A Kleiman–Bertini theorem for sheaf tensor products

Abstract

Fix a variety X X with a transitive (left) action by an algebraic group G G . Let E \mathcal {E} and F \mathcal {F} be coherent sheaves on X X . We prove that for elements g g in a dense open subset of G G , the sheaf T o r i X ( E , g F ) \mathcal {T}\hspace {-.7ex}or^X_i(\mathcal {E}, g \mathcal {F}) vanishes for all i > 0 i > 0 . When E \mathcal {E} and F \mathcal {F} are structure sheaves of smooth subschemes of X X in characteristic zero, this follows from the Kleiman–Bertini theorem; our result has no smoothness hypotheses on the supports of E \mathcal {E} or F \mathcal {F} , or hypotheses on the characteristic of the ground field.