A comparison of Hochschild homology in algebraic and smooth settings

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