Hostname: page-component-78c5997874-mlc7c Total loading time: 0 Render date: 2024-11-14T05:20:54.253Z Has data issue: false hasContentIssue false
  • English
  • Français

K-theoretic exceptional collections at roots of unity

Published online by Cambridge University Press:  11 May 2010

A. Polishchuk
A. Polishchuk
Affiliation:
Department of Mathematics, University of Oregon, Eugene, OR 97405, [email protected]
Get access

Abstract

Using cyclotomic specializations of equivariant K-theory with respect to a torus action we derive congruences for discrete invariants of exceptional objects in derived categories of coherent sheaves on a class of varieties that includes Grassmannians and smooth quadrics. For example, we prove that if , where the ni's are powers of a fixed prime number p, then the rank of an exceptional object on X is congruent to ±1 modulo p.

Keywords

Exceptional collectionequivariant K-theoryLefschetz formula
Type
Research Article
Information
Journal of K-Theory , Volume 7 , Issue 1 , February 2011 , pp. 169 - 201
Copyright
Copyright © ISOPP 2010

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.)

References

1.Beilinson, A., Coherent sheaves on and problems in linear algebra, Functional Anal. Appl. 12 (1978), no. 3, 214–216.Google Scholar
2.Beilinson, A., Bernstein, J., Deligne, P., Faisceaux pervers, Astérisque, 100, Soc. Math. France, Paris, 1982.Google Scholar
3.Bourbaki, N., Éléments de mathématique. Groupes et algébres de Lie. Ch. IV–VI. Hermann, Paris, 1968.Google Scholar
4.Bridgeland, T., t-structures on some local Calabi-Yau varieties, J. Algebra 289 (2005), no. 2, 453–483.Google Scholar
5.Elagin, A., Semiorthogonal decompositions of derived categories of equivariant coherent sheaves, preprint math.AG/0809.5166.Google Scholar
6.Gorodentsev, A., Kuleshov, S., Helix theory, Mosc. Math. J. 4 (2004), no. 2, 377–440.Google Scholar
7.Inaba, M., Toward a definition of moduli of complexes of coherent sheaves on a projective scheme, J. Math. Kyoto Univ. 42 (2002), no. 2, 317–329.Google Scholar
8.Kambayashi, T., Projective representation of algebraic linear groups of transformations, American Journal of Math. 88, no. 1,199–205.Google Scholar
9.Kapranov, M., On the derived category of coherent sheaves on Grassmann manifolds, Math USSR Izvestiya 24 (1985), 183–192.Google Scholar
10.Kapranov, M., Derived category of coherent bundles on a quadric, Functional Anal. Appl. 20 (1986), no. 2, 141–142.Google Scholar
11.Kostant, B., Kumar, S., T-equivariant K-theory of generalized flag varieties, J. DIff. Geom. 32 (1990), 549–603.Google Scholar
12.Kuleshov, S., Orlov, D., Exceptional sheaves on del Pezzo surfaces, Izvestiya Math. 44 (1995), no. 3, 479–513.Google Scholar
13.Kuznetsov, A., Hyperplane sections and derived categories, Izvestiya Math. 70 (2006), no. 3, 447–547.Google Scholar
14.Lieblich, M., Moduli of complexes on a proper morphism, J. Algebraic Geom. 15 (2006), no. 1, 175–206.Google Scholar
15.Macdonald, I. G., Symmetric functions and Hall polynomials, Oxford University Press, New York, 1979.Google Scholar
16.Merkurjev, A. S., Comparison of the equivariant and the standard K-theory of algebraic varieties, St. Petersburg Math. J. 9 (1998), no. 4, 815–850.Google Scholar
17.Nielsen, H. A., Diagonalizably linearized coherent sheaves, Bull. Soc. Math. France 102 (1974), 85–97.Google Scholar
18.Nogin, D., Helices on some Fano threefolds: constructivity of semiorthogonal bases of K0, Ann. Sci. Čcole Norm. Sup. (4) 27 (1994), no. 2, 129–172.Google Scholar
19.Popov, V. L., Picard groups of homogeneous spaces of linear algebraic groups and one-dimensional homogeneous vector fiberings, Math. USSR Izvestiya 8 (1974), 301–327.Google Scholar
20.Samokhin, A., The derived category of coherent sheaves on , Russian Math. Surveys 56 (2001), 592–594.Google Scholar
21.Segal, G., Equivariant K-theory, IHES Publ. Math. 34 (1968) 129–151.Google Scholar
22.Vezzosi, G., Vistoli, A., Higher algebraic K-theory of group actions with finite stabilizers, Duke Math. J. 113 (2002), no. 1, 1–55.Google Scholar