Noetherian Local Domain Research Articles

Existence of almost Cohen–Macaulay algebras implies the existence of big Cohen–Macaulay algebras

In [1], the dagger closure is extended over finitely generated modules over Noetherian local domain (R,m) and it is proved to be a Dietz closure. In this short note we show that it also satisfies the ‘Algebra axiom’ of [9] and this leads to the following result of this paper: For a complete Noetherian local domain, if it is contained in an almost Cohen–Macaulay domain, then there exists a balanced big Cohen–Macaulay algebra over it.

On the vanishing of local cohomology of the absolute integral closure in positive characteristic

The aim of this paper is to extend the main result of C. Huneke and G. Lyubeznik in [Adv. Math. 210 (2007), 498–504] to the class of rings that are images of Cohen–Macaulay local rings. Namely, let R be a local Noetherian domain of positive characteristic that is an image of a Cohen–Macaulay local ring. We prove that all local cohomology of R (below the dimension) maps to zero in a finite extension of the ring. As a direct consequence we obtain that the absolute integral closure of R is a big Cohen–Macaulay algebra. Since every excellent local ring is an image of a Cohen–Macaulay local ring, this result is a generalization of the main result of M. Hochster and Huneke in [Ann. of Math. 135 (1992), 45–79] with a simpler proof.

Excellent normal local domains and extensions of Krull domains

We consider properties of extensions of Krull domains such as flatness that involve behavior of extensions and contractions of prime ideals. Let (R,m) be an excellent normal local domain with field of fractions K, let y be a nonzero element of m and let R⁎ denote the (y)-adic completion of R. For elements τ1,…,τs of yR⁎ that are algebraically independent over R, we construct two associated Krull domains: an intersection domain A:=K(τ1,…τs)∩R⁎ and its approximation domain B; see Setting 2.2.If in addition R is countable with dimR≥2, we prove that there exist elements τ1,…,τs,… as above such that, for each s∈N, the extension R[τ1,…,τs]↪R⁎[1/y] is flat; equivalently, B=A and A is Noetherian. Using this result we establish the existence of a normal Noetherian local domain B such that: B dominates R; B has (y)-adic completion R⁎; and B contains a height-one prime ideal p such that R⁎/pR⁎ is not reduced. Thus B is not a Nagata domain and hence is not excellent.We present several theorems involving the construction. These theorems yield examples where B⊊A and A is Noetherian while B is not Noetherian; and other examples where B=A and A is not Noetherian.

A characterization of local Noetherian domains which admit only finitely many star operations: The infinite residue field case

We characterize local Noetherian domains R with infinite residue field which admit only finitely many star (prime) operations. We further show that a positive integer n is equal to the number of star operations on such an R if and only if n=2,4 or an odd positive integer.

One-dimensional bad Noetherian domains

Local Noetherian domains arising as local rings of points of varieties or in the context of algebraic number theory are analytically unramified, meaning their completions have no nontrivial nilpotent elements. However, looking elsewhere, many sources of analytically ramified local Noetherian domains have been exhibited over the last seventy five years. We give a unified approach to a number of such examples by describing classes of DVRs which occur as the normalization of an analytically ramified local Noetherian domain, as well as some that do not occur as such a normalization. We parameterize these examples, or at least large classes of them, using the module of Kahler differentials of a relevant field extension.

Quasi-one-fibered ideals of order one in dimension two

Simple complete $\M$-primary ideals are fundamental in the Zariski-Lipman theory of complete ideals in a two-dimensional regular local ring $(R,\M)$. Complete $\M$-primary ideals of order one constitute a particular class of such ideals containing, for example, all the first neighborhood ideals of $R$. A number of interesting properties of complete $\M$-primary ideals of order one have been proved by several authors. For example, these ideals have only one Rees valuation (and hence are one-fibered) and they have the very simple form $(x_1^n,x_2)R$, $n\in\n_{+}$, with $x_1,x_2$ a minimal ideal basis of $\M$. In the present paper we investigate how far some results concerning these complete $\M$-primary ideals of order one can be extended to complete quasi-one-fibered $\M$-primary ideals of order one in a natural generalization of $R$: a two-dimensional normal Noetherian local domain with algebraically closed residue field and the associated graded ring an integrally closed domain.

Muhly local domains and Zariski's theory of complete ideals

Let $(R,\M)$ be a two-dimensional Muhly local domain, that is, a two-dimensional integrally closed Noetherian local domain with algebraically closed residue field and with the associated graded ring $\text{gr}_{\M}R$ an integrally closed domain. In this paper we show that a number of fundamental results of Zariski's theory of complete ideals in two-dimensional regular local rings are not necessarily valid in $R$. However, if the associated graded ring $\text{gr}_{\M}R$ satisfies an additional assumption as in work of Muhly and Sakuma, then we are able to show that ``any product of contracted ideals is contracted'' holds in $R$ if and only if $R$ has minimal multiplicity.

Generic formal fibers and analytically ramified stable rings

AbstractLetAbe a local Noetherian domain of Krull dimensiond. Heinzer, Rotthaus, and Sally have shown that if the generic formal fiber ofAhas dimensiond– 1, thenAis birationally dominated by a 1-dimensional analytically ramified local Noetherian ring having residue field finite over the residue field ofA. We explore further this correspondence between prime ideals in the generic formal fiber and 1-dimensional analytically ramified local rings. Our main focus is on the case where the analytically ramified local rings are stable, and we show that in this case the embedding dimension of the stable ring reflects the embedding dimension of a prime ideal maximal in the generic formal fiber, thus providing a measure of how far the generic formal fiber deviates from regularity. A number of characterizations of analytically ramified local stable domains are also given.

Generic formal fibers and analytically ramified stable rings

AbstractLet A be a local Noetherian domain of Krull dimension d. Heinzer, Rotthaus, and Sally have shown that if the generic formal fiber of A has dimension d – 1, then A is birationally dominated by a 1-dimensional analytically ramified local Noetherian ring having residue field finite over the residue field of A. We explore further this correspondence between prime ideals in the generic formal fiber and 1-dimensional analytically ramified local rings. Our main focus is on the case where the analytically ramified local rings are stable, and we show that in this case the embedding dimension of the stable ring reflects the embedding dimension of a prime ideal maximal in the generic formal fiber, thus providing a measure of how far the generic formal fiber deviates from regularity. A number of characterizations of analytically ramified local stable domains are also given.

Prescribed subintegral extensions of local Noetherian domains

We show how subintegral extensions of certain local Noetherian domains S can be constructed with specified invariants including reduction number, Hilbert function, multiplicity and local cohomology. The construction behaves analytically like Nagata idealization but rather than a ring extension of S, it produces a subring R of S such that R⊆S is subintegral.

Noncommutative Fitting invariants and improved annihilation results

To each finitely presented module M over a commutative ring R, one can associate an R-ideal FitR(M) which is called the (zeroth) Fitting ideal of M over R, and which is always contained in the R-annihilator of M. In an earlier article, the second-named author generalized this notion by replacing R with a (not necessarily commutative) 𝔬-order Λ in a finite-dimensional separable algebra, where 𝔬 is an integrally closed complete commutative noetherian local domain. To obtain annihilators, one has to multiply the Fitting invariant of a (left) Λ-module M by a certain ideal ℋ(Λ) of the centre of Λ. In contrast to the commutative case, this ideal can be properly contained in the centre of Λ. In the present article, we determine explicit lower bounds for ℋ(Λ) in many cases. Furthermore, we define a class of ‘nice’ orders Λ over which Fitting invariants have several useful properties such as good behaviour with respect to direct sums of modules, computability in a certain sense and ℋ(Λ) being the best possible.

Uniform bounds on symbolic powers

We study the question of obtaining uniform bounds for the growth of symbolic powers of ideals in Noetherian rings. First we use a generalized version of Artinʼs strong approximation theorem to obtain uniform bounds for prime ideals in the integral closure of a finite Galois field extension of the fraction field of an excellent Henselian regular local ring. As a corollary, we obtain a similar result for a complete Noetherian local domains. We also obtain linear uniform bounds for the growth of symbolic powers in certain normal subrings of equicharacteristic regular rings under mild conditions.

Integrally closed rings in birational extensions of two-dimensional regular local rings

AbstractLet D be an integrally closed local Noetherian domain of Krull dimension 2, and let f be a nonzero element of D such that fD has prime radical. We consider when an integrally closed ring H between D and Df is determined locally by finitely many valuation overrings of D. We show such a local determination is equivalent to a statement about the exceptional prime divisors of normalized blow-ups of D and, when D is analytically normal, this property holds for D if and only if it holds for the completion of D. This latter fact, along with MacLane's notion of key polynomials, allows us to prove that in some central cases where D is a regular local ring and f is a regular parameter of D, then H is determined locally by a single valuation. As a consequence, we show that if H is also the integral closure of a finitely generated D-algebra, then the exceptional prime ideals of the extension H/D are comaximal. Geometrically, this translates into a statement about intersections of irreducible components in the closed fiber of the normalization of a proper birational morphism.

Noetherian domains which admit only finitely many star operations

In this paper, we study Noetherian domains which admit only finitely many star operations. We show that such domains (which are not fields) must have Krull dimension one, and we effectively reduce to the local case. For a one-dimensional local Noetherian domain (R,M), M nonprincipal, M−1 is an overring of R, and M−1/M is naturally an R/M-vector space. We succeed in counting the number of star operations on R in several cases when dimR/MM−1/M=3.

Extending a valuation centred in a local domain to the formal completion

Let (R; m; k) be a local noetherian domain with field of fractions K and R_v a valuation ring, dominating R (not necessarily birationally). Let v|K be the restriction of v to K; by definition, v|K is centered at R. Let \hat{R} denote the m-adic completion of R. In the applications of valuation theory to commutative algebra and the study of singularities, one is often induced to replace R by its m-adic completion \hat{R} and v by a suitable extension \hat{v} to \hat{R}/P for a suitably chosen prime ideal P, such that P \cap R = (0). The purpose of this paper is to give, assuming that R is excellent, a systematic description of all such extensions \hat{v} and to identify certain classes of extensions which are of particular interest for applications.

Trace Properties and Integral Domains, II

An integral domain R has the radical trace property, or is an RTP domain, if I(R: I) is a radical ideal for each nonzero noninvertible ideal I. A related property is LTP: I(R: I)R P = PR P for each minimal prime P of I(R: I). It is clear that each RTP domain is an LTP domain, but whether the two are equivalent is open except in certain special cases. The ideal I(R: I) is the trace of I and I is a trace ideal if I(R: I) = I. A new characterization for RTP domains is established for Noetherian domains, Mori domains, and Prüfer domains. In these special cases, R is an RTP domain if and only if IB(R: IB) = I for each trace ideal I and each ideal B of (R: I) that contains I (here (R: I) is a ring as I is a trace ideal). In the Prüfer case, having IJ(R: IJ) = I for each trace ideal I and each ideal J of R containing I is enough to have the radical trace property. On the other hand, an example is given of a one-dimensional local Noetherian domain R that is not an RTP domain but does satisfy IJ(R: IJ) = I for each trace ideal I and each ideal J of R that contains I.

A few examples of local rings, I

In this paper, we first recall and apply the fundamental techniques of constructing bad Noetherian local domains, due to C. Rotthaus, T. Ogoma, R. C. Heitmann, and M. Brodmann and C. Rotthaus, to show several basic examples: (1) a three-dimensional Nagata normal local domain, which is a complete intersection, whose regular locus is not open; (2) a three-dimensional Henselian Nagata normal local domain, which is not catenary. Next we present a unified version of Brodmann and Rotthaus’s and Ogoma’s methods in order to obtain a particular local domain A with a specified prime element x such that the local domain A/xA is the bad Noetherian local domain given above: (3) a three-dimensional unmixed local domain A that has xA=p∈Spec(A) such that A/p is not unmixed. Finally we follow Ogoma’s construction of factorial local domains whose completions are designated complete local domains. Then, we gather some examples of bad factorial local domains.

Quasi-one-fibered ideals in two-dimensional Muhly local domains

In this paper, a two-dimensional integrally closed Noetherian local domain ( R , M ) with algebraically closed residue field is said to be a Muhly local domain, if the associated graded ring is an integrally closed domain. Using the theory of degree functions developed by Rees and Sharp, and the Zariski–Lipman theory of complete ideals in two-dimensional regular local rings, we study M -primary ideals of R having among their Rees valuations exactly one valuation ≠ ord R . We therefore call these ideals quasi-one-fibered. Some consequences are deduced.

Semigroups of valuations on local rings, II

Given a noetherian local domain $R$ and a valuation $\nu$ of its field of fractions which is nonnegative 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 than those resulting from the previous paper of the first author. Given an ordered embedding $\Gamma\subset ({{\bf R}}^h)_{\hbox{\rm lex}}$, where $h$ is the rank of $\nu$, we also study the shape in ${{\bf R}}^h$ 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 $\nu(R\setminus \{0\})$.

Homological properties of the perfect and absolute integral closures of Noetherian domains

For a Noetherian local domain R let R + be the absolute integral closure of R and let R ∞ be the perfect closure of R, when R has prime characteristic. In this paper we investigate the projective dimension of residue rings of certain ideals of R + and R ∞. In particular, we show that any prime ideal of R ∞ has a bounded free resolution of countably generated free R ∞-modules. Also, we show that the analogue of this result is true for the maximal ideals of R +, when R has residue prime characteristic. We compute global dimensions of R + and R ∞ in some cases. Some applications of these results are given.