Regularity Criterion of Noetherian Local Rings of Prime Characteristic

Let R be a local ring of essentially finite type over a field k of characteristic p > 0. We introduce the concept of pn-bases for an integer n > 0, and the notions of an n-admissible field for R/k and of a (t)-admissible field for R/k. For such a local ring R we give regularity criteria in terms of the higher differential algebra and of a pn-basis, respectively.