Noetherian Local Domain Research Articles

On Symmetric Numerical Semigroups

A numerical semigroup is a subset S of N such that is closed under sums, contains the zero and generates Z as a group. From this definition Ž w x. Ž one obtains see, for example, 2 that S has a conductor C i.e., the . maximum among all the natural numbers not belonging to S . A numerical semigroup S is called symmetric if for every integer z f S, C y z g S. The study of the subsemigroups of N is a classical problem, which is equivalent to study of the sets of natural solutions of linear equations with w x coefficients in N. From 1970 and after the works 5, 6 and others, the study of the subsemigroups of N is also motivated by its applications to algebraic geometry. An example arises when we consider an analytically irreducible one-dimensional Noetherian local domain R, the integral closure V of R in the quotient field of R is a discrete valuation ring, if we assume that R and V have the same residue class field and we denote by Ž . the valuation attached to V, then S s R is a numerical semigroup. R Then properties on the semigroup S imply properties of the ring R. For R w x Ž w x. example, 6 proves that R is Gorenstein see 1 if and only if S is R symmetric. The close relation between numerical semigroups and monomial curves has produced the usage of algebraic geometry terminology in the field of numerical semigroups. For example, the smallest integer m greater than zero in S is usually called the multiplicity of S, the cardinal e of a minimal system of generators for S is called the embedding dimension of S, and a finite presentation for S is called a relation for S.

Torsion in the tensor product of an ideal with its inverse

Let (R, m) be a one-dimensional Noetherian local domain. Assume the integral closure is a discrete valuation ring with maximal ideal . 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 is always non-zero. In this paper we will show that if both m and I are generated by powers ofx, the residue fields of R and are the same and the multiplicity of R is at most seven, then . It follows that has torsion under these conditions.

Formal fibers and complete homomorphic images

Let ( R , m ) (R,{\mathbf {m}}) be an excellent normal local Henselian domain, and suppose that q {\mathbf {q}} is a prime ideal in R R of height > 1 > 1 . We show that, if R / q R/{\mathbf {q}} is not complete, then there are infinitely many height one prime ideals p ⊆ q R ^ {\mathbf {p}} \subseteq {\mathbf {q}}\hat R of R ^ \hat R with p ∩ R = 0 {\mathbf {p}} \cap R = 0 ; in particular, the dimension of the generic formal fiber of R R is at least one. This result may in fact indicate that a much stronger relationship between maximal ideals in the formal fibers of an excellent Henselian local ring and its complete homomorphic images is possibly satisfied. The second half of the paper is concerned with a property of excellent normal local Henselian domains R R with zero-dimensional formal fibers. We show that for such an R R one has the following good property with respect to intersection: for any field L L such that Q ( R ) ⊆ L ⊆ Q ( R ^ ) \mathcal {Q}(R) \subseteq L \subseteq \mathcal {Q}(\hat R) , the ring L ∩ R ^ L \cap \hat R is a local Noetherian domain which has completion R ^ \hat R .

Formal fibers and birational extensions

Suppose (R, m) is a local Noetherian domain with quotient field K and m-adic completion Ȓ. It is well known that the fibers of the morphism Spec(Ȓ) ₒ Spec(R), i.e., the formal fibers of R, encode important information about the structure of R. Perhaps the most important condition in Grothendieck’s 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/mC is a finite R-module.

Grothendieck Groups for Hypersurface Rings

Let $R$ and $S$ be commutative complete noetherian local Gorenstein domains. If the category of finitely generated maximal Cohen-Macaulay modules over $R$ and $S$ are stably equivalent and the equivalence commutes with the first syzygy functors, then we show that the Grothendieck groups for $R$ and $S$ are isomorphic. In particular, we apply this result to hypersurface rings.

Grothendieck groups for hypersurface rings

Let R R and S S be commutative complete noetherian local Gorenstein domains. If the category of finitely generated maximal Cohen-Macaulay modules over R R and S S are stably equivalent and the equivalence commutes with the first syzygy functors, then we show that the Grothendieck groups for R R and S S are isomorphic. In particular, we apply this result to hypersurface rings.

Extension closed reflexive modules

Throughout this note, R stands for a ring with identity, and all modules are unital R-modules. In this note, we ask when the class of all finitely generated reflexive (torsionless) left modules is closed under extensions. This is not always the case, even if R is left and right noetherian (cf. Bass [2, Lemma 3.2] and [3, p. 12]). As will be shown, in case R is a commutat ive noetherian local domain of Krull dimension two, R is a CohenMacaulay ring whenever the class of the finitely generated reflexive modules is closed under extensions. Our main aim is to show that if R is a left and right noetherian ring then both the class of all finitely generated torsionless left modules and the class of all finitely generated reflexive left modules are closed under extensions if and only if for i = 1, 2 the functor Ext , 1 (Ext~ ( , R), R) vanishes on the finitely generated right modules. Note that the latter condition is satisfied if R is a commutat ive Gorenstein ring (see Bass [3]). We will show also that if R is a left and right noetherian ring with a minimal injective resolution 0 ~ RR ~ Eo ~ E1 , . . . such that weak dim E l < i + 1 for i = 0, 1 then both the class of finitely generated torsionless left modules and the class of finitely generated reflexive left modules are closed under extensions. In the proof, it will be shown that if R is a left and right noetherian ring with inj dim RR _--< 2 then the class of the finitely generated reflexive left modules is closed under extensions. On the other hand, even if R is a left and right noetherian ring with inj dim RR = inj dim R R < 2, the class of all finitely generated torsionless left modules is not necessarily closed under extensions. In what follows, we denote by ( )* both the R-dual functors, and for a given module M we denote by eu: M ~ M** the usual evaluation map and by E(M) the injective envelope of M. Recall that a module M is said to be torsionless if eu is a monomorph i sm and to be reflexive if eu is an isomorphism. Note that a module M is torsionless if and only if it is cogenerated by R. We denote by mod R (resp. mod R ~ the category of the finitely presented left (resp. right) modules. Note that if R is left noetherian then every finitely generated left module is finitely presented.

Pythagorean real curve germs

Let k be a real closed field. A real curve germ over k is a real one-dimensional Noetherian local integral domain with residual field k. A Noetherian local ring A with maximal ideal m and completion  is an AP-ring if for every system of polynomials F∈A[Y]s, Y=(Y1,⋯,Yr), for every formal solution ŷ∈Âr of F=0, and for every integer λ≥0, there exists a solution y∈Ar of F=0 such that y≡ŷ mod mλ Â. A real AP-curve is a real curve germ which is an AP-ring. The Pythagoras number p(A) of A is the least p, 1≤p≤+∞, such that each sum of squares in A is a sum of p squares. The author proves that for any real AP-curve A (over a real closed field) the derived normal ring Ā of A and the completion  of A are real curve germs and p(A)≤p(Â)<∞, p(Ā)=1. The value semigroup of a real AP-curve is a numerical semigroup, that is, an additive subsemigroup of the nonnegative integers, whose complement is finite. The main theorem classifies real AP-curves A which are Pythagorean (that is, p(A)=1) by their value semigroup Γ: Every real AP-curve with value semigroup Γ is non-Pythagorean if and only if there are q,p1,p2 ∈ Γ with q<p1≤p2 such that p1+p2−q∉Γ. Moreover, for a given numerical semigroup Γ the author proves: Every real AP-curve with value semigroup Γ is Pythagorean if and only if for each q∈Γ, p∈E with q<p, one has (q+c)/2≤p. Here c denotes the least positive integer such that Γ contains each p≥c and E is some specified subset of Γ. The paper ends with some applications: A Gorenstein real AP-curve is Pythagorean if and only if its multiplicity is ≤2. A monomial real AP-curve is Pythagorean if and only if it is Arf. There is a list of all Pythagorean real algebroid curves of multiplicity ≤5.

The parametric blowing-up of two-dimensional local domains

Let (A, M) be a local Noetherian integral domain of dimension two and X = Spec A. For an ideal IA the graded ring RA(I) = noIn denotes the Rees algebra of A with respect to I. The projective scheme Y = Proj RA(I) is called the blowing-up of X (resp. A) along I. Then there exists a proper mapping : YX. The preimage Z = 1(V(I)) is called the exceptional fibre. Note that induces an isomorphism

A Peculiar Unmixed Domain

We construct a local noetherian domain $R$ containing a nonzero prime element $U$ and satisfying: (i) The completion $\hat R$ of $R$ is a domain. (ii) $U\hat R$ has an embedded prime divisor.

Order ideals of minimal generators

Let R R be a local noetherian domain with algebraically closed residue field and let M M be a finitely generated module of rank r r which is not free. Then there is some minimal generator x x of M M such that the ideal of images of x x under maps of M M to R R has height at most r r .

On a problem of Kaplansky

For a ring R, let ▪ denote its reduced Grothendieck group. The purpose of this note is to establish (i) every f.g. torsion, or torsion-free, Abelian group is ▪ for some local Noetherian domain R, (ii) for every finitely generated Abelian group G, there exists a local Noetherian domain R for which ▪ is a finitely generated group H that contains a direct summand isomorphic to G. These are partial answers to the problem of Kaplansky: Given an Abelian group G, is there a local Noetherian domain R with ▪ ≈ G ([2, p. 68]).

A note on the hilbertfunction of a one-dimensional Cohen-Macaulay ring

In this note we give examples of one-dimensional noetherian local integral domains with the property that the number of generators of the square of the maximal ideal is less than the embedding dimension. On the other hand we show that if $$(R,m)$$ is a local one-dimensional Cohen-Macaulay ring, then $$(m^n )$$ for all n such that μ $$(m^n )$$ < m(R). Here μ denotes the minimal number of generators of an ideal and m(R) the multiplicity of R. A similar statement was conjectured by D. Sally in [5], Our main result 2.1 generalizes Prop. 2.6. in [6].

The Width of a Module

An R-module N is said to have finite width n if n is the smallest integer such that for any set of n + 1 elements of N, at least one of the elements is in the submodule generated by the remaining n. The width of N over R will be denoted by W(R, N).The notion of width was introduced by Brameret [2, p. 3605]. However, Cohen [3] investigated rings of finite rank, which, in the case that R is a local Noetherian domain, is equivalent to width (Proposition 1.6). He showed that finite width of R was both equivalent to R having Krull dimension one, and to R having the restricted minimum condition (Theorem 1.12).

On the Chain Problem of Prime Ideals

There is a problem called the chain problem of prime ideals, which asks, when 0 is a Noetherian local integral domain, whether the length of an arbitrary maximal chain of prime ideals in 0 is equal to rank 0 or not.