An example of a non-Fourier–Mukai functor between derived categories of coherent sheaves

  • Alice Rizzardo
  • Michel Van den BerghEmail author
  • Amnon Neeman


Orlov’s famous representability theorem asserts that any fully faithful exact functor between the bounded derived categories of coherent sheaves on smooth projective varieties is a Fourier–Mukai functor. In this paper we show that this result is false without the fully faithfulness hypothesis. We also show that our functor does not lift to the homotopy category of spectral categories if the ground field is \({\mathbb Q}\).

Mathematics Subject Classification

13D09 18E30 14A22 



The authors thank Greg Stevenson and Adam-Christiaan Van Roosmalen for a number of interesting discussions which reinforced our interest in the problem.

Alice Rizzardo would like to thank the University of Hasselt for its hospitality during two recent visits to Belgium. She would also like to thank MSRI for providing a welcoming environment during the Noncommutative Geometry and Representation Theory semester, which encouraged many fruitful conversations, some of which got this project started. She is grateful to Jon Pridham for providing very useful ideas and guidance for “Appendix B”, and to Julian Holstein for help with understanding stable infinity categories. Finally, she would like to thank her advisor, Johan de Jong, for originally suggesting she look into exact functors between derived categories in the non fully faithful case.

Michel Van den Bergh thanks the International School for Advanced Studies (SISSA) in the beautiful city of Trieste for the wonderful working conditions it provides.

The authors thank Andrei Căldăraru for useful discussions concerning derived self intersections and Fourier–Mukai functors. They are deeply grateful to Wendy Lowen for allowing the use of some ideas developed during an ongoing cooperation with the second author.

Finally the authors thank Amnon Neeman for providing them with an alternative and much more general proof for the extension of (1.1).


  1. 1.
    Alonso Tarrío, L., Jeremías López, A., Lipman, J.: Local homology and cohomology on schemes. Ann. Sci. École Norm. Sup. (4) 30(1), 1–39 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Arinkin, D., Caldararu, A.: When is the self-intersection of a subvariety a fibration? Adv. Math. 231(2), 815–842 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Ballard, M.R.: Equivalences of derived categories of sheaves on quasi-projective schemes. arXiv:0905.3148 [math.AG]
  4. 4.
    Beilinson, A.A., Bernstein, J., Deligne, P.: Faisceaux Pervers, Astérisque, vol. 100. Société Mathématique de France, Paris (1983)zbMATHGoogle Scholar
  5. 5.
    Benson, D., Krause, H., Schwede, S.: Introduction to Realizability of Modules Over Tate Cohomology, Representations of Algebras and Related Topics, Fields Institute Communications, vol. 45, pp. 81–97. American Mathematical Society, Providence (2005)zbMATHGoogle Scholar
  6. 6.
    Blumberg, A.J., Gepner, D., Tabuada, G.: A universal characterization of higher algebraic \(K\)-theory. Geom. Topol. 17(2), 733–838 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Böckstedt, M., Neeman, A.: Homotopy limits in triangulated categories. Compos. Math. 86, 209–234 (1993)MathSciNetzbMATHGoogle Scholar
  8. 8.
    Bondal, A., Kapranov, M.: Representable functors, Serre functors, and reconstructions. Izv. Akad. Nauk SSSR Ser. Mat. 53(6), 1183–1205 (1989). 1337Google Scholar
  9. 9.
    Bondal, A., Van den Bergh, M.: Generators and representability of functors in commutative and noncommutative geometry. Mosc. Math. J. 3(1), 1–36 (2003). 258MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Canonaco, A., Orlov, D., Stellari, P.: Does full imply faithful? J. Noncommut. Geom. 7(2), 357–371 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Canonaco, A., Stellari, P.: Fourier–Mukai functors in the supported case. Compos. Math. 150(8), 1349–1383 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Canonaco, A., Stellari, P.: A tour about existence and uniqueness of dg enhancements and lifts. J. Geom. Phys. 122, 28–52 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Canonaco, A., Stellari, P.: Twisted Fourier–Mukai functors. Adv. Math. 212(2), 484–503 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Canonaco, A., Stellari, P.: Fourier-Mukai functors: a survey. Derived categories in algebraic geometry. EMS Ser. Congr. Rep., Eur. Math. Soc., Zürich, 27–60 (2012)Google Scholar
  15. 15.
    Canonaco, A., Stellari, P.: Non-uniqueness of Fourier–Mukai kernels. Math. Z. 272(1–2), 577–588 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    De Deken, O., Lowen, W.: On deformations of triangulated models. Adv. Math. 243, 330–374 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Gerstenhaber, M., Schack, S.D.: The cohomology of presheaves of algebras. I. Presheaves over a partially ordered set. Trans. Am. Math. Soc. 310(1), 135–165 (1988)MathSciNetzbMATHGoogle Scholar
  18. 18.
    Hartshorne, R.: Residues and duality. Lecture notes in mathematics, vol. 20. Springer, Berlin (1966)CrossRefGoogle Scholar
  19. 19.
    Hartshorne, R.: Algebraic Geometry. Springer, Berlin (1977)CrossRefzbMATHGoogle Scholar
  20. 20.
    Kapranov, M.M.: On the derived categories of coherent sheaves on some homogeneous spaces. Invent. Math. 92(3), 479–508 (1988)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Kawamata, Y.: Equivalences of derived categories of sheaves on smooth stacks. Am. J. Math. 126(5), 1057–1083 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Keller, B.: Introduction to \({A}\)-infinity algebras and modules. Homol. Homotopy Appl. 3(1), 1–35 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Knörrer, H.: Cohen–Macaulay modules on hypersurface singularities I. Invent. Math. 116, 147–164 (1985)MathSciNetGoogle Scholar
  24. 24.
    Kuznetsov, A.: Height of exceptional collections and hochschild cohomology of quasiphantom categories. arXiv:1211.4693 [math.AG]
  25. 25.
    Lefèvre-Hasegawa, K.: Sur les \(A_\infty \)-catégories, Ph.D. thesis, Université Paris, p. 7 (2003)Google Scholar
  26. 26.
    Loday, J.-L.: Cyclic Homology, 2nd edn. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 301, Springer, Berlin (1998). Appendix E by María O. Ronco, Chapter 13 by the Author in Collaboration with Teimuraz PirashviliGoogle Scholar
  27. 27.
    Lowen, W.: Hochschild cohomology, the characteristic morphism and derived deformations. Compos. Math. 144(6), 1557–1580 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Lowen, W.: The curvature problem for formal and infinitesimal deformations. arXiv:1505.03698 [math.KT]
  29. 29.
    Lowen, W.: Hochschild cohomology of abelian categories and ringed spaces. Adv. Math. 198(1), 172–221 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Lowen, W.: A Hochschild cohomology comparison theorem for prestacks. Trans. Am. Math. Soc. 363(2), 969–986 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    Lunts, V.A., Orlov, D.O.: Uniqueness of enhancement for triangulated categories. J. Am. Math. Soc. 23(3), 853–908 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    Lunts, V.A., Schnürer, O.M.: New enhancements of derived categories of coherent sheaves and applications. J. Algebra 446, 203–274 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  33. 33.
    Lurie, J.: Higher Algebra. Accessed 27 Jan 2019
  34. 34.
    Neeman, A.: 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(5), 547–566 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  35. 35.
    Neeman, A.: Some new axioms for triangulated categories. J. Algebra 139, 221–255 (1991)MathSciNetCrossRefzbMATHGoogle Scholar
  36. 36.
    Neeman, A.: Stable homotopy as a triangulated functor. Invent. Math. 109(1), 17–40 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  37. 37.
    Neeman, A.: The Grothendieck duality theorem via Bousfield’s techniques and Brown representability. J. Am. Math. Soc. 9, 205–236 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  38. 38.
    Neeman, A.: Triangulated Categories, Annals of Mathematics Studies, vol. 148. Princeton University Press, Princeton, NJ (2001)Google Scholar
  39. 39.
    Orlov, D.: Smooth and proper noncommutative schemes and gluing of DG categories. Adv. Math. 302, 59–105 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  40. 40.
    Orlov, D.: Equivalences of derived categories and \(K3\) surfaces. J. Math. Sci. (New York) 84(5), 1361–1381 (1997). (algebraic geometry, 7)MathSciNetCrossRefzbMATHGoogle Scholar
  41. 41.
    Positselski, L.: Weakly curved A-infinity algebras over a topological local ring. arXiv:1202.2697 [math.CT]
  42. 42.
    Positselski, L.: Derived equivalences as derived functors. J. Lond. Math. Soc. (2) 43(1), 37–48 (1991)MathSciNetzbMATHGoogle Scholar
  43. 43.
    Rizzardo, A.: On the existence of Fourier–Mukai kernels. Math. Z. 287(1–2), 155–179 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  44. 44.
    Rizzardo, A.: Representability of cohomological functors over extension fields. J. Noncommut. Geom. 11(4), 1267–1287 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  45. 45.
    Rizzardo, Alice, Van den Bergh, Michel: Scalar extensions of derived categories and non-Fourier–Mukai functors. Adv. Math. 281, 1100–1144 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  46. 46.
    Rudyak, Y.B.: On Thom Spectra, Orientability, and Cobordism, Springer Monographs in Mathematics. Springer, Berlin (1998). (with a foreword by Haynes Miller)zbMATHGoogle Scholar
  47. 47.
    Schwede, S.: Symmetric Spectra. Accessed 27 Jan 2019
  48. 48.
    Shipley, B.: \(H\mathbb{Z}\)-algebra spectra are differential graded algebras. Am. J. Math. 129(2), 351–379 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  49. 49.
    Swan, R.G.: Hochschild cohomology of quasiprojective schemes. J. Pure Appl. Algebra 110(1), 57–80 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  50. 50.
    Tabuada, G.: Generalized spectral categories, topological Hochschild homology and trace maps. Algebr. Geom. Topol. 10(1), 137–213 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  51. 51.
    Toën, B.: The homotopy theory of \(dg\)-categories and derived Morita theory. Invent. Math. 167(3), 615–667 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  52. 52.
    van de Ven, A.: A property of algebraic varieties in complex projective spaces. Colloque de Géométrie Différentielle Globale (Bruxelles, 1958), Centre Belge Rech. Math., Louvain, 1959, pp. 151–152. MR 0116361 (22 #7149)Google Scholar
  53. 53.
    Vologodsky, V.: Triangulated endofunctors of the derived category of coherent sheaves which do not admit dg liftings. arXiv:1604.08662 [math.AG]

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  • Alice Rizzardo
    • 1
  • Michel Van den Bergh
    • 2
    Email author
  • Amnon Neeman
    • 3
  1. 1.School of MathematicsThe University of EdinburghEdinburghUK
  2. 2.Universiteit HasseltDiepenbeekBelgium
  3. 3.Centre for Mathematics and Its Applications, Mathematical Sciences InstituteThe Australian National UniversityCanberraAustralia

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