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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 {\it 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 $$\Gamma$$-construction, and apply this to give additional nonsingularity criteria.
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