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: 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 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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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.