$L$-smooth factorization for Noetherian $F$-finite rings
Fuente:
arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866929678328004608 |
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| author | Blickle, Manuel Fink, Daniel |
| author_facet | Blickle, Manuel Fink, Daniel |
| contents | We show that any homomorphism between Noetherian $F$-finite rings can be factored into a regular morphism between Noetherian $F$-finite rings followed by a surjection. This result establishes an analog of the 'smooth-by-surjective' factorization for finite type maps. As part of our analysis, we observe that for maps of Noetherian $F$-finite rings, regularity and formal smoothness are both equivalent to $L$-smoothness, meaning that the cotangent complex, as in the smooth case, is a locally free module of finite rank concentrated in degree zero.
Our findings may also be viewed as a relative version of Gabber's final remark in \citep{Gab04}, which states that any Noetherian $F$-finite ring is a quotient of a regular Noetherian $F$-finite ring. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2501_09437 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | $L$-smooth factorization for Noetherian $F$-finite rings Blickle, Manuel Fink, Daniel Commutative Algebra Algebraic Geometry 13A35, 13B10, 13B35, 13D03 We show that any homomorphism between Noetherian $F$-finite rings can be factored into a regular morphism between Noetherian $F$-finite rings followed by a surjection. This result establishes an analog of the 'smooth-by-surjective' factorization for finite type maps. As part of our analysis, we observe that for maps of Noetherian $F$-finite rings, regularity and formal smoothness are both equivalent to $L$-smoothness, meaning that the cotangent complex, as in the smooth case, is a locally free module of finite rank concentrated in degree zero. Our findings may also be viewed as a relative version of Gabber's final remark in \citep{Gab04}, which states that any Noetherian $F$-finite ring is a quotient of a regular Noetherian $F$-finite ring. |
| title | $L$-smooth factorization for Noetherian $F$-finite rings |
| topic | Commutative Algebra Algebraic Geometry 13A35, 13B10, 13B35, 13D03 |
| url | https://arxiv.org/abs/2501.09437 |