Advertisement

Mathematische Zeitschrift

, Volume 287, Issue 1–2, pp 155–179 | Cite as

On the existence of Fourier–Mukai functors

  • Alice RizzardoEmail author
Article
  • 150 Downloads

Abstract

A theorem by Orlov states that any equivalence \(F:D^{b}_{\mathrm {Coh}}(X) \rightarrow D^{b}_{\mathrm {Coh}}(Y)\) between the bounded derived categories of coherent sheaves of two smooth projective varieties X and Y is isomorphic to a Fourier–Mukai transform \(\Phi _{E}(-)=R\pi _{2*}(E\mathop {\otimes }\limits ^{L} L\pi _1^{*}(-))\), where the kernel E is in \(D^{b}_{\mathrm {Coh}}(X\times Y)\). In the case of an exact functor which is not necessarily fully faithful, we compute some sheaves that play the role of the cohomology sheaves of the kernel, and that are isomorphic to the latter whenever an isomorphism \(F\cong \Phi _{E}\) exists. We then exhibit a class of functors that are not full or faithful and still satisfy the above result.

Keywords

Fourier–Mukai functor Orlov’s theorem 

Mathematics Subject Classification

13D09 18E30 

Notes

Acknowledgements

This paper is derived from part of the author’s PhD thesis. The author thanks her thesis advisor Aise Johan de Jong for suggesting the problem as well as providing invaluable guidance over the years. The last draft of this paper was written while at SISSA, in Trieste.

References

  1. 1.
    Bondal, A., Van den Bergh, M.: Generators and representability of functors in commutative and noncommutative geometry. Mosc. Math. J. 3(1), 1–36 (2003)MathSciNetzbMATHGoogle Scholar
  2. 2.
    Canonaco, A., Orlov, D.O., Stellari, P.: Does full imply faithful? J. Noncommut. Geom. 7(2), 357–371 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Canonaco, A., Stellari, P.: Twisted Fourier–Mukai functors. Adv. Math. 212(2), 484–503 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Canonaco, A., Stellari, P.: Non-uniqueness of Fourier–Mukai kernels. Math. Z. 272, 577–588 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Dold, A.: Zur Homotopietheorie der Kettenkomplexe. Math. Ann. 140, 278–298 (1960)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Grothendieck, A.: Éléments de géométrie algébrique. II. Étude globale élémentaire de quelques classes de morphismes. Inst. Hautes Études Sci. Publ. Math. No. 8 (1961)Google Scholar
  7. 7.
    Orlov, D.O.: Equivalences of derived categories and K3 surfaces. J. Math. Sci. (New York) 84(5), 1361–1381 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Rizzardo, A., Van den Bergh, M.: An example of a non-Fourier–Mukai Functor Between Derived Categories of Coherent Sheaves Preprint. Available at arxiv:1410.4039
  9. 9.
    Rizzardo, A., Van den Bergh, M.: Scalar extensions of derived categories and non-fourier-mukai functors. Adv. Math. 281, 1100–1144 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Serre, J.-P.: Faisceaux algébriques cohérents, vol. 61. The Annals of Mathematics , Second Series (2007)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2016

Authors and Affiliations

  1. 1.SISSATriesteItaly

Personalised recommendations