Almost complete intersections and factorial rings

Abstract

It is our purpose in this paper to investigate the relationship between two classes of ideals in regular local rings R: prime ideals P such that R/P is factorial, and prime ideals Q which are minimally generated by htQ + 1 elements. These latter ideals are called almost complete intersections. Peskine and Szpiro [ 121 say two unmixed ideals Z and J in a regular local ring R are (geometrically) linked if Z and .Z have no common minimal prime ideals and if there is an R-sequence xi ,..., x, such that (xi ,..., x,) = Zn.Z. We will delete the word “geometricaly” throughout this paper; linkage will always be used in the sense above. Murthy [ 1 l] essentially showed that if R is a regular local ring and P is a prime ideal such that R/P is factorial, then P is linked to an almost complete intersection. In Section 1 we provide a new and simple proof of this result using a generalization of a theorem of Hartshorne [2] concerning the connectedness of Spec(R). Recall that a ring R is said to satisfy Serre’s condition S, if the following condition holds: