A comparison of Hochschild homology in algebraic and smooth settings
| Autor: | Kazhdan, David, Solleveld, Maarten |
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| Rok vydání: | 2022 |
| Předmět: | |
| Druh dokumentu: | Working Paper |
| Popis: | 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 $C^\infty (V)^G$. Comment: V2: compactness assumptions in section 3 lifted. V3: various corrections and improvements in the proofs in sections 1 and 2, new section with examples |
| Databáze: | arXiv |
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