A general homological Kleiman-Bertini theorem

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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 Tor sheaves of F against G-orbits vanish, then for generic g is an element of G, the sheaf TorjX(g F, E) 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.

Original languageEnglish
Pages (from-to)597-609
Number of pages13
JournalAlgebra & Number Theory
Issue number5
Publication statusPublished - 2009


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


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