2021/06/03 by Melvin Hochster, Hochster, Melvin, Jack Jeffries +1
Mathematics · #13B02 #13B10 #13B40 #13C11 #Advanced Differential Equations and Dynamical Systems #Algebraic Geometry and Number Theory #Commutative Algebra (math.AC) #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology
paper · pdf · doi:10.48550/arxiv.2106.01996
openalex publication_date 2021/06/03 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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.