Frobenius splitting, strong F-regularity, and small Cohen-Macaulay modules

Abstract

Let M M be a finitely generated module over an (F-finite local) ring R R of prime characteristic p > 0 p >0 . Let e M {}^e\!M denote the result of restricting scalars using the map F e : R → R F^e\colon R \to R , the e e\, th iteration of the Frobenius endomorphism. Motivated in part by the fact that in certain circumstances the splitting of e M {}^e\!M as e e grows can be used to prove the existence of small (i.e., finitely generated) maximal Cohen-Macaulay modules, we study splitting phenomena for e M {}^e\!M from several points of view. In consequence, we are able to prove new results about when one has such splittings that generalize results previously known only in low dimension, we give new characterizations of when a ring is strongly F-regular, and we are able to prove new results on the existence of small maximal Cohen-Macaulay modules in the multi-graded case. In addition, we study certain corresponding questions when the ring is no longer assumed F-finite and purity is considered in place of splitting. We also answer a question, raised by Datta and Smith, by showing that a regular Noetherian domain, even in dimension 2, need not be very strongly F-regular.