Notes in Mathematics, Vol. 270. Springer, Berlin (1972). Séminaire de Géométrie Algébrique du Bois-Marie 1963–1964 (SGA 4), Dirigé par M. Artin, A. Grothendieck et J. L. Verdier. Avec la collaboration de N. Bourbaki, P. Deligne et B. Saint-Donat

  44. The Stacks Project Authors. The Stacks Project (2016). http://stacks.math.columbia.edu

  45. Toën, B.: The homotopy theory of $dg$-categories and derived Morita theory. Invent. Math. 167(3), 615–667 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  46. Verdier, J.-L.: Dualité dans la cohomologie des espaces localement compacts, Exp. No. 300. In Séminaire Bourbaki: Vol. 1965/1966, Exposés 295–312, pp. viii+293. W. A. Benjamin, Inc., New York (1966)

  47. Weibel, C.A.: Homotopy algebraic $K$-theory. In Algebraic K-theory and algebraic number theory (Honolulu, HI, 1987), Volume 83 of Contemporary Mathematics, pp. 461–488. American Mathematical Society, Providence (1989)

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Acknowledgements

We thank Valery Lunts for many inspiring discussions. He was hoping very much that a theory as presented in this work should exist. We thank Michael Mandell for discussions and Emily Riehl and Michael Shulman for useful correspondence concerning model categories. We thank Timothy Logvinenko, Hanno Becker, Alexander Efimov, James Gillespie, Greg Stevenson, Pierre-Yves Gaillard, Lorenzo Ramero and Amnon Neeman for useful discussions. Hanno Becker and Jan Weidner shared an observation which led to Lemma 4.4. Frédéric Déglise answered a question concerning Theorem 4.8. We thank the referee for very detailed comments, in particular for drawing our attention to set-theoretical problems concerning functor categories, and for suggesting a more intrinsic definition of the 2-multicategory \({{\text {ENH}}}_{\mathsf {k}}\) of dg enhancements. The author was supported by a postdoctoral fellowship of the DFG, and by SPP 1388 and SFB/TR 45 of the DFG.

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  1. Mathematisches Institut, Universität Bonn, Endenicher Allee 60, 53115, Bonn, Germany

    Olaf M. Schnürer

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Schnürer, O.M. Six operations on dg enhancements of derived categories of sheaves. Sel. Math. New Ser. 24, 1805–1911 (2018). https://doi.org/10.1007/s00029-018-0392-4

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