A relative homology criteria of smoothness
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arXiv
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| Format: | Preprint |
| Published: |
2024
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| _version_ | 1866915587413770240 |
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| author | Iusenko, Kostiantyn Marcos, Eduardo do Nascimento Pretti, Victor do Valle |
| author_facet | Iusenko, Kostiantyn Marcos, Eduardo do Nascimento Pretti, Victor do Valle |
| contents | We investigate the relationship between smoothness and the relative global dimension of a ring extension. We prove that a smooth commutative algebra $A$ over $B$ has finite relative global dimension $\text{gdim}(A,B)$. Conversely, under a mild condition on $B$, the finiteness of $\text{gdim}(A,B)$ implies that the map $B \to A$ is smooth. We also relate the relative global dimension to the usual global dimension of the fibers of $B \to A$, and establish a formula for the relative global dimension of tensor products of extensions. Finally, we present examples and an alternative characterization of smoothness in terms of relative Hochschild homology. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2404_08534 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A relative homology criteria of smoothness Iusenko, Kostiantyn Marcos, Eduardo do Nascimento Pretti, Victor do Valle Commutative Algebra K-Theory and Homology 18G25(Primary), 13D05, 13B10 (Secundary) We investigate the relationship between smoothness and the relative global dimension of a ring extension. We prove that a smooth commutative algebra $A$ over $B$ has finite relative global dimension $\text{gdim}(A,B)$. Conversely, under a mild condition on $B$, the finiteness of $\text{gdim}(A,B)$ implies that the map $B \to A$ is smooth. We also relate the relative global dimension to the usual global dimension of the fibers of $B \to A$, and establish a formula for the relative global dimension of tensor products of extensions. Finally, we present examples and an alternative characterization of smoothness in terms of relative Hochschild homology. |
| title | A relative homology criteria of smoothness |
| topic | Commutative Algebra K-Theory and Homology 18G25(Primary), 13D05, 13B10 (Secundary) |
| url | https://arxiv.org/abs/2404.08534 |