Relative Inverse Limit Perfection of Derived Commutative Rings
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arXiv
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| Format: | Preprint |
| Published: |
2025
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| _version_ | 1866915339360534528 |
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| author | Fink, Daniel |
| author_facet | Fink, Daniel |
| contents | We study the relative Frobenius map associated with a map of derived commutative rings over a field of positive characteristic. As part of this, we examine a relative analog of perfectness and construct a relative inverse limit perfection which, under suitable conditions on the base, serves as a right adjoint to the inclusion of relatively perfect algebras into the category of all algebras. Specializing to animated rings, we investigate relative versions of semiperfectness and F-finiteness, and use these to show that any map of F-finite animated rings factors into a free map of finite type, followed by a relatively perfect map, followed by a surjective map. We also show that, for a morphism of Noetherian F-finite rings, the vanishing of the cotangent complex implies that the morphism is relatively perfect. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2506_10626 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Relative Inverse Limit Perfection of Derived Commutative Rings Fink, Daniel Commutative Algebra 13A35, 13B10, 13D03, 13D09, 14A30, 14B25 We study the relative Frobenius map associated with a map of derived commutative rings over a field of positive characteristic. As part of this, we examine a relative analog of perfectness and construct a relative inverse limit perfection which, under suitable conditions on the base, serves as a right adjoint to the inclusion of relatively perfect algebras into the category of all algebras. Specializing to animated rings, we investigate relative versions of semiperfectness and F-finiteness, and use these to show that any map of F-finite animated rings factors into a free map of finite type, followed by a relatively perfect map, followed by a surjective map. We also show that, for a morphism of Noetherian F-finite rings, the vanishing of the cotangent complex implies that the morphism is relatively perfect. |
| title | Relative Inverse Limit Perfection of Derived Commutative Rings |
| topic | Commutative Algebra 13A35, 13B10, 13D03, 13D09, 14A30, 14B25 |
| url | https://arxiv.org/abs/2506.10626 |