Regular Overrings of Regular Local Rings

It is shown that the set of regular local rings of dimension n n containing an integral domain D D , having the same quotient field as D D , and ordered by containment satisfies the descending chain condition. The set of regular local rings of dimension n n contained in a regular local ring of dimension m , n > m m,n > m , need not satisfy the ascending chain condition, as is shown by example.

In this paper, it is shown that the value semigroup of a prime divisor of the second kind on a 2 2 -dimensional regular local ring is symmetric. Further, a necessary and sufficient condition for two prime divisors of the second kind on a 2 2 -dimensional regular local ring to have the same value semigroup is obtained.

The tree of quadratic transforms of a regular local ring of dimension two

1. Let L be a noetherian domain that is integrally closed in its quotient field F. To each ideal A of L is assigned an ideal A,a' the integral closure or completion of A, that consists of all elements x E L for which an equation of the form, xn + a1xn + * + an = 0, ai eAr, holds. If Q is the set of all valuations v of F such that the associated valuation ring R, contains L, then Aa coincides with the ideal Ab = {x; v(x) > v(A), V v E Q}, [2; 3]. An ideal A is said to be complete if A = Aa, and the set of all complete ideals is denoted by F(L). If A and B belong to F(L), the product AB may not belong to F(L), but the completion of the product (AB)a does. Hence a binary composition x is defined on F(L) by the condition, A x B = (AB)a. Under this composition F(L) is a commutative semigroup with an identity in which the cancellation law holds [2]. In case L is a two dimensional regular local ring, Zariski has shown [4, Appendix 5] that (1(L), x) is a Gaussian semigroup, and that the composition x is ordinary product. In this paper we study the case in which L is a two dimensional normal local domain which is subject to conditions less stringent than regularity. (See ?2 below.) It is shown that modulo a simple equivalence relation the semigroup (FO(L), x) is Gaussian, where Fo(L) is the subset of F(L) that consists of primary ideals belonging to the maximal ideal of L. However (FO(L), x) is not Gaussian in an absolute sense for in simple examples it is seen that the maximal ideal M of L is an irreducible element of (F(L), x) that is not prime. (Here we are using the semigroup terminology of Jacobson [1, Chapter IV].) Our methods are direct extensions of those of Zariski. In case L is regular, the form ring associated with L and the sequence of powers of the maximal ideal M is a polynomial ring over a field and Zariski's arguments are based in part on the fact that such a ring is a unique factorization domain. In our case the form ring is an integrally closed noetherian domain, and we obtain results analogous to Zariski's by using the Artin theory of factorization in the sense of quasi-equality that is valid in such domains.

Let V be a complete regular local (Noetherian) ring and let f ∈ V be a nonunit. We find necessary and sufficient conditions for to be the completion with respect to the maximal ideal of an integral domain of the form where S is a regular local ring whose completion with respect to its maximal ideal is V. In addition, if contains the rationals, we give necessary and sufficient conditions for to be the completion of an excellent local domain of the form , where f ∈ S, and S is a regular local ring whose completion is V.

Cores of ideals in $2$-dimensional regular local rings.

Contractedness of 𝔪-primary integrally closed ideals played a central role in the development of Zariski's theory of integrally closed ideals in two-dimensional regular local rings (R, 𝔪). In such rings, the contracted 𝔪-primary ideals are known to be characterized by the property that I: 𝔪 = I: x for some x ∈ 𝔪 ∖𝔪2. We call the ideals with this property full ideals and compare this class of ideals with the classes of 𝔪-full ideals, basically full ideals, and contracted ideals in higher dimensional regular local rings. The 𝔪-full ideals are easily seen to be full. In this article, we find a sufficient condition for a full ideal to be 𝔪-full. We also show the equivalence of the properties full, 𝔪-full, contracted, integrally closed, and normal, for the class of parameter ideals. We then find a sufficient condition for a basically full parameter ideal to be full.

In this paper we define and explore the analytic spread ℓ ( I ) $\ell (\mathcal {I})$ of a filtration in a local ring. We show that, especially for divisorial and symbolic filtrations, some basic properties of the analytic spread of an ideal extend to filtrations, even when the filtration is non-Noetherian. We also illustrate some significant differences between the analytic spread of a filtration and the analytic spread of an ideal with examples. In the case of an ideal I $I$ , we have the classical bounds ht ( I ) ⩽ ℓ ( I ) ⩽ dim R $\mbox{ht}(I)\leqslant \ell (I)\leqslant \dim R$ . The upper bound ℓ ( I ) ⩽ dim R $\ell (\mathcal {I})\leqslant \dim R$ is true for filtrations I $\mathcal {I}$ , but the lower bound is not true for all filtrations. We show that for the filtration I $\mathcal {I}$ of symbolic powers of a height two prime ideal p $\mathfrak {p}$ in a regular local ring of dimension three (a space curve singularity), so that ht ( I ) = 2 $\mbox{ht}(\mathcal {I}) =2$ and dim R = 3 $\dim R=3$ , we have that 0 ⩽ ℓ ( I ) ⩽ 2 $0\leqslant \ell (\mathcal {I})\leqslant 2$ and all values of 0, 1 and 2 can occur. In the cases of analytic spread 0 and 1 the symbolic algebra is necessarily non-Noetherian. The symbolic algebra is non-Noetherian if and only if ℓ ( p ( n ) ) = 3 $\ell (\mathfrak {p}^{(n)})=3$ for all symbolic powers of p $\mathfrak {p}$ and if and only if ℓ ( I a ) = 3 $\ell (\mathcal {I}_a)=3$ for all truncations I a $\mathcal {I}_a$ of I $\mathcal {I}$ .

A commutative Noetherian local ring of finite global dimension is a regular local ring and the structure of such a ring is well described 1n many texts (see §1.9). It is perhaps surprising that very little seems to be known about the corresponding non-commutat1ve rings and it is our aim 1n this thesis to examine some of the ways in which the commutative theory can, and cannot be extended to a right Noetherian local ring of finite right global dimension. Chapter 1 contains basic definitions and, in Chapter 2, results are obtained on the projective and Injective dimensions of modules over right Noetherian local rings. Commutative regular local rings are domains and we begin Chapter 3 by considering the question of when a right Noetherian local ring R of finite right global dimension is a prime ring. An example of Stafford's shows that R is not necessarily prime; however, by examining the Nilpotent radical of R, we are able to extend results of Ramras [57] and Walker [75] and show that R is indeed prime when certain prime ideals are localisable. In Chapter 4 we consider the lattice of prime ideals of R when R is an AR-r1ng and provide theorems which generalise some of the basic results from the theory of commutative regular local rings. Section 4.3 contains examples which not only illustrate points arising in Chapters 3 and 4 but also show that some of the techniques which are mainstays of the commutative theory, fall dramatically in a non-commutat1ve setting. In Chapter 5, we generalise the concepts of regular sequences and Cohen Macaulay rings enabling us to prove, in §5.2 and chapter 6, that a right Noetherian local ring of finite right global dimension, which 1s integral over its centre, is a prime ring and exhibits many properties similar to those enjoyed by commutative regular local rings. Examples are provided which show that some alternative approaches are not applicable to the rings considered in these chapters. Regular local rings are Gorensteln rings and, as such, are the subject of an elegant structure theorem due to Bass [5]. In Chapter 7, we generalise his theorem to the situation of rings Integral over their centres. Each chapter begins with a summary of the results contained in that chapter.

Let $Q$ be a regular local ring of dimension $3$. We show how to trim a Gorenstein ideal in $Q$ to obtain an ideal that defines a quotient ring that is close to Gorenstein in the sense that its Koszul homology algebra is a Poincare duality algebra $P$ padded with a nonzero graded vector space on which $P_{\ge 1}$ acts trivially. We explicitly construct an infinite family of such rings.

We explore the question of when the Rees algebra R(I)=⊕n≥0In of I is an almost Gorenstein graded ring, where R is a 2-dimensional regular local ring and I is a contracted ideal of R. Recently, we showed that R(I) is an almost Gorenstein graded ring for every integrally closed ideal I of R. The main results of the present article show that if I is a contracted ideal with o(I)≤2, then R(I) is an almost Gorenstein graded ring, while if o(I)≥3, then R(I) is not necessarily an almost Gorenstein graded ring, even though I is a contracted stable ideal. Thus both affirmative and negative answers are given.

Almost complete intersections and factorial rings

Cartan maps, clean rings, and unique factorization

Let $R$ be a Noetherian local ring with the maximal ideal $\mathfrak{m}$. Assume that $R$ contains ideals $I$ and $J$ satisfying the conditions (1) $I\subseteq J$, (2) $I : \mathfrak{m}\nsubseteq J$, and (3) $J$ is $\mathfrak{m}$-full, that is $\mathfrak{m}J : x = J$ for some $x \in \mathfrak{m}$. Then the theorem says that $R$ is a regular local ring, if the projective dimension $\mathrm{pd}_{R}I$ of $I$ is finite. Let $\mathfrak{q} = (a_{1}, a_{2}, \ldots , a_{t})R$ be an ideal in a Noetherian local ring $R$ generated by a maximal $R$-regular sequence $a_{1}, a_{2}, \ldots , a_{t}$ and let $\bar{\mathfrak{q}}$ denote the integral closure of $\mathfrak{q}$. Then, thanks to the theorem applied to the ideals $I = \mathfrak{q} : \mathfrak{m}$ and $J = \mathfrak{\bar{q}}$, it follows that $I^{2} = \mathfrak{q}I$, unless $R$ is a regular local ring. Consequences are discussed.

Let \((S, \mathfrak {n})\) be a regular local ring and let \(I \subseteq \mathfrak {n}^{2} \) be a perfect ideal of S. Sharp upper bounds on the minimal number of generators of I are known in terms of the Hilbert function of R = S/I. Starting from information on the ideal I, for instance the minimal number of generators, a difficult task is to determine good bounds on the minimal number of generators of the leading ideal I∗ which defines the tangent cone of R or to give information on its graded structure. Motivated by papers of S. C. Kothari and S. Goto et al. concerning the leading ideal of a complete intersection I = (f, g) in a regular local ring, we present results provided ht (I) = 2. If I is a complete intersection, we prove that the Hilbert function of R determines the graded Betti numbers of the leading ideal and, as a consequence, we recover most of the results of the previously quoted papers. The description is more complicated if ν(I) > 2 and a careful investigation can be provided when ν(I) = 3. Several examples illustrating our results are given.