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

Manuel Blickle, Daniel Fink

TL;DR

The paper proves that any morphism between Noetherian $F$-finite rings over $\mathbb{F}_p$ factors as an $L$-smooth map between Noetherian $F$-finite rings followed by a surjection, extending the smooth-by-surjective paradigm to positive characteristic. It introduces and analyzes $L$-smoothness via the cotangent complex, showing that for Noetherian $F$-finite maps regularity, formal smoothness, and $L$-smoothness coincide, with $L$-smooth maps being flat and stable under composition. The main contributions are (i) a factorization for maps of regular rings into an $L$-smooth step followed by a regular immersion, (ii) a general construction reducing arbitrary maps to such a factorization via pushouts and Adams completion, and (iii) an Adams-completion description that identifies the intermediate ring and cotangent data, linking derived and classical completions. Together, these results extend structural factorization principles to a broad class of rings in characteristic $p$ and connect cotangent-complex finiteness with $F$-finiteness, echoing Gabber’s remarks on regular presentations.

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.

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

TL;DR

The paper proves that any morphism between Noetherian -finite rings over factors as an -smooth map between Noetherian -finite rings followed by a surjection, extending the smooth-by-surjective paradigm to positive characteristic. It introduces and analyzes -smoothness via the cotangent complex, showing that for Noetherian -finite maps regularity, formal smoothness, and -smoothness coincide, with -smooth maps being flat and stable under composition. The main contributions are (i) a factorization for maps of regular rings into an -smooth step followed by a regular immersion, (ii) a general construction reducing arbitrary maps to such a factorization via pushouts and Adams completion, and (iii) an Adams-completion description that identifies the intermediate ring and cotangent data, linking derived and classical completions. Together, these results extend structural factorization principles to a broad class of rings in characteristic and connect cotangent-complex finiteness with -finiteness, echoing Gabber’s remarks on regular presentations.

Abstract

We show that any homomorphism between Noetherian -finite rings can be factored into a regular morphism between Noetherian -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 -finite rings, regularity and formal smoothness are both equivalent to -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 -finite ring is a quotient of a regular Noetherian -finite ring.
Paper Structure (5 sections, 29 theorems, 45 equations)

This paper contains 5 sections, 29 theorems, 45 equations.

Table of Contents

  1. Introduction
  2. L-smoothness of ring maps
  3. The factorization for regular rings
  4. The factorization in general
  5. A Description via the Adams completion

Key Result

Theorem 1.1

Any map $R\xrightarrow{\ \ }S$ of Noetherian $F$-finite $\mathbb{F}_p$-algebras admits a factorization \begin{tikzcd} & {T} \arrow[rd, "\psi"] & \\ {R} \arrow[ru, "\varphi"] \arrow[rr] & & {S} \end{tikzcd}in which $T$ is a Noetherian $F$-finite $\mathbb{F}_p

Theorems & Definitions (70)

  • Theorem 1.1
  • Theorem 2.1
  • proof
  • Definition 2.2
  • Lemma 2.3
  • Example 2.4
  • Example 2.5
  • Lemma 2.6
  • proof
  • Proposition 2.7
  • ...and 60 more