Suppose (R, m) is a local Noetherian domain with quotient field K and m-adic completion Ȓ. It is well known that the fibers of the morphism Spec(Ȓ) ₒ Spec(R), i.e., the formal fibers of R, encode important information about the structure of R. Perhaps the most important condition in Grothendieck’s definition of R being excellent is that the formal fibers of R be geometrically regular. Indeed, a local Noetherian ring is excellent provided it is universally catenary and has geometrically regular formal fibers [G, (7.8.3), page 214]. But the structure of the formal fibers of R is often difficult to determine. We are interested here in bringing out the interrelatedness of properties of the generic formal fiber of R with the existence of certain local Noetherian domains C birationally dominating R and having C/mC is a finite R-module.

Let (R, m) be a normal local domain of dimension n > 1. Suppose that R is analytically irreducible, i.e. that ii, the m-adic completion of R, is a domain. Given a valuation domain V birationally dominating R, what can be said about valuation domains W which extend V and birationally dominate Ai? In [l, Lemma 131, Abhyankar shows that V has an extension W which birationally dominates k. Here we discuss the uniqueness of such an extension. For this, we will assume, in addition, that i is normal, i.e., that R is analytically normal and that R satisfies a property, defined below, which is weaker than excellence. Matsumura’s work on dimensions of formal fibers was a motivating influence for the questions we consider on extending valuations from R to ff. In [lo], Matsumura asks if the dimension of the generic formal fiber, i.e., the fiber over 0 in the embedding R+ fi’, can be a positive integer other than 0, y1 2 or n 1. Since extensions of rank 1 valuations to i which grow in rank produce nonzero primes in the generic formal fiber, we initiated a study of these extensions hoping to shed some light on Matsumura’s question.’

In this study of complete, or integrally closed, ideals in a two-dimensional regular local ring (R, m), Zariski established a one-to-one correspondence between prime divisors of R, i.e. rank 1 discrete valuations v birationally dominating R with residue field of transcendence degree 1 over R/m, and m-primary simple complete ideals Iv in R; cf. [17] and [18]. In this correspondence, the blow-up of such an ideal has unique exceptional prime and the localization at this prime is the valuation ring of a prime divisor of R. In this paper, we will study such ideals in a more general setting, so we begin by recalling some definitions and background results.

The aim of this paper is to examine, primarily from an algebraic point of view, the structure of a 2-dimensional normal local domain (S, n) which birationally dominates a 2-dimensinal regular local ring (R, m). From the geometric point of view, the sine qua non is Lipman’s paper [Lr] on rational singularities, for it follows from one of the early results in that paper that S has a rational singularity. Using elementary algebraic techniques we are able to recover much information concerning the structure of S. We have particularly focused on the fact (from Lipman [Ll] and Artin [Ar]) that S must have minimal multiplicity. We hope this approach will aid in understanding the algebra of rational singularities and in the exploration of many open questions in dimension two and in higher dimensions. The framework of the paper is set up as follows. In the first section we use basic algebraic techniques such as analytic independence and Zariski’s Main Theorem to prove, for example, that S has a regular height 1 prime if R/m is infinite. If S is not regular, we may assume that R is “maximally regular” in S and when this is the case we show, for instance, that the

There are relatively few classes of local rings (R, m) for which the question of the rationality of the Poincaré serieswhere k = R/m, has been settled. (For an example of a local ring with non-rational Poincaré series see the recent paper by D. Anick, “Construction of loop spaces and local rings whose Poincaré—Betti series are nonrational”, C. R. Acad. Sc. Paris 290 (1980), 729-732.) In this note, we compute the Poincaré series of a certain family of local Cohen-Macaulay rings and obtain, as a corollary, the rationality of the Poincaré series of d-dimensional local Gorenstein rings (R, m) of embedding dimension at least e + d – 3, where e is the multiplicity of R. It follows that local Gorenstein rings of multiplicity at most five have rational Poincaré series.

Journal of the London Mathematical SocietyVolume s2-20, Issue 1 p. 19-26 Notes and Papers Stretched Gorenstein Rings Judith D. Sally, Judith D. Sally Department of Mathematics, Northwestern University, Evanston, Illinois 60201, U.S.A.Search for more papers by this author Judith D. Sally, Judith D. Sally Department of Mathematics, Northwestern University, Evanston, Illinois 60201, U.S.A.Search for more papers by this author First published: August 1979 https://doi.org/10.1112/jlms/s2-20.1.19Citations: 40AboutPDF ToolsRequest permissionExport citationAdd to favoritesTrack citation ShareShare Give accessShare full text accessShare full-text accessPlease review our Terms and Conditions of Use and check box below to share full-text version of article.I have read and accept the Wiley Online Library Terms and Conditions of UseShareable LinkUse the link below to share a full-text version of this article with your friends and colleagues. Learn more.Copy URL Share a linkShare onFacebookTwitterLinkedInRedditWechat Citing Literature Volumes2-20, Issue1August 1979Pages 19-26 RelatedInformation

Let (R, m) be a local ring with associated graded ring RlmφmJm^ζBmVw?Θ •••.This paper deals with the problem of finding properties of R which lead to good properties in grR.There are two main results in this paper which give techniques for recognizing when the maximal homogeneous ideal of grR contains regular elements.Applications of these results give examples of local Cohen-Macaulay rings which have Cohen-Ma caulay associated graded rings.

In [1] Bass proved the following theorem.Let A be a 1-dimensional Noetherian ring with finite integral closure. 1 If every ideal of A can be generated by two elements, then every ideal is projective over its endomorphism ring.Bass also conjectured that the converse holds.Theorem 2.4 below shows that this is indeed the case.