A comparison of Hochschild homology in algebraic and smooth settings

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