A general homological Kleiman–Bertini theorem

Sierra, Susan

doi:10.2140/ant.2009.3.597

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A general homological Kleiman–Bertini theorem

Susan J. Sierra

Vol. 3 (2009), No. 5, 597–609

DOI: 10.2140/ant.2009.3.597

Abstract

Let $G$ be a smooth algebraic group acting on a variety $X$ . Let $ℱ$ and $ℰ$ be coherent sheaves on $X$ . We show that if all the higher $T o r$ sheaves of $ℱ$ against $G$ -orbits vanish, then for generic $g \in G$ , the sheaf $T {o r}_{j}^{X} (g ℱ, ℰ)$ vanishes for all $j \geq 1$ . This generalizes a result of Miller and Speyer for transitive group actions and a result of Speiser, itself generalizing the classical Kleiman–Bertini theorem, on generic transversality, under a general group action, of smooth subvarieties over an algebraically closed field of characteristic 0.

Keywords

generic transversality, homological transversality, Kleiman's theorem, group action

Mathematical Subject Classification 2000

Primary: 14L30

Secondary: 16S38

Milestones

Received: 9 March 2009

Accepted: 21 July 2009

Published: 9 November 2009

Authors

Susan J. Sierra
Department of Mathematics University of Washington Seattle, WA 98195 United States
http://www.math.washington.edu/~sjsierra/

Vol. 3, No. 5, 2009