Judith D. Sally

Birational extensions in dimension two and integrally closed ideals

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

Stable rings and a problem of Bass

1. Stable ideals and stable rings. Let A be a commutative Noetherian ring with 1. In analogy with Lipman s terminology in [3], we call an ideal I of a ring A stable if I is projective over its endomorphism ring, EndA(I). The ring A is stable if every ideal of A is stable. Since the relationship between stability and multiplicity is one key factor in the proof of the theorem, we gather some of the needed facts here. Let I be an ideal of A which contains a nonzero divisor. Then EndA(I) £ ( / : ƒ ) = {x e K\xl ç ƒ}, where K is the total quotient ring of A. Thus End^(7) is a subring of the integral closure of A in K and I is an ideal of End^J). Suppose that A is a 1-dimensional local Macaulay ring and that I is an ideal containing a nonzero divisor. Then ƒ is a stable ideal of A if and only if ƒ is a principal ideal of the semilocal ring EndA(I). Therefore, if ƒ is stable, End^(/) = A, the ring obtained from A by blowing up I (cf. [3]). For reference, we state as Proposition 1.1 the characterization of stable ideals due to Lipman [3]. The notation is as follows : ju(7) denotes the multiplicity of an ideal J and X(B) denotes the length of an A -module B.

Formal fibers and birational extensions

Suppose (i?, m) is a local Noetherian domain with quotient field K and m-adic completion R . It is well known that the fibers of the morphism Spec(i?) —* Spec(R), i.e., the formal fibers of R, encode important information about the structure of R. Perhaps the most important condition in Grothendiecks 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 / m C is a finite i?-module. The possible dimensions of the formal fibers of a local Noetherian ring are considered in [Ma2] and in [R3]. Following Matsumura in [Ma2], we use a(A) to denote the maximal dimension of a formal fiber of a local Noetherian ring A If (/?, m) is a local Noetherian domain with quotient field K and m-adic completion R, then Matsumura shows [Ma2, Corollary 1, page 262] that a(R) is the dimension of the generic formal fiber R[K\. He also observes that if R is of positive dimension, then a(R) r > 1 are integers, and xu... >xn are indeterminates, then two interesting examples considered in [Ma2] are the rings

Extensions of valuations to the completion of a local domain

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.’

Integrally Closed Projectively Equivalent Ideals

In this paper, I will be a regular ideal in a Noetherian ring R and Ī will denote the integral closure of I. If J is another ideal of I. and J are said to be projectively equivalent if there are positive integers n and m with \(\overline {{I^n}} \, = \,\overline {{J^m}} \). Projective equivalence was introduced by Samuel in [S], and further developed by Nagata in [N]. The concept of projective equivalence is rather interesting, but there appears to be much about it which is not known. We hope to dispel some of the darkness. We study the set of integrally closed ideals which are projectively equivalent to I, and show that this set is rather well behaved. For instance, it is linearly ordered by inclusion, and ”eventually periodic”, a phrase we will make more precise in the next paragraph. An interesting corollary to our work is that there is a fixed positive integer d such that for any ideal J projectively equivalent to I, there is a positive integer n with \(\overline {{I^n}} {\mkern 1mu} = {\mkern 1mu} \overline {{J^d}} \).

One-Fibered Ideals

This is essentially an expository paper about m-primary ideals I in a local ring (R, m) which have a single exceptional prime, i. e., whose closed fiber in the normalization \(\overline {R\left( I \right)} \) of the blow-up R(I) is irreducible. Stated slightly differently, it is about m-primary ideals which have a single associated Rees valuation. If R is analytically unramified then the existence of such an ideal in R implies that R is analytically irreducible. It is conjectured that, conversely, every analytically irreducible local domain has such an ideal.

Non-splitting of prime divisors

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 I v 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.