  Consider a complex affine variety $\tilde V$ and a real analytic
Zariski-dense submanifold V of $\tilde V$. We compare modules over the ring $O
(\tilde V)$ of regular functions on $\tilde V$ with modules over the ring
$C^\infty (V)$ of smooth complex valued functions on V.
  Under a mild condition on the tangent spaces, we prove that $C^\infty (V)$ is
flat as a module over $O (\tilde V)$. From this we deduce a comparison theorem
for the Hochschild homology of finite type algebras over $O (\tilde V)$ and the
Hochschild homology of similar algebras over $C^\infty (V)$.
  We also establish versions of these results for functions on $\tilde V$
(resp. V) that are invariant under the action of a finite group G. As an
auxiliary result, we show that $C^\infty (V)$ has finite rank as module over
$C^\infty (V)^G$.
