@MISC{Miller06akleiman–beritini, author = {Ezra Miller and David and E Speyer}, title = {A KLEIMAN–BERITINI THEOREM FOR SHEAF TENSOR PRODUCTS}, year = {2006} }

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Abstract

Abstract. Fix a variety X with a transitive (left) action by an algebraic group G. Let E and F be coherent sheaves on X. We prove that for elements g in a dense open subset of G, the sheaf Tor X i (E, gF) vanishes for all i> 0. When E and 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 or hypotheses on the characteristic of the ground field. All schemes in this note are Noetherian over an arbitrary base field k. In particular, we make no restrictions on the characteristic. The aim of this note is to prove the following result. Theorem. Let X be a reduced variety with a transitive left action of an algebraic group G. Let E and F be coherent sheaves on X and, for g ∈ G, let gF denote the pushforward of F along multiplication by g. Then there is a Zariski open subset U of G such that, for all k-rational points g ∈ U, the sheaf Tori(E, gF) is 0 for all i> 0. If k is infinite, and G is connected as well as affine, and either (1) G is reductive or (2) k is perfect, then U always contains a k-rational point of G [Bor91, Corollary 18.3].