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
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}\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.


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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
Email: speyer@umich.edu

DOI: https://doi.org/10.1090/S1056-3911-07-00479-1
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