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In their work on differential operators in positive characteristic, Smith and Van den Bergh define and study the derived functors of differential operators; these arise naturally as obstructions to differential operators reducing to positive characteristic. In this paper, we provide formulas for the ring of differential operators as well as these derived functors of differential operators. We apply these descriptions to show that differential operators behave well under reduction to positive characteristic under certain hypotheses. We show that these functors also detect a number of interesting properties of singularities. [ABSTRACT FROM AUTHOR]

In a polynomial ring over a perfect field, the symbolic powers of a prime ideal can be described via differential operators: a classical result by Zariski and Nagata says that the n-th symbolic power of a given prime ideal consists of the elements that vanish up to order n on the corresponding variety. However, this description fails in mixed characteristic. In this paper, we use p-derivations, a notion due to Buium and Joyal, to define a new kind of differential powers in mixed characteristic, and prove that this new object does coincide with the symbolic powers of prime ideals. This seems to be the first application of p-derivations to commutative algebra. [ABSTRACT FROM AUTHOR]

Levasseur and Stafford described the rings of differential operators on various classical invariant rings of characteristic zero; in each of the cases that they considered, the differential operators form a simple ring. Towards an attack on the simplicity of rings of differential operators on invariant rings of linearly reductive groups over the complex numbers, Smith and Van den Bergh asked if differential operators on the corresponding rings of positive prime characteristic lift to characteristic zero differential operators. We prove that, in general, this is not the case for determinantal hypersurfaces, as well as for Pfaffian and symmetric determinantal hypersurfaces. We also prove that, with very few exceptions, these hypersurfaces—and, more generally, classical invariant rings—do not admit a mod p 2 lift of the Frobenius endomorphism. [ABSTRACT FROM AUTHOR]

The F -signature of a local ring of prime characteristic is a numerical invariant that detects many interesting properties. For example, this invariant detects (non)singularity and strong F -regularity. However, it is very difficult to compute. Motivated by different aspects of the F -signature, we define a numerical invariant for rings of characteristic zero or p > 0 that exhibits many of the useful properties of the F -signature. We also compute many examples of this invariant, including cases where the F -signature is not known. We also obtain a number of results on symbolic powers and Bernstein-Sato polynomials. [ABSTRACT FROM AUTHOR]