A comparison of Hochschild homology in algebraic and smooth settings

Abstract

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 $C^{\infty }(V{)}^G$.

arXiv commentV2: compactness assumptions in section 3 lifted. V3: various corrections and improvements in the proofs in sections 1 and 2, new section with examples