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.