  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 in G, the sheaf Tor^X_j(gF, E) vanishes for
all j >0. 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.
