Hostname: page-component-cd9895bd7-hc48f Total loading time: 0 Render date: 2024-12-29T02:50:51.635Z Has data issue: false hasContentIssue false
  • English
  • Français

A derived equivalence for a degree 6 del Pezzo surface over an arbitrary field

Published online by Cambridge University Press:  20 January 2011

M. Blunk ,
S.J. Sierra  and
S. Paul Smith
M. Blunk
Affiliation:
Department of Mathematics, The University of British Columbia Room 121, 1984 Mathematics Road Vancouver, B.C. V6T 1Z2, Canada, [email protected]
S.J. Sierra
Affiliation:
Princeton University, Department of Mathematics, Fine Hall, Washington Road Princeton, NJ 08544-1000, USA, [email protected]
S. Paul Smith
Affiliation:
Department of Mathematics, Box 354350, Univ. Washington Seattle, WA 98195, USA, [email protected]
Get access

Abstract

Let S be a degree six del Pezzo surface over an arbitrary field F. Motivated by the first author's classification of all such S up to isomorphism [3] in terms of a separable F-algebra B×Q×F, and by his K-theory isomorphism Kn(S) ≅ Kn(B×Q×F) for n ≥ 0, we prove an equivalence of derived categories

where A is an explicitly given finite dimensional F-algebra whose semisimple part is B×Q×F.

Keywords

Derived equivalencedel Pezzo surfacearbitrary field
Type
Research Article
Information
Journal of K-Theory , Volume 8 , Issue 3 , December 2011 , pp. 481 - 492
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.Auslander, M., Reiten, I., and Smalø, S., Representation Theory of Artin Algebras, Camb. Studies in Adv. Math., No. 36, 1995.Google Scholar
2.Baer, D., Tilting sheaves in representation theory of algebras, Manus. Math., 60 (1988) 323–347.CrossRefGoogle Scholar
3.Blunk, M., Del Pezzo surfaces of degree 6 over an arbitrary field, J. Algebra, 323 (2010), no. 1, 42–58CrossRefGoogle Scholar
4.Bondal, A. and Van den Bergh, M., Generators and representability of functors in commutative and noncommutative geometry, Moscow Mathematical Journal, 3 (2003), no. 1, 1–36.CrossRefGoogle Scholar
5.Cox, D.A., The homogeneous coordinate ring of a toric variety, J. Alg. Geom., 4 (1995), no. 1, 17–50.Google Scholar
6.Hille, L. and Van den Bergh, M., Fourier-Mukai transforms, pp. 147-173 in Handbook of Tilting Theory, Lond. Math. Soc. Lect. Note Series (No. 332), Eds. Hügel, L.A., Happel, D., and Krause, H., Camb. Univ. Press, 2007.Google Scholar
7.Huybrechts, D., Fourier-Mukai transforms in algebraic geometry, Oxford Univ. Press, 2006.CrossRefGoogle Scholar
8.Neeman, A., The connection between the K-theory localization theorem of Thomason, Trobaugh and Yao and the smashing subcategories of Bousfield and Ravanel, Ann. Sci. de l' Éc. Norm. Sup., 25 (1992), no. 5, 547–566.CrossRefGoogle Scholar
9.Smith, S.P., Computation of the Grothendieck and Picard groups of a toric DM stack χ, by using a homogeneous coordinate ring for χ, Glasgow Math. Jour., to appear. arXiv:0806.0192v2.Google Scholar
10.Thomason, R. W. and Trobaugh, T., Higher algebraic K-theory of schemes and of derived categories, pp. 247-435 in The Grothendieck Festschrift, Vol. III, Progr. Math., 88, Birkhäuser Boston, Boston, MA, 1990Google Scholar