First neighborhood complete ideals in two-dimensional Muhly local domains

Abstract

a b s t r a c tComplete ideals adjacent to the maximal ideal of a two-dimensional regular local ring (called first neighborhood complete ideals) have been studied by S. Noh.Here these ideals are studied in the more general case of a two-dimensional Muhly local domain, i.e., an integrally closed Noetherian local domain with algebraically closed residue field and the associated graded ring an integrally closed domain.It is shown that certain properties of such a local ring R (e.g. being regular or being a rational singularity in case the embedding dimension is 3) can be characterized in terms of the first neighborhood complete ideals.A necessary and sufficient condition for an immediate prime divisor of R to possess an asymptotically irreducible ideal is derived. If, in addition, R is a rational singularity, we are able to give a description of such an ideal.

Muhly local domains and Zariski's theory of complete ideals

“…Here the inclusion A w ⊂ (x 3 , y, z) follows from the fact that A w is the integral closure of (x 3 , y) and that (x 3 , y, z) is integrally closed (see Lemma 2.2 (ii) in [3]). This chain has the following properties:…”

First neighborhood complete ideals in two-dimensional Muhly local domains are projectively full

“…We now discuss a few results about complete ideals adjacent to M in a twodimensional Muhly local domain (R, M). A more complete treatment of this matter can be found in the forthcoming paper [5]. First we will make clear how to produce complete ideals adjacent to M, namely by taking the inverse transform of the maximal ideal of the immediate quadratic transforms of (R, M) (terminology is explained below).…”

“…(v) We know that (R , M ) is the only immediate base point of I and M is its transform in R . This implies that T (IM) = {v M , w}; henceT (I) ⊂ {v M , w}.Since the two-dimensional Muhly local domain (R, M) is not regular, we have v M ∈ T (I) because of Theorem 3.3 in[5]. Finally, w ∈ T (I) since otherwise I would be a simple complete M-primary ideal with T (I) = {v M }, implying that I = M, a contradiction.Next we prove that the complete M-primary ideals adjacent to M obtained by considering the inverse transform in R of the maximal ideal of the immediate transforms of R are in fact all the complete M-primary ideals adjacent to M. More precisely we have the following result.…”

Integrally Closed Ideals and Rees Valuation

“…Let Min.(m/? [//]) = (ßi, ... ,.Qt\, ctnd let v¡ he the Rees valuation of V, := R[lt]Q, n K. Then the following conditions are equivalent:(1) ^ is regular for all i with \ < i < t,(2) For any reduction {a,b)R of I, there exist elements c,^, ..., c, (;/. / such that a, b, c,y ..., c,^ is a minimal generating set of I and v-Xci) > t;,(fl) = vXb) for i = \,..., Í and j -1,..., «; (3) There exist a reduction {a,b)R of I and elements C/, ..., Ci in I such that a, b, c,.|, ..., c, _^ is a minimal generating set of I and vXc/.)…”