Some Remarks on One-Dimensional Local Domains

Abstract

Let R be a one-dimensional (noetherian) local domain with field of quotients Q. Then any ring extension S of R in Q is obtained as a ring of quotients of some integral extension C of R. Here, if S is local and if C can be chosen to be finite over R, then we call S an K-locality. If R is analytically ramified, then .R does not satisfy the finiteness condition for integral extensions in Q (cf. [3], p. 122, Exercise 1). In other words, .R possesses at least one latent singularity with respect to a certain analytic branch of R which can not be resolved by any quadratic dilatations. The purpose of this note is to give a necessary and sufficient condition for the finiteness of ring extensions S as .R-modules, and to prove a characterization of ^-localities by making use of the concept of latent multiplicity found in [L] and [4] (more detailed accounts of this theory may be found in [3]).