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

TL;DR

This paper proves that homomorphisms between Noetherian F-finite rings can be factored into a regular morphism followed by a surjection, establishing an analog of smooth-by-surjective factorization and linking regularity with L-smoothness.

Contribution

It introduces a factorization of ring homomorphisms into regular and surjective parts and shows the equivalence of regularity, formal smoothness, and L-smoothness for these rings.

Findings

Homomorphisms factor into regular morphisms followed by surjections.

Regularity and formal smoothness are equivalent to L-smoothness in this context.

Any Noetherian F-finite ring is a quotient of a regular Noetherian F-finite ring.

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.