The D-module Structure of F-split Rings

The purpose of this note is to point out an interesting connection between the structure of a commutative, Noetherian ring of prime characteristic as a (left) module over its ring of diierential operators and various well studied properties such as F-purity, F-regularity, and strong F-regularity. Theorem 2.2 establishes the rst connections between the D-module structure of rings of characteristic p and the theory of tight closure introduced by Hochster and Huneke HH1]. This theorem can also be viewed as a partial answer to a characteristic p version of the question raised by Levasseur and Staaord LS, 0.13.3]: \When is R a simple D(R) module?" For F-split rings R of low dimension, the answer is that R is simple as a D(R) module if and only if all ideals of R are tightly closed. (A ring is F-split means that the Frobenius map raising elements to their p th powers splits.) In the last section, we discuss some connections with pseudorational local rings, and indicate some of the diiculty in passing between the characteristic zero and characteristic p cases. Throughout this note, R will denote a reduced commutative, Noetherian ring of prime characteristic p > 0. We make the assumption that R is a nitely generated algebra (or equivalently, module) over its subring R p of p th powers. This mild assumption implies that R is excellent K], and is satissed, for instance, whenever R is nitely generated over a perfect eld or is a complete local ring with a perfect residue eld. The symbol I denotes an arbitrary ideal of R, and for any such I, the symbol I q] denotes the ideal of R generated by the q th powers of the elements (equivalently, the generators) of I, where q = p e is an integer power of p. Test ideals are central to the study of tight closure. We will introduce a variant notion of the idea of a test ideal for R deened by Hochster and Huneke in HH1]. We rst summarize some of the terminology associated with the study of tight closure.