[Paper Review] A Jacobian criterion for nonsingularity in mixed characteristic

Melvin Hochster, Jack Jeffries|arXiv (Cornell University)|Jun 3, 2021

Advanced Differential Equations and Dynamical Systems9 references4 citations

TL;DR

This paper establishes a Jacobian criterion for nonsingularity in mixed characteristic by introducing a matrix that combines partial derivatives and $p$-derivations, enabling the identification of the singular locus in algebras over complete local rings of mixed characteristic. The key contribution is a regularity criterion using the freeness of a new module, the universal perivation module $\widetilde{\Omega}_{R|\mathbb{Z}}$, which generalizes classical Jacobian conditions to mixed characteristic settings without requiring F-finiteness of the residue field.

ABSTRACT

We give a version of the usual Jacobian characterization of the defining ideal of the singular locus in the equal characteristic case: the new theorem is valid for essentially affine algebras over a complete local algebra over a mixed characteristic discrete valuation ring. The result makes use of the minors of a matrix that includes a row coming from the values of a $p$-derivation. To study the analogue of modules of differentials associated with the mixed Jacobian matrices that arise in our context, we introduce and investigate the notion of a perivation, which may be thought of, roughly, as a linearization of the notion of $p$-derivation. We also develop a mixed characteristic analogue of the positive characteristic $Γ$-construction, and apply this to give additional nonsingularity criteria.

Motivation & Objective

• To extend the classical Jacobian criterion for nonsingularity to mixed characteristic rings, where standard smoothness-based methods fail due to the lack of equivalence between regularity and smoothness.
• To address the challenge of detecting singularities in algebras over discrete valuation rings of mixed characteristic, particularly when the residue field is not F-finite.
• To develop a new algebraic framework—perivation and the universal perivation module—capable of capturing differential-like behavior in mixed characteristic.
• To construct a mixed characteristic analogue of the positive characteristic $\Gamma$-construction, called the $\widetilde{\Gamma}$-construction, to eliminate F-finiteness hypotheses in regularity criteria.

Proposed method

• Introduce the notion of a perivation as a linearization of $p$-derivations, generalizing derivations and $p$-derivations in mixed characteristic.
• Define the universal perivation module $\widetilde{\Omega}_{R|A}$ as the cokernel of a matrix combining $p$-derivatives and $p$-th powers of partial derivatives.
• Construct the $\widetilde{\Gamma}$-construction as a mixed characteristic analogue of the $\Gamma$-construction, using lifts of $p$-bases to extend algebras in a way that preserves regularity and commutes with base change.
• Use the $\widetilde{\Gamma}$-construction to lift the regularity criterion to rings with non-F-finite residue fields by taking a direct limit over cofinite subsets of a $p$-base.
• Apply the $\widetilde{\Gamma}$-construction to reduce the problem to a setting where the residue field is F-finite, allowing the use of the perivation module criterion.
• Establish a correspondence between the singular locus of $R$ and that of $R^{\widetilde{\Gamma}}$, ensuring that regularity is preserved under the construction.

Experimental results

Research questions

• RQ1 How can the classical Jacobian criterion for nonsingularity be generalized to algebras over mixed characteristic discrete valuation rings where smoothness does not imply regularity?
• RQ2 What is the role of $p$-derivations in characterizing singularities in mixed characteristic, and how can they be incorporated into a differential module framework?
• RQ3 Can a mixed characteristic analogue of the $\Gamma$-construction be developed to eliminate F-finiteness assumptions in regularity criteria?
• RQ4 How does the universal perivation module $\widetilde{\Omega}_{R|\mathbb{Z}}$ relate to the module of Kähler differentials in equal characteristic settings?
• RQ5 Under what conditions is the freeness of $\widetilde{\Omega}_{R^{\widetilde{\Gamma}}|\mathbb{Z}}$ equivalent to the regularity of $R$?

Key findings

• The singular locus of a local ring $R$ essentially of finite type over a complete $V$-algebra in mixed characteristic is characterized by the vanishing of the $h \times h$ minors of a matrix combining $p$-derivatives and $p$-th powers of partial derivatives.
• The universal perivation module $\widetilde{\Omega}_{R|\mathbb{Z}}$ is defined as the cokernel of the Jacobian-type matrix involving $p$-derivations and $p$-th powers of partial derivatives, generalizing the module of Kähler differentials.
• For a local ring $R$ with $p \in \mathfrak{m}$, $R$ is regular if and only if $\widetilde{\Omega}_{R|\mathbb{Z}}$ is free of rank $\dim(R) + \log_p[k:k^p]$ when the residue field is F-finite.
• The $\widetilde{\Gamma}$-construction provides a purely inseparable, faithfully flat extension $R^{\widetilde{\Gamma}}$ of $R$ that preserves the singular locus and allows the removal of F-finiteness assumptions.
• For any sufficiently small cofinite subset $\widetilde{\Gamma} \subseteq \widetilde{\Lambda}$, the ring $R$ is regular if and only if $\widetilde{\Omega}_{R^{\widetilde{\Gamma}}|\mathbb{Z}}$ is free of rank $\dim(R) + \log_p[k(R^{\widetilde{\Gamma}}):k(R^{\widetilde{\Gamma}})^p]$, even when the residue field is not F-finite.
• The construction ensures that the regular locus of $R$ and $R^{\widetilde{\Gamma}}$ agree on the fiber over $p$, and the criterion is invariant under base change to local $V$-algebras with complete ideals.