A comparison of Hochschild homology in algebraic and smooth settings

TL;DR

This paper compares Hochschild homology in algebraic and smooth contexts for complex varieties and real submanifolds, establishing flatness and homology comparison theorems under certain geometric conditions.

Contribution

It proves flatness of smooth functions over regular functions and provides a comparison theorem for Hochschild homology between algebraic and smooth settings.

Findings

C^ ablafty (V) is flat over O ( ilde V) under mild tangent space conditions.

Hochschild homology of algebras over O ( ilde V) and C^ ablafty (V) are comparable.

C^ ablafty (V) has finite rank as a module over its G-invariant subalgebra.

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