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arXiv · 2501.09437

$L$-smooth factorization for Noetherian $F$-finite rings

Manuel Blickle·Daniel Fink
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Abstract

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.

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Manuel Blickle, Daniel Fink. 2025-01-16. $L$-smooth factorization for Noetherian $F$-finite rings. https://arxiv.org/abs/2501.09437

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