Uniform Behaviour of the Frobenius closures of ideals generated by regular sequences

This paper is concerned with ideals in a commutative Noetherian ring R of prime characteristic. The main purpose is to show that the Frobenius closures of certain ideals of R generated by regular sequences exhibit a desirable type of 'uniform' behaviour. The principal technical tool used is a result, proved by R. Hartshorne and R. Speiser in the case where R is local and contains its residue field which is perfect, and subsequently extended to all local rings of prime characteristic by G. Lyubeznik, about a left module over the skew polynomial ring R[x, f] (associated to R and the Frobenius homomorphism f, in the indeterminate x) that is both x-torsion and Artinian over R. (c) 2005 Elsevier Inc. All rights reserved.