Mamoru Furuya

Regularity of Analytic Algebra of Prime Characteristic

Let A be an analytic algebra over a field k of characteristic p > 0. In this article, for an analytic k-algebra we introduce the concept of analytic pn-basis which generalizes the pn-basis defined in [1], and the concept of an pn-admissible field for an algebraic function field, we give regularity criteria and absolute regularity criteria for an analytic algebra A/k in terms of the higher differential algebra and analytic pn-basis. The results are partial extension of our previous results for affine algebras to the case of analytic algebras (cf. [1, 3]), and these are partial generalization of results of Orbanz in the analytic case (cf. [9]).

Some Regularity Criteria for Affine Semilocal Rings

The purpose of this article is to study an existence of p-bases and mainly to give some regularity criteria for semilocal rings essentially of finite type over fields.

-adic -basis and regular local ring

We introduce the concept of m-adic p-basis as an extension of the concept of p-basis. Let (S, m) be a regular local ring of prime characteristic p and R a ring such that S ⊃ R ⊃ S p . Then we prove that R is a regular local ring if and only if there exists an m-adic p-basis of S/R and R is Noetherian.

Differential Criterion of Complete Regular Local Rings

Abstract In the paper by Furuya and Niitsuma [Furuya, M., Niitsuma, H. (2002a). On m -adic higher differentials and regularities of Noetherian complete local rings. J. Math. Kyoto Univ. 42(1):33–40], we gave a regularity criterion of complete Noetherian local rings in terms of the concept of m -adic higher differentials with some assumptions of separability on the residue fields of the local rings. The aim of this paper is to try to weaken “the separability assumptions” in a premise of the theorems in Furuya and Niitsuma (2002a).