Hostname: page-component-89b8bd64d-rbxfs Total loading time: 0 Render date: 2026-05-07T10:31:58.409Z Has data issue: false hasContentIssue false
  • English
  • Français

Fourier–Mukai functors in the supported case

Part of: (Co)homology theory Abelian categories

Published online by Cambridge University Press:  30 June 2014

Alberto Canonaco  and
Paolo Stellari
Alberto Canonaco
Affiliation:
Dipartimento di Matematica ‘F. Casorati’, Università degli Studi di Pavia, Via Ferrata 1, 27100 Pavia, Italy email alberto.canonaco@unipv.it
Paolo Stellari
Affiliation:
Dipartimento di Matematica ‘F. Enriques’, Università degli Studi di Milano, Via Cesare Saldini 50, 20133 Milano, Italy email paolo.stellari@unimi.it
Get access Rights & Permissions [Opens in a new window]

Abstract

We prove that exact functors between the categories of perfect complexes supported on projective schemes are of Fourier–Mukai type if the functor satisfies a condition weaker than being fully faithful. We also get generalizations of the results in the literature in the case without support conditions. Some applications are discussed and, along the way, we prove that the category of perfect supported complexes has a strongly unique enhancement.

Keywords

derived categoriessupported sheavesFourier–Mukai functors

MSC classification

Secondary: 14F05: Sheaves, derived categories of sheaves and related constructions 18E10: Exact categories, abelian categories 18E30: Derived categories, triangulated categories

Information

Type
Research Article
Information
Compositio Mathematica , Volume 150 , Issue 8 , August 2014 , pp. 1349 - 1383
Copyright
© The Author(s) 2014 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

References

Ballard, M. R., Sheaves on local Calabi–Yau varieties, Preprint (2008), arXiv:0801.3499.Google Scholar
Ballard, M. R., Equivalences of derived categories of sheaves on quasi-projective schemes, Preprint (2009), arXiv:0905.3148.Google Scholar
Bayer, A. and Macrì, E., The space of stability conditions on the local projective plane, Duke Math. J. 160 (2011), 263–322.Google Scholar
Bondal, A., Larsen, M. and Lunts, V., Grothendieck ring of pretriangulated categories, Int. Math. Res. Not. IMRN 29 (2004), 1461–1495.CrossRefGoogle Scholar
Bondal, A. and Van den Bergh, M., Generators and representability of functors in commutative and noncommutative geometry, Mosc. Math. J. 3 (2003), 1–36.CrossRefGoogle Scholar
Bridgeland, T., Stability conditions on triangulated categories, Ann. of Math. (2) 166 (2007), 317–346.Google Scholar
Canonaco, A., Orlov, D. and Stellari, P., Does full imply faithful?, J. Noncommut. Geom. 7 (2013), 357–371.Google Scholar
Canonaco, A. and Stellari, P., Uniqueness of dg enhancements for categories of compact objects, available at: http://users.unimi.it/stellari/Research/Papers/compact.pdf.Google Scholar
Canonaco, A. and Stellari, P., Twisted Fourier–Mukai functors, Adv. Math. 212 (2007), 484–503.Google Scholar
Canonaco, A. and Stellari, P., Non-uniqueness of Fourier–Mukai kernels, Math. Z. 272 (2012), 577–588.Google Scholar
Canonaco, A. and Stellari, P., Fourier–Mukai functors: a survey, EMS Series of Congress Reports (Eur. Math. Soc., Zurich, 2013), 27–60.Google Scholar
Drinfeld, V., DG quotients of DG categories, J. Algebra 272 (2004), 643–691.CrossRefGoogle Scholar
Fu, C. and Yang, D., The Ringel–Hall Lie algebra of a spherical object, J. Lond. Math. Soc. (2) 85 (2012), 511–533.Google Scholar
Hartshorne, R., Algebraic geometry, Graduate Texts in Mathematics, vol. 52 (Springer, New York, 1977).CrossRefGoogle Scholar
Huybrechts, D. and Lehn, M., The geometry of moduli spaces of sheaves (Cambridge Unversity Press, Cambridge, 2010).CrossRefGoogle Scholar
Ishii, A., Ueda, K. and Uehara, H., Stability conditions on A n-singularities, J. Differential Geom. 84 (2010), 87–126.CrossRefGoogle Scholar
Ishii, A. and Uehara, H., Autoequivalences of derived categories on the minimal resolutions of A n-singularities on surfaces, J. Differential Geom. 71 (2005), 385–435.Google Scholar
Kashiwara, M. and Schapira, P., Categories and sheaves, Grundlehren der Mathematischen Wissenschaften, vol. 332 (Springer, Berlin, 2006).Google Scholar
Kawamata, Y., Equivalences of derived categories of sheaves on smooth stacks, Amer. J. Math. 126 (2004), 1057–1083.CrossRefGoogle Scholar
Keller, B., Chain complexes and stable categories, Manuscripta Math. 67 (1990), 379–417.Google Scholar
Keller, B., Derived categories and their uses, in Handbook of algebra, Vol. 1 (North-Holland, Amsterdam, 1996), 671–701.Google Scholar
Keller, B., On differential graded categories, in International Congress of Mathematicians, Vol. II (European Mathematical Society, Zürich, 2006), 151–190.Google Scholar
Keller, B., Yang, D. and Zhou, G., The Hall algebra of a spherical object, J. Lond. Math. Soc. 80 (2009), 771–784.Google Scholar
Kontsevich, M., Homological algebra of mirror symmetry, in Proceedings of the International Congress of Mathematicians (Zurich, 1994), ed. Chatterji, S. D. (Birkhäuser, Basel, 1995), 120–139.Google Scholar
Kuznetsov, A. and Lunts, V., Categorical resolutions of irrational singularities, Preprint (2012), arXiv:1212.6170.Google Scholar
Lin, K. and Pomerleano, D., Global matrix factorizations, Preprint (2011), arXiv:1101.5847.Google Scholar
Lipman, J., Lectures on local cohomology and duality, in Local cohomology and its applications, Lecture Notes in Pure and Applied Mathematics, vol. 226 (Marcel Dekker, New York, 2001), 39–89.Google Scholar
Lunts, V. and Orlov, D., Uniqueness of enhancements for triangulated categories, J. Amer. Math. Soc. 23 (2010), 853–908.Google Scholar
Neeman, A., The Grothendieck duality theorem via Bousfield’s techniques and Brown representability, J. Amer. Math. Soc. 9 (1996), 205–236.Google Scholar
Orlov, D., Equivalences of derived categories and K3 surfaces, J. Math. Sci. 84 (1997), 1361–1381.Google Scholar
Rouquier, R., Dimensions of triangulated categories, J. K-Theory 1 (2008), 193–258.Google Scholar
Toën, B., The homotopy theory of dg-categories and derived Morita theory, Invent. Math. 167 (2007), 615–667.Google Scholar