Skip to main content
SpringerLink
Log in

Geometric Phantom Categories

  • Published:
Publications mathématiques de l'IHÉS Aims and scope Submit manuscript

Abstract

In this paper we give a construction of phantom categories, i.e. admissible triangulated subcategories in bounded derived categories of coherent sheaves on smooth projective varieties that have trivial Hochschild homology and trivial Grothendieck group. We also prove that these phantom categories are phantoms in a stronger sense, namely, they has trivial K-motives and, hence, all their higher K-groups are trivial too.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. V. Alexeev and D. Orlov, Derived categories of Burniat surfaces and exceptional collections, preprint (2012), arXiv:1208.4348.

  2. P. Berthelot, A. Grothendieck, and L. Illusie, Théorie des intersections et théorème de Riemann–Roch, in Séminaire de Géométrie Algébrique du Bois-Marie 1966–1967 (SGA 6), Lecture Notes in Mathematics, vol. 225, 1971.

    Google Scholar 

  3. S. Bloch, Lectures on Algebraic Cycles, Duke Univ. Math. Series IV, 1980.

    MATH  Google Scholar 

  4. Ch. Böhning, H.-Ch. Graf von Bothmer, and P. Sosna, On the derived category of the classical Godeaux surface, preprint (2012), arXiv:1206.1830.

  5. Ch. Böhning, H.-Ch. Graf von Bothmer, L. Katzarkov, and P. Sosna, Determinantal Barlow surfaces and phantom categories, preprint (2012), arXiv:1210.0343.

  6. A. Bondal and M. Kapranov, Representable functors, Serre functors, and reconstructions, Izv. Akad. Nauk SSSR, Ser. Mat., 53 (1989), 1183–1205, 1337.

    Google Scholar 

  7. A. Bondal and M. Kapranov, Enhanced triangulated categories, Mat. Sb., 181 (1990), 669–683.

    MATH  Google Scholar 

  8. A. Bondal and D. Orlov, Semiorthogonal decomposition for algebraic varieties, preprint MPIM 95/15 (1995), arXiv:math.AG/9506012.

  9. W. Fulton, Intersection Theory, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 2, Springer, Berlin, 1984.

    Book  MATH  Google Scholar 

  10. S. Galkin and E. Shinder, Exceptional collection on the Beauville surface, preprint (2012), arXiv:1210.3339.

  11. S. Gorchinskiy and V. Guletskii, Motives and representability of algebraic cycles on threefolds over a field, J. Algebr. Geom., 21 (2012), 347–373.

    Article  MathSciNet  MATH  Google Scholar 

  12. H. Inose and M. Mizukami, Rational equivalence of 0-cycles on some surfaces of general type with p g =0, Math. Ann., 244 (1979), 205–217.

    Article  MathSciNet  MATH  Google Scholar 

  13. B. Kahn, J. P. Murre, and C. Pedrini, On the transcendental part of the motive of a surface, in London Math. Soc. Lecture Notes Ser., vol. 344, pp. 143–202, Cambridge University Press, Cambridge, 2004.

    Google Scholar 

  14. B. Keller, Invariance and localization for cyclic homology of DG algebras, J. Pure Appl. Algebra, 123 (1998), 223–273.

    Article  MathSciNet  MATH  Google Scholar 

  15. B. Keller, On differential graded categories, in International Congress of Mathematicians, vol. II, pp. 151–190, Eur. Math. Soc., Zürich, 2006.

    Google Scholar 

  16. M. Kontsevich and Y. Soibelman, Notes on A ∞-algebras, A ∞-categories and non-commutative geometry, in Homological Mirror Symmetry, Lecture Notes in Phys., vol. 757, pp. 153–219, Springer, Berlin, 2009.

    Chapter  Google Scholar 

  17. A. Kuznetsov, Hochschild homology and semiorthogonal decompositions, preprint (2009), arXiv:0904.4330.

  18. A. Kuznetsov, Height of exceptional collections and Hochschild cohomology of quasiphantom categories, preprint (2012), arXiv:1211.4693.

  19. V. Lunts and D. Orlov, Uniqueness of enhancement for triangulated categories, J. Am. Math. Soc., 23 (2010), 853–908.

    Article  MathSciNet  MATH  Google Scholar 

  20. V. A. Lunts and O. M. Schnürer, Smoothness of equivariant derived categories, preprint (2012), arXiv:1205.3132.

  21. Yu. I. Manin, Correspondences, motifs and monoidal transformations, Math. USSR Sb., 6 (1968), 439–470.

    Article  Google Scholar 

  22. M. Marcolli and G. Tabuada, From exceptional collections to motivic decompositions via noncommutative motives, preprint (2012), arXiv:1202.6297.

  23. A. S. Merkurjev and A. A. Suslin, K-cohomology of Severi-Brauer varieties and the norm residue homomorphism, Math. USSR, Izv., 21 (1983), 307–340.

    Article  Google Scholar 

  24. D. Orlov, Derived categories of coherent sheaves and equivalences between them, Russ. Math. Surv., 58 (2003), 511–591.

    Article  MATH  Google Scholar 

  25. D. Quillen, Algebraic K-theory I, Lect. Notes Math., 341 (1973), 85–147.

    Article  MathSciNet  Google Scholar 

  26. A. Scholl, Classical motives, in Proc. Sympos. Pure Math., vol. 55, pp. 163–187, Am. Math. Soc., Providence, 1994.

    Google Scholar 

  27. G. Tabuada, Invariants additifs de dg-catégories, Int. Math. Res. Not., 53 (2005), 3309–3339.

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  29. B. Toën and M. Vaquié, Moduli of objects in dg-categories, Ann. Sci. École Norm. Sup. (4), 40 (2007), 387–444.

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Algebra and Number Theory Section, Steklov Mathematical Institute RAS, Gubkin str. 8, Moscow, 119991, Russia

    Sergey Gorchinskiy

  2. Algebraic Geometry Section, Steklov Mathematical Institute RAS, Gubkin str. 8, Moscow, 119991, Russia

    Dmitri Orlov

Authors
  1. Sergey Gorchinskiy
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Dmitri Orlov
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Dmitri Orlov.

Additional information

S.G. was partially supported by RFBR grants 11-01-00145, 12-01-31506, 12-01-33024, MK-4881.2011.1, NSh grant 5139.2012.1, by AG Laboratory HSE, RF gov. grant, ag. 11.G34.31.0023. D.O. was partially supported by RFBR grants 10-01-93113, 11-01-00336, 11-01-00568, NSh grant 5139.2012.1, by AG Laboratory HSE, RF gov. grant, ag. 11.G34.31.0023.

About this article

Cite this article

Gorchinskiy, S., Orlov, D. Geometric Phantom Categories. Publ.math.IHES 117, 329–349 (2013). https://doi.org/10.1007/s10240-013-0050-5

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10240-013-0050-5

Keywords

Search

Navigation