A criterion for log regularity via log Frobenius-Witt differentials

Abstract: T. Saito introduced FW-derivations and the modules of FW-differentials. He gave a regularity criterion in terms of the modules of FW-differentials. In this paper, we introduce logarithmic analogues of FW-derivations and the modules of FW-differentials. We study basic properties of them and give a logarithmic regularity criterion in terms of the modules of logarithmic FW-differentials.

• The paper introduces a log Frobenius-Witt differential framework that characterizes log regularity in prelog rings in positive and mixed characteristics.
• It develops functoriality, base change, and completion theorems, enabling explicit structure computations for monoid algebras.
• The work unifies regularity criteria in both logarithmic and classical settings while suggesting links to derived and motivic invariants.

Log Regularity Characterized by Logarithmic Frobenius-Witt Differentials

Introduction

This paper establishes a new criterion for log regularity of prelogarithmic rings based on the properties of modules of logarithmic Frobenius-Witt (FW) differentials. Building on the theory of FW-derivations introduced by Saito and prior work of Hochster–Jeffries linking FW-differentials to regularity in mixed characteristic, Takeuchi augments this paradigm to logarithmic geometry. The central focus is to define log FW-differentials, analyze their structure, and use their algebraic properties to provide an equivalence between freeness conditions for these modules and log regularity analogous to the classical situation.

Background and Main Results

Let $p$ be a prime and $(R,Q,\alpha)$ a local prelog ring, with $R$ Noetherian local, residue field $k$ of characteristic $p$, and $Q$ a monoid mapping to $R$. In the classical commutative case, Saito gave a regularity criterion: for a local ring $R$ with $F$-finite residue field $k$ and $(R,Q,\alpha)$0, $(R,Q,\alpha)$1 is regular if and only if the FW-differential module $(R,Q,\alpha)$2 (resp. its reduction mod $(R,Q,\alpha)$3) is free of rank $(R,Q,\alpha)$4 [MR4412577]. Hochster and Jeffries independently gave an analogous result using their universal perivation module [MR4855866].

Takeuchi extends these results to the log setting by defining logarithmic FW-derivations and the corresponding module $(R,Q,\alpha)$5. The main theorem is the following equivalence:

• For $(R,Q,\alpha)$6 as above with $(R,Q,\alpha)$7 integral, $(R,Q,\alpha)$8 fine and saturated, and $(R,Q,\alpha)$9 $R$0-finite with $R$1,
  1. $R$2 is a free $R$3-module of rank $R$4,
  2. $R$5 has $R$6-dimension $R$7,
  3. $R$8 is log regular,

  then $R$9; further, if $k$0 is $k$1-finite, all three are equivalent.

This "logarithmic Jacobian criterion" thus precisely characterizes log regularity in terms of the structure of log FW-differentials.

Technical Construction and Proof Outline

Logarithmic Frobenius-Witt Differentials

Given a prelog ring $k$2, a log FW-derivation is a pair $k$3 with $k$4 an FW-derivation (non-additive in characteristic $k$5) and $k$6 a monoid homomorphism, satisfying the compatibility $k$7 for $k$8. The module of universal log FW-differentials, $k$9, represents the functor of log FW-derivations.

Crucially, properties such as functoriality, base change, and behavior under completion are established, and connections to log cotangent complexes (as in Olsson and Gabber) are anticipated.

Log Regularity

The definition of log regularity follows Kato-Ogus: $p$0 is log regular if

1. $p$1 is a regular local ring, where $p$2 is the ideal generated by $p$3,
2. $p$4.

Key structural results, such as the description of log regular rings in terms of completions of monoid algebras (Kato’s theorem [MR3838359]), are used repeatedly.

Main Theorem Proof

The proof proceeds via several reductions and structural theorems:

• Reduction to the case where $p$5 is sharp, fine, and saturated, using splitting arguments.
• Analysis of the log FW-differential module in terms of $p$6 and $p$7 components, leveraging Saito’s and Kato’s methods.
• The direct sum decomposition of the module on the residue field, aligning the rank/dimension computation exactly with the combinatorial data of $p$8 and $p$9.
• In both equal and mixed characteristic, structural theorems (using Cohen rings, power series, and monoid algebra completions) permit explicit module-theoretic control over $Q$0 and the effect of killing $Q$1.

The delicate passage to completions and handling torsion phenomena is managed using base change and faithfully flat descent.

Notable Claims and Implications

• Equivalence of Freeness and Log Regularity: The main theorem asserts that the module-theoretic freeness of $Q$2 is equivalent to log regularity, both at the level of $Q$3-modules and on the residue field. This provides a precise and computable Jacobian-type condition in the log setting, subsuming Saito’s and Hochster–Jeffries’ results as special cases.
• The theory incorporates mixed characteristic, giving an explicit generator description for the ideal in the completed monoid algebra, with strong control over the basis of $Q$4 modulo such ideals.
• The module of log FW-differentials is shown to be finitely generated under mild finiteness hypotheses ($Q$5-finiteness and finite generation of $Q$6).
• The result is compatible with the passage from prelog to log structures.

Theoretical Significance and Further Directions

This criterion for log regularity via log FW-differentials positions these modules as canonical invariants in logarithmic geometry, mirroring the cotangent complex in the classical case. The paper notes, and points toward, an expected deeper connection with the logarithmic cotangent complex of Olsson–Gabber [MR2195148] and the possibility of a more refined obstruction theory for log smoothness and $Q$7-adic Hodge-theoretic applications.

Practically, the results provide a module-theoretic approach to detecting log regularity in both equal and mixed characteristic, which is crucial for the study of singularities in logarithmic schemes and for $Q$8-adic cohomology theories.

Potential directions include:

• Explicit computation of log FW-differentials in geometric examples (e.g., toric singularities).
• Exploiting these modules in the study of crystalline or prismatic cohomology.
• Formulating analogous criteria for log smoothness, factoring through the log cotangent complex.
• Exploring connections to de Rham–Witt and prismatic structures.

Conclusion

Takeuchi’s work provides a thorough module-theoretic characterization of log regularity in terms of the (logarithmic) Frobenius-Witt differential module. This closes the gap between commutative and logarithmic algebra in the context of regularity criteria, opens new computational methods for the study of log singularities, and lays foundational groundwork for further relations with logarithmic derived invariants. The module $Q$9 emerges as the logarithmic analog of the Jacobian criterion—an essential tool for future developments in arithmetic and logarithmic geometry.

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Reference:

"A criterion for log regularity via log Frobenius-Witt differentials" (2604.17394)