A Jacobian criterion for nonsingularity in mixed characteristic
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arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2021
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| _version_ | 1866916394834067456 |
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| author | Hochster, Melvin Jeffries, Jack |
| author_facet | Hochster, Melvin Jeffries, Jack |
| contents | We give a version of the usual Jacobian characterization of the defining ideal of the singular locus in the equal characteristic case: the new theorem is valid for essentially affine algebras over a complete local algebra over a mixed characteristic discrete valuation ring. The result makes use of the minors of a matrix that includes a row coming from the values of a $p$-derivation. To study the analogue of modules of differentials associated with the mixed Jacobian matrices that arise in our context, we introduce and investigate the notion of a perivation, which may be thought of, roughly, as a linearization of the notion of $p$-derivation. We also develop a mixed characteristic analogue of the positive characteristic $Γ$-construction, and apply this to give additional nonsingularity criteria. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2106_01996 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | A Jacobian criterion for nonsingularity in mixed characteristic Hochster, Melvin Jeffries, Jack Commutative Algebra 13B02, 13B10, 13B40, 13C11 We give a version of the usual Jacobian characterization of the defining ideal of the singular locus in the equal characteristic case: the new theorem is valid for essentially affine algebras over a complete local algebra over a mixed characteristic discrete valuation ring. The result makes use of the minors of a matrix that includes a row coming from the values of a $p$-derivation. To study the analogue of modules of differentials associated with the mixed Jacobian matrices that arise in our context, we introduce and investigate the notion of a perivation, which may be thought of, roughly, as a linearization of the notion of $p$-derivation. We also develop a mixed characteristic analogue of the positive characteristic $Γ$-construction, and apply this to give additional nonsingularity criteria. |
| title | A Jacobian criterion for nonsingularity in mixed characteristic |
| topic | Commutative Algebra 13B02, 13B10, 13B40, 13C11 |
| url | https://arxiv.org/abs/2106.01996 |