Noetherian Local Domain Research Articles

W-Divisoriality in Polynomial Rings

We extend the Bass–Matlis characterization of local Noetherian divisorial domains to the non-Noetherian case. This result is then used to study the following question: If a domain D is w-divisorial, that is, if each w-ideal of D is divisorial, then is D[X] automatically w-divisorial? We show that the answer is yes if D is either integrally closed or Mori.

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

Let ( R , M ) (R, \mathcal {M}) be 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. Motivated by recent work of Ciuperca, Heinzer, Ratliff and Rush on projectively full ideals, we prove that every complete ideal adjacent to the maximal ideal M \mathcal {M} is projectively full.

First neighborhood complete ideals in two-dimensional Muhly local domains

Complete 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.

A class of projectively full ideals in two-dimensional Muhly local domains

Let ( R , M ) be a regular local domain of dimension d ⩾ 2 and let x 1 , … , x d be a regular system of parameters. Then, Ciuperca, Heinzer, Ratliff Jr., and Rush have proved that every ideal of the form I = ( x 1 n , x 2 , … , x d ) R with n ∈ N + , is projectively full. If ( R , M ) is 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), then we are able to prove a similar result for every minimal ideal basis x 1 , … , x d of M such that x 1 ∉ rad ( x 2 , … , x d ) .

Semigroups of valuations on local rings

Given a noetherian local domain $R$ and a valuation $\nu$ of its field of fractions which is non negative on $R$, we derive some very general bounds on the growth of the number of distinct valuation ideals of $R$ corresponding to values lying in certain parts of the value group $\Gamma$ of $\nu$. We show that this growth condition imposes restrictions on the semigroups $\nu(R\setminus \{0\})$ for noetherian $R$ which are stronger that those resulting from the previous paper \cite{C2} of the first author. Given an ordered embedding $\Gamma\subset ({\mathbf R}^h)_{\hbox{\rm lex}}$, where $h$ is the rank of $\nu$, we also study the shape in ${\mathbf R}^h$ of the parts of $\Gamma$ which appear naturally in this study. We give examples which show that this shape can be quite wild in a way which does not depend on the embedding and suggest that it is a good indicator of the complexity of the semigroup $\nu(R\setminus \{0\})$.

Algebraic series and valuation rings over nonclosed fields

Suppose that k is an arbitrary field. Let k [ [ x 1 , … , x n ] ] be the ring of formal power series in n variables with coefficients in k . Let k ¯ be the algebraic closure of k and σ ∈ k ¯ [ [ x 1 , … , x n ] ] . We give a simple necessary and sufficient condition for σ to be algebraic over the quotient field of k [ [ x 1 , … , x n ] ] . We also characterize valuation rings V dominating an excellent Noetherian local domain R of dimension 2, and such that the rank increases after passing to the completion of a birational extension of R . This is a generalization of the characterization given by M. Spivakovsky [Valuations in function fields of surfaces, Amer. J. Math. 112 (1990) 107–156] in the case when the residue field of R is algebraically closed.

Invariants associated with ideals in one-dimensional local domains

Let R be a one-dimensional local Noetherian domain with maximal ideal m , quotient field K and residue field R / m : = k . We assume that the integral closure R ¯ of R in its quotient field K is a DVR and a finite R-module. We assume also that the field k is isomorphic to the residue field of R ¯ . For I a proper ideal of R, denote the inverse of I by I ∗ ; that is, I ∗ is the set ( R : K I ) of elements of K that multiply I into R. We investigate two numerical invariants associated to a proper ideal I of R that have previously come up in the literature from various points of view. The two invariants are: (1) the difference between the composition lengths of I ∗ / R and R / I , and (2) the difference between the product, when the composition length of R / I is multiplied by the composition length of m ∗ / R , and the length of I ∗ / R . We show that these two differences can be expressed in terms of the type sequence of R, a finite sequence of positive integers related to the natural valuation inherited from R ¯ .

Absolute bounds on the number of generators of Cohen-Macaulay ideals of height two

For a Noetherian local domain A, there exists an upper bound Nτ (A) on the minimal number of generators of any height two ideal a for whichA/a is Cohen-Macaulay of type τ . If A contains an infinite field, then we may take Nτ (A) := (τ + 1)ehom(A), where ehom(A) is the homological multiplicity of A.

Absolute integral closure in positive characteristic

Let R be a local Noetherian domain of positive characteristic. A theorem of Hochster and Huneke [M. Hochster, C. Huneke, Infinite integral extensions and big Cohen–Macaulay algebras, Ann. of Math. 135 (1992) 53–89] states that if R is excellent, then the absolute integral closure of R is a big Cohen–Macaulay algebra. We prove that if R is the homomorphic image of a Gorenstein local ring, then all the local cohomology (below the dimension) of such a ring maps to zero in a finite extension of the ring. As a result there follow an extension of the original result of Hochster and Huneke to the case in which R is a homomorphic image of a Gorenstein local ring, and a considerably simpler proof of this result in the cases where the assumptions overlap, e.g., for complete Noetherian local domains.

On linearity of finitely generated R-analytic groups

In this paper we prove that if R is a commutative Noetherian local pro-p domain of characteristic 0, then every finitely generated R-standard group is linear.

The Poincaré series associated with finitely many monomial valuations

Let $(R,M)$ be a Noetherian local domain, $F$ its quotient field and $v$ a valuation of $F$ centred at $R$. In the hope of understanding the valuation $v$, a useful tool is the so-called graded algebra relative to $v$, $gr_{v}R$. Its study can be understood as an attempt to approach the problem of classifying valuations. On the other hand, $gr_{v}R$ is also an object of interest since it can be provided with a geometrical meaning [15].

On the structure of the value semigroup of a valuation

Let v be a valuation of the quotient field of a noetherian local domain R. Assume that v is centered at R. This paper studies the structure of the value semigroup of v, S. Ideals defining toric varieties can be defined from the graded algebra K[T] of cancellative commutative finitely generated semigroups such that T n (-T) = {0}. The value semigroup of a valuation S need not be finitely generated but we prove that S n (-S) = {0} and so, the study in this paper can also be seen as a generalization to infinite dimension of that of toric varieties. In this paper, we prove that K[S] can be regarded as a module over an infinitely dimensional polynomial ring A v . We show a minimal graded resolution of K[S] as A v -module and we give an explicit method to obtain the syzygies of K[S] as A v -module. Finally, it is shown that free resolutions of K[S] as A v -module can be obtained from certain cell complexes related to the lattice associated to the kernel of the map A v → K[S].

Some Results on Tight Closure and Completion

We construct two examples of nonexcellent local Noetherian domains which demonstrate that tight closure and completion do not commute. The first example is a local normal domain A with a height one principal prime ideal P such that ( PÁ )* ≠ P * Á . We also construct an example of a complete local normal Gorenstein domain which is not F-regular but is the completion of an F-regular local ring.

Conormal Geometry of Maximal Minors

Let A be a Noetherian local domain, N be a finitely generated torsion-free module, and M a proper submodule that is generically equal to N. Let A[N] be an arbitrary graded overdomain of A generated as an A-algebra by N placed in degree 1. Let A[M] be the subalgebra generated by M. Set C≔Proj(A[M]) and r≔dimC. Form the (closed) subset W of Spec(A) of primes p where A[N]p is not a finitely generated module over A[M]p, and denote the preimage of W in C by E. We prove this: (1) dimE=r−1 if either (a) Nis free and A[N] is the symmetric algebra, or (b) W is nonempty and A is universally catenary, and (2) E is equidimensional if (a) holds and A is universally catenary. Our proof was inspired by some recent work of Gaffney and Massey, which we sketch; they proved (2) when A is the ring of germs of a complex-analytic variety, and applied it to improve a characterization of Thom's Af-condition in equisingularity theory. From (1), we recover, with new proofs, the usual height inequality for maximal minors and an extension of it obtained by the authors in 1992. From the latter, we recover the authors' generalization to modules of Böger's criterion for integral dependence of ideals. Finally, we introduce an application of (1), being made by Thorup, to the geometry of the dual variety of a projective variety, and use it to obtain an interesting example where the conclusion of (1) fails and A[N] is a finitely generated module over A[M].

Arrangements of ideals in a commutative algebra

In this note we introduce the notion of 1-complete arrangements of ideals, and, taking into account an arrangement of prime divisors of a radical principal ideal, we give a sufficient condition for a noetherian local domain to be normal.

Deformation of [formula omitted]-purity and [formula omitted]-regularity

For a Noetherian local domain (R,m,K), it is an open question whether strong F-regularity deforms. We provide an affirmative answer to this question when the canonical module satisfies certain additional assumptions. The techniques used here involve passing to a Gorenstein ring, using an anti-canonical cover.

Noetherian Domains Inside a Homomorphic Image of a Completion

Over the past 60 years, important examples of Noetherian domains have been constructed using power series, homomorphic images, and intersections. In these constructions it is often crucial that the resulting domain is computable as a directed union. In this article we analyse this construction and show that the Noetherian property for the associated directed union is equivalent to a flatness condition. LetRbe a Notherian integral domain with fraction fieldL. Letxbe a nonzero nonunit ofRand letR* denote the (x)-adic completion ofR. SupposeIis an ideal ofR* with the propertyp∩R=(0) for eachp∈Ass(R*/I).Theorem. The embeddingR↪(R*/I)xis flat if and only ifA≔L∩(R*/I) is Noetherian and is realizable as a localization of a subring ofRx=R[1/x]. We present several examples where this flatness condition holds; one of these examples is a local Noetherian domain that is not universally catenary, but has geometrically regular formal fibers.

Ratliff–Rush Closures and Coefficient Modules

Let (R,m) be ad-dimensional Noetherian local domain. SupposeMis a finitely generated torsion-freeR-module and supposeFis a freeR-module containingM. In analogy with a result of Ratliff and Rush [Indiana Univ. Math. J.27(1978), 929–934] concerning ideals, we define and prove existence and uniqueness of theRatliff–RushclosureofMinF. We also discuss properties of Ratliff–Rush closure.In addition to the preceding assumptions, supposeF/Mhas finite length as anR-module. Then we define theBuchsbaum–RimpolynomialofMinF. In analogy with the work of K. Shah [Trans. Amer. Math. Soc.327(1991), 373–384], we definecoefficientmodulesofMinF. Under the assumption thatRis quasi-unmixed, we prove existence and uniqueness of coefficient modules ofMinF.

Noetherian Rings between a Semilocal Domain and Its Completion

We consider a general technique for constructing local Noetherian integral domains. LetRbe a semilocal Noetherian domain with Jacobson radicalmand field of fractionsK. Letybe a nonzero element ofmand letR* be the (y)-adic completion ofR. For elements τ1,…,τs∈yR* algebraically independent overK, we obtain a necessary and sufficient condition forA≔K(τ1,…,τs)∩R* to be simultaneously Noetherian and a directed union of localized polynomial rings insvariables overR. We specify conditions in order that excellence be preserved, and we use the construction to obtain a non-Noetherian ringAof the formK(τ)∩R* which is a directed union of localized polynomial rings overR.

The number of generators for the product of an ideal with its inverse

Let (R,m) be a commutative one-dimensional Noetherian local domain. Assume the integral closure satisfying asymptotic condition where b is a strictly positive C∞ function on the unit sphere in R2. is a discrete valvation ring and the residue fields of R and are the same. Let I be a non-principal ideal of R. A question, motivated by the work of Auslander in the early 1960's, is whether the torsion submodule of I ⊗R I−1 is always non-zero. In this paper we investigate the stronger condition, μR(I)μR(I−1) greater than sign μR(II −1)(where μ denotes the number of generators). We will show this inequality holds whenever the multiplicity of R is at most four.