Journal of Algebraic Geometry

Journal of Algebraic Geometry

Online ISSN 1534-7486; Print ISSN 1056-3911



A Kleiman-Bertini theorem for sheaf tensor products

Authors: Ezra Miller and David E Speyer
Journal: J. Algebraic Geom. 17 (2008), 335-340
Published electronically: July 2, 2007
MathSciNet review: 2369089
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Abstract | References | Additional Information

Abstract: Fix a variety $ X$ with a transitive (left) action by an algebraic group $ G$. Let $ \mathcal{E}$ and $ \mathcal{F}$ be coherent sheaves on $ X$. We prove that for elements $ g$ in a dense open subset of $ G$, the sheaf $ \mathcal{T}\hspace{-.7ex}\mathit{or}^X_i(\mathcal{E}, g \mathcal{F})$ vanishes for all $ i > 0$. When $ \mathcal{E}$ and $ \mathcal{F}$ are structure sheaves of smooth subschemes of $ X$ in characteristic zero, this follows from the Kleiman-Bertini theorem; our result has no smoothness hypotheses on the supports of $ \mathcal{E}$ or  $ \mathcal{F}$, or hypotheses on the characteristic of the ground field.

References [Enhancements On Off] (What's this?)

  • [Bor91] A. Borel, Linear Algebraic Groups, Graduate Texts in Mathematics, Vol. 126, Springer-Verlag, New York, 1991. MR 1102012 (92d:20001)
  • [Bri02] M. Brion, Positivity in the Grothendieck group of complex flag varieties, Journal of Algebra 258 (2002), no. 1, 137-159. MR 1958901 (2003m:14017)
  • [Bri05] M. Brion, Lectures in the geometry of flag varieties, in Topics in cohomological studies of algebraic varieties, Trends in Mathematics, Birkhäuser, Boston, 2005, pp. 33-85. MR 2143072 (2006f:14058)
  • [Buc02] A. S. Buch, A Littlewood-Richardson rule for the $ K$-theory of Grassmannians, Acta. Math. 189 (2002), no. 1, 37-78. MR 1946917 (2003j:14062)
  • [Eis95] D. Eisenbud, Commutative algebra with a view toward algebraic geometry, Graduate Texts in Mathematics, Vol. 150, Springer-Verlag, New York, 1995. MR 1322960 (97a:13001)
  • [Gro65] A. Grothendieck, Éléments de géométrie algébrique. IV. Étude locale des schémas et des morphismes de schémas, II, Inst. Hautes Études Sci. Publ. Math., Vol. 24, 1965. MR 0199181 (33:7330)
  • [Gro66] A. Grothendieck, Éléments de géométrie algébrique. IV. Étude locale des schémas et des morphismes de schémas, III, Inst. Hautes Études Sci. Publ. Math., Vol. 28, 1966. MR 0217086 (36:178)
  • [Har77] Robin Hartshorne, Algebraic geometry, Graduate Texts in Mathematics, Vol. 52, Springer-Verlag, New York, 1977. MR 0463157 (57:3116)
  • [Kle74] S. Kleiman, The transversality of a generic translate, Compositio Math. 28 (1974) 287-297. MR 0360616 (50:13063)
  • [Las90] A. Lascoux, Anneau de Grothendieck de la variété de drapeaux, The Grothendieck Festschrift, Vol. III, 1-34, Progr. Math., 88, Birkhäuser Boston, Boston, MA, 1990. MR 1106909 (92j:14064)
  • [Spr98] M. Spreafico, Axiomatic theory for transversality and Bertini type theorems, Archiv der Mathematik 70 (1998), 407-424. MR 1612610 (99f:14008)
  • [Wei94] C. Weibel An introduction to homological algebra, Cambridge Studies in Advanced Mathematics, Vol. 38, Cambridge University Press, Cambridge, 1994. MR 1269324 (95f:18001)

Additional Information

Ezra Miller
Affiliation: School of Mathematics, University of Minnesota, Minneapolis, Minnesota

David E Speyer
Affiliation: Department of Mathematics, University of Michigan, Ann Arbor, Michigan

Received by editor(s): March 4, 2006
Received by editor(s) in revised form: January 30, 2007
Published electronically: July 2, 2007
Additional Notes: The first author gratefully acknowledges support from NSF CAREER award DMS-0449102 and a University of Minnesota McKnight Land-Grant Professorship. The second author is a Clay Research Fellow and is pleased to acknowledge the support of the Clay Mathematics Institute. This paper originated in a visit of David E Speyer to the University of Minnesota, and he is grateful for their excellent hospitality.

American Mathematical Society