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.

Mathematics Subject Classification

14F05 16E45 18G10 

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Notes

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.

References

  1. 1.
    Beke, T.: Sheafifiable homotopy model categories. Math. Proc. Camb. Philos. Soc. 129(3), 447–475 (2000)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Bondal, A.I., Kapranov, M.M.: Enhanced triangulated categories. Mat. Sb. 181(5), 669–683 (1990)MATHGoogle Scholar
  3. 3.
    Bondal, A.I., Larsen, M., Lunts, V.A.: Grothendieck ring of pretriangulated categories. Int. Math. Res. Not. 29, 1461–1495 (2004)MathSciNetCrossRefMATHGoogle Scholar
  4. 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. 5.
    Cisinski, D.-C., Déglise, F.: Local and stable homological algebra in Grothendieck abelian categories. Homol. Homot. Appl. 11(1), 219–260 (2009)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Cirone, E.R.: A strictly-functorial and small dg-enhancement of the derived category of perfect complexes (2017). arXiv:1502.06573v2 Google Scholar
  7. 7.
    Cisinski, D.-C.: Localizing an arbitrary additive category. MathOverflow (2010). http://mathoverflow.net/q/44155 (version: 2010-10-29)
  8. 8.
    Dugger, D., Hollander, S., Isaksen, D.C.: Hypercovers and simplicial presheaves. Math. Proc. Camb. Philos. Soc. 136(1), 9–51 (2004)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Drinfeld, V.: DG quotients of DG categories. Revised Publication (2008). arXiv:math/0210114v7 Google Scholar
  10. 10.
    Gillespie, J.: The flat model structure on complexes of sheaves. Trans. Am. Math. Soc. 358(7), 2855–2874 (2006)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Gillespie, J.: Kaplansky classes and derived categories. Math. Z. 257(4), 811–843 (2007)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Ganter, N., Kapranov, M.: Representation and character theory in 2-categories. Adv. Math. 217(5), 2268–2300 (2008)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Gabber, O., Ramero, L.: Foundations for almost ring theory. Preprint, Release 7 (2017). arXiv:math/0409584v12 Google Scholar
  14. 14.
    Groth, M.: Monoidal derivators and additive derivators (2012). arXiv:1203.5071v1 Google Scholar
  15. 15.
    Guillermou, S.: dg-methods for Microlocalization. Publ. Res. Inst. Math. Sci. 47(1), 99–140 (2011)MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Hirschhorn, P.S.: Model Categories and Their Localizations, Volume 99 of Mathematical Surveys and Monographs. American Mathematical Society, Providence (2003)Google Scholar
  17. 17.
    Hörmann, F.: Six functor formalisms and fibered multiderivators (2017). arXiv:1603.02146v2 Google Scholar
  18. 18.
    Hovey, M.: Model Categories, Volume 63 of Mathematical Surveys and Monographs. American Mathematical Society, Providence (1999)Google Scholar
  19. 19.
    Kelly, G.M.: Basic concepts of enriched category theory. Repr. Theory Appl. Categ. 10, vi+137 (2005)MathSciNetMATHGoogle Scholar
  20. 20.
    Keller, B.: On differential graded categories. In International Congress of Mathematicians. Vol. II, pp. 151–190. European Mathematical Society, Zürich (2006)Google Scholar
  21. 21.
    Kashiwara, M., Schapira, P.: Sheaves on Manifolds, Volume 292 of Grundlehren der Mathematischen Wissenschaften. Springer, Berlin (1994)Google Scholar
  22. 22.
    Kashiwara, M., Schapira, P.: Categories and Sheaves, Volume 332 of Grundlehren der Mathematischen Wissenschaften. Springer, Berlin (2006)Google Scholar
  23. 23.
    Kuznetsov, A.: Height of exceptional collections and Hochschild cohomology of quasiphantom categories. J. Reine Angew. Math. 708, 213–243 (2015)MathSciNetMATHGoogle Scholar
  24. 24.
    Lawson, T.: Localization of enriched categories and cubical sets (2016). arXiv:1602.05313v1 Google Scholar
  25. 25.
    Leinster, T.: Higher Operads, Higher Categories, Volume 298 of London Mathematical Society Lecture Note Series. Cambridge University Press, Cambridge (2004)CrossRefGoogle Scholar
  26. 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)Google Scholar
  27. 27.
    Lunts, V.A., Orlov, D.O.: Uniqueness of enhancement for triangulated categories. J. Am. Math. Soc. 23(3), 853–908 (2010)MathSciNetCrossRefMATHGoogle Scholar
  28. 28.
    Lunts, V.A., Schnürer, O.M.: Smoothness of equivariant derived categories. Proc. Lond. Math. Soc. 108(5), 1226–1276 (2014)MathSciNetCrossRefMATHGoogle Scholar
  29. 29.
    Lunts, V.A., Schnürer, O.M.: New enhancements of derived categories of coherent sheaves and applications. J. Algebra 446, 203–274 (2016)MathSciNetCrossRefMATHGoogle Scholar
  30. 30.
    Lunts, V.A.: Categorical resolution of singularities. J. Algebra 323(10), 2977–3003 (2010)MathSciNetCrossRefMATHGoogle Scholar
  31. 31.
    Lurie, J.: Higher Topos Theory, Volume 170 of Annals of Mathematics Studies. Princeton University Press, Princeton (2009)Google Scholar
  32. 32.
    Liu, Y., Zheng, W.: Enhanced six operations and base change theorem for Artin stacks (2017). arXiv:1211.5948v3 Google Scholar
  33. 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. 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)MATHGoogle Scholar
  35. 35.
    Nadler, D.: Microlocal branes are constructible sheaves. Sel. Math. (N.S.) 15(4), 563–619 (2009)MathSciNetCrossRefMATHGoogle Scholar
  36. 36.
    Polishchuk, A., van den Bergh, M.: Semiorthogonal decompositions of the categories of equivariant coherent sheaves for some reflection groups (2017). arXiv:1503.04160v4 Google Scholar
  37. 37.
    Riehl, E.: Categorical Homotopy Theory, Volume 24 of New Mathematical Monographs. Cambridge University Press, Cambridge (2014)CrossRefGoogle Scholar
  38. 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)Google Scholar
  39. 39.
    Shulman, M.: Homotopy limits and colimits and enriched homotopy theory (2009). arXiv:math/0610194v3 Google Scholar
  40. 40.
    Spaltenstein, N.: Resolutions of unbounded complexes. Compos. Math. 65(2), 121–154 (1988)MathSciNetMATHGoogle Scholar
  41. 41.
    Schnürer, O.M., Soergel, W.: Proper base change for separated locally proper maps. Rend. Semin. Mat. Univ. Padova 135, 223–250 (2016)MathSciNetCrossRefMATHGoogle Scholar
  42. 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-DonatGoogle Scholar
  43. 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-DonatGoogle Scholar
  44. 44.
    The Stacks Project Authors. The Stacks Project (2016). http://stacks.math.columbia.edu
  45. 45.
    Toën, B.: The homotopy theory of $dg$-categories and derived Morita theory. Invent. Math. 167(3), 615–667 (2007)MathSciNetCrossRefMATHGoogle Scholar
  46. 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)Google Scholar
  47. 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)Google Scholar

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Authors and Affiliations

  1. 1.Mathematisches InstitutUniversität BonnBonnGermany

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