A Kleiman–Bertini theorem for sheaf tensor products

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