  The classical case of Schur--Weyl duality states that the actions of the
group algebras of $GL_n$ and $S_d$ on the $d^{th}$-tensor power of a free
module of finite rank centralize each other. We show that Schur--Weyl duality
holds for commutative rings where enough scalars can be chosen whose non-zero
differences are invertible. This implies all the known cases of Schur--Weyl
duality so far. We also show that Schur--Weyl duality fails for $\mathbb{Z}$
and for any finite field when $d$ is sufficiently large.
