Regularity of Noetherian local rings

Abstract

Let (R, m) be a Noetherian local ring of prime characteristic p. In this paper, as an extension of the concept of p n -basis, we introduce the notion of m -adic p n -basis, and we show that R/ R p n has an m -adic p n -basis for every n (n=1,2,…) if and only if R is a regular local ring. For any Noetherian local ring, further we introduce the concept of m -adic free graded algebra and we give a regularity criterion in terms of the higher differential algebra, and others.