Vol. 3, No. 5, 2009

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A general homological Kleiman–Bertini theorem

Susan J. Sierra

Vol. 3 (2009), No. 5, 597–609

Let G be a smooth algebraic group acting on a variety X. Let and be coherent sheaves on X. We show that if all the higher Tor sheaves of against G-orbits vanish, then for generic g G, the sheaf TorjX(g,) vanishes for all j 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.

generic transversality, homological transversality, Kleiman's theorem, group action
Mathematical Subject Classification 2000
Primary: 14L30
Secondary: 16S38
Received: 9 March 2009
Accepted: 21 July 2009
Published: 9 November 2009
Susan J. Sierra
Department of Mathematics
University of Washington
Seattle, WA 98195
United States