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Abstract We prove a differential version of the Artin–Rees lemma with the use of Noetherian differential operators. As a consequence, we obtain several uniformity results for nonreduced rings. 

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. 

Abstract We initiate the study of the resolution of singularities properties of Nash blowups over fields of prime characteristic. We prove that the iteration of normalized Nash blowups desingularizes normal toric surfaces. We also introduce a prime characteristic version of the logarithmic Jacobian ideal of a toric variety and prove that its blowup coincides with the Nash blowup of the variety. As a consequence, the Nash blowup of a, not necessarily normal, toric variety of arbitrary dimension in prime characteristic can be described combinatorially. 

The Bernstein-Sato polynomial is an important invariant of an element or an ideal in a polynomial ring or power series ring of characteristic zero, with interesting connections to various algebraic and topological aspects of the singularities of the vanishing locus. Work of Mustaţă, later extended by Bitoun and the third author, provides an analogous Bernstein-Sato theory for regular rings of positive characteristic. In this paper, we extend this theory to singular ambient rings in positive characteristic. We establish finiteness and rationality results for Bernstein-Sato roots for large classes of singular rings, and relate these roots to other classes of numerical invariants defined via the Frobenius map. We also obtain a number of new results and simplified arguments in the regular case. 

Abstract In this manuscript, we prove the Bernstein inequality and develop the theory of holonomic $$D$$-modules for rings of invariants of finite groups in characteristic zero, and for strongly $$F$$-regular finitely generated graded algebras with finite $$F$$-representation type in prime characteristic. In each of these cases, the ring itself, its localizations, and its local cohomology modules are holonomic. We also show that holonomic $$D$$-modules, in this context, have finite length, and we prove the existence of Bernstein–Sato polynomials in characteristic zero. We obtain these results using a more general version of Bernstein filtrations.