A comparison of Hochschild homology in algebraic and smooth settings
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
| Veröffentlicht: |
2022
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| _version_ | 1866929276712910848 |
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| author | Kazhdan, David Solleveld, Maarten |
| author_facet | Kazhdan, David Solleveld, Maarten |
| contents | 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 $C^\infty (V)^G$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2202_06573 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | A comparison of Hochschild homology in algebraic and smooth settings Kazhdan, David Solleveld, Maarten Commutative Algebra 13D07, 13J10, 16E40 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 $C^\infty (V)^G$. |
| title | A comparison of Hochschild homology in algebraic and smooth settings |
| topic | Commutative Algebra 13D07, 13J10, 16E40 |
| url | https://arxiv.org/abs/2202.06573 |