A KleimanâBertini theorem for sheaf tensor products
Authors:
Ezra Miller and David E Speyer
Journal:
J. Algebraic Geom. 17 (2008), 335-340
DOI:
https://doi.org/10.1090/S1056-3911-07-00479-1
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 $\mathcal {T}\hspace {-.7ex}or^X_i(\mathcal {E}, g \mathcal {F})$ 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 KleimanâBertini 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
- Armand Borel, Linear algebraic groups, 2nd ed., Graduate Texts in Mathematics, vol. 126, Springer-Verlag, New York, 1991. MR 1102012
- Michel Brion, Positivity in the Grothendieck group of complex flag varieties, J. Algebra 258 (2002), no. 1, 137â159. Special issue in celebration of Claudio Procesiâs 60th birthday. MR 1958901, DOI https://doi.org/10.1016/S0021-8693%2802%2900505-7
- Michel Brion, Lectures on the geometry of flag varieties, Topics in cohomological studies of algebraic varieties, Trends Math., BirkhĂ€user, Basel, 2005, pp. 33â85. MR 2143072, DOI https://doi.org/10.1007/3-7643-7342-3_2
- Anders Skovsted Buch, A Littlewood-Richardson rule for the $K$-theory of Grassmannians, Acta Math. 189 (2002), no. 1, 37â78. MR 1946917, DOI https://doi.org/10.1007/BF02392644
- David Eisenbud, Commutative algebra, Graduate Texts in Mathematics, vol. 150, Springer-Verlag, New York, 1995. With a view toward algebraic geometry. MR 1322960
- A. Grothendieck, ĂlĂ©ments de gĂ©omĂ©trie algĂ©brique. IV. Ătude locale des schĂ©mas et des morphismes de schĂ©mas. II, Inst. Hautes Ătudes Sci. Publ. Math. 24 (1965), 231 (French). MR 199181
- A. Grothendieck, ĂlĂ©ments de gĂ©omĂ©trie algĂ©brique. IV. Ătude locale des schĂ©mas et des morphismes de schĂ©mas. III, Inst. Hautes Ătudes Sci. Publ. Math. 28 (1966), 255. MR 217086
- Robin Hartshorne, Algebraic geometry, Springer-Verlag, New York-Heidelberg, 1977. Graduate Texts in Mathematics, No. 52. MR 0463157
- Steven L. Kleiman, The transversality of a general translate, Compositio Math. 28 (1974), 287â297. MR 360616
- Alain Lascoux, Anneau de Grothendieck de la variĂ©tĂ© de drapeaux, The Grothendieck Festschrift, Vol. III, Progr. Math., vol. 88, BirkhĂ€user Boston, Boston, MA, 1990, pp. 1â34 (French). MR 1106909, DOI https://doi.org/10.1007/978-0-8176-4576-2_1
- Maria Luisa Spreafico, Axiomatic theory for transversality and Bertini type theorems, Arch. Math. (Basel) 70 (1998), no. 5, 407â424. MR 1612610, DOI https://doi.org/10.1007/s000130050213
- Charles A. Weibel, An introduction to homological algebra, Cambridge Studies in Advanced Mathematics, vol. 38, Cambridge University Press, Cambridge, 1994. MR 1269324
References
- A. Borel, Linear Algebraic Groups, Graduate Texts in Mathematics, Vol. 126, Springer-Verlag, New York, 1991. MR 1102012 (92d:20001)
- M. Brion, Positivity in the Grothendieck group of complex flag varieties, Journal of Algebra 258 (2002), no. 1, 137â159. MR 1958901 (2003m:14017)
- M. Brion, Lectures in the geometry of flag varieties, in Topics in cohomological studies of algebraic varieties, Trends in Mathematics, BirkhĂ€user, Boston, 2005, pp. 33â85. MR 2143072 (2006f:14058)
- A. S. Buch, A Littlewood-Richardson rule for the $K$-theory of Grassmannians, Acta. Math. 189 (2002), no. 1, 37â78. MR 1946917 (2003j:14062)
- 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)
- A. Grothendieck, ĂlĂ©ments de gĂ©omĂ©trie algĂ©brique. IV. Ătude locale des schĂ©mas et des morphismes de schĂ©mas, II, Inst. Hautes Ătudes Sci. Publ. Math., Vol. 24, 1965. MR 0199181 (33:7330)
- A. Grothendieck, ĂlĂ©ments de gĂ©omĂ©trie algĂ©brique. IV. Ătude locale des schĂ©mas et des morphismes de schĂ©mas, III, Inst. Hautes Ătudes Sci. Publ. Math., Vol. 28, 1966. MR 0217086 (36:178)
- Robin Hartshorne, Algebraic geometry, Graduate Texts in Mathematics, Vol. 52, Springer-Verlag, New York, 1977. MR 0463157 (57:3116)
- S. Kleiman, The transversality of a generic translate, Compositio Math. 28 (1974) 287â297. MR 0360616 (50:13063)
- A. Lascoux, Anneau de Grothendieck de la variĂ©tĂ© de drapeaux, The Grothendieck Festschrift, Vol. III, 1â34, Progr. Math., 88, BirkhĂ€user Boston, Boston, MA, 1990. MR 1106909 (92j:14064)
- M. Spreafico, Axiomatic theory for transversality and Bertini type theorems, Archiv der Mathematik 70 (1998), 407â424. MR 1612610 (99f:14008)
- 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
Email:
ezra@math.umn.edu
David E Speyer
Affiliation:
Department of Mathematics, University of Michigan, Ann Arbor, Michigan
MR Author ID:
663211
Email:
speyer@umich.edu
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.