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A Jacobian criterion for nonsingularity in mixed characteristic

Melvin Hochster, Jack Jeffries

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.

A Jacobian criterion for nonsingularity in mixed characteristic

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 -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 -derivation. We also develop a mixed characteristic analogue of the positive characteristic -construction, and apply this to give additional nonsingularity criteria.

Paper Structure

This paper contains 14 sections, 21 theorems, 69 equations.

Table of Contents

  1. Introduction
  2. p-derivations and perivations
  3. $p$-derivations
  4. Derivations of inseparability
  5. $p$-derivations modulo $p^2$
  6. Perivations
  7. Universal perivation modules
  8. Jacobian criterion
  9. Theorem A
  10. Elementary proof
  11. Theorem B
  12. Lifting the $\Gamma$-construction to mixed characteristic
  13. The $\Gamma$-construction in positive prime characteristic.
  14. The $\widetilde{\Gamma}$-construction in mixed characteristic.

Key Result

Lemma 2.6

Let $(A,\delta)$ be a $\delta$-ring, with $p$ a regular element on $A$, and let $S$ be a smooth $A$-algebra. Then there exists a $p$-derivation mod $p^2$ on $S$ that extends $\delta$.

Theorems & Definitions (61)

  • Definition 2.2
  • Example 2.3
  • Remark 2.4
  • proof
  • Definition 2.5
  • Lemma 2.6
  • proof
  • Example 2.7
  • Lemma 2.8
  • proof
  • ...and 51 more