Complete Ideals Research Articles

Remark on Rees valuations of finitely supported complete monomial ideals in a regular local ring

Let be an equicharacteristic regular local ring of dimension and and consider the sequence of d-dimensional local monomial quadratic transformations where is a local monomial quadratic transformation of Ri for We prove that where and is the special -simple complete ideal associated with Let I and J be finitely supported complete -primary monomial ideals in R. We show that if the set of base points of the ideal IJ is linearly ordered, then

Results on the behavior of the Ratliff–Rush operation and the depth of the associated graded ring

We give a full characterization of the unique Ratliff–Rush complete ideals in the polynomial rings R with two variables over a field. For a class of monomial ideals in R, we investigate the Ratliff–Rush behavior of powers of these ideals and also the depth of their associated graded ring. Namely, we give some sufficient and some necessary conditions for this depth to be positive. The results of this paper allow us to answer several questions of [13].

CRITICAL BINOMIAL IDEALS OF NORTHCOTT TYPE

AbstractIn this paper, we study a family of binomial ideals defining monomial curves in the n-dimensional affine space determined by n hypersurfaces of the form $x_i^{c_i} - x_1^{u_{i1}} \cdots x_n^{u_{1n}}$ in $\Bbbk [x_1, \ldots , x_n]$ with $u_{ii} = 0, \ i\in \{ 1, \ldots , n\}$ . We prove that the monomial curves in that family are set-theoretic complete intersections. Moreover, if the monomial curve is irreducible, we compute some invariants such as genus, type and Frobenius number of the corresponding numerical semigroup. We also describe a method to produce set-theoretic complete intersection semigroup ideals of arbitrary large height.

Noetherian intersections of regular local rings of dimension two

Let D be a 2-dimensional regular local ring and let Q(D) denote the quadratic tree of 2-dimensional regular local overrings of D. We describe the desingularization of projective models over D both algebraically in terms of the saturation of complete ideals and order-theoretically in terms of the quadratic tree Q(D). We prove that this describes all the Noetherian rings that are intersections of rings in Q(D).

Monomial generators of complete planar ideals

We provide an algorithm that computes a set of generators for any complete ideal in a smooth complex surface. More interestingly, these generators admit a presentation as monomials in a set of maximal contact elements associated to the minimal log-resolution of the ideal. Furthermore, the monomial expression given by our method is an equisingularity invariant of the ideal. As an outcome, we provide a geometric method to compute the integral closure of a planar ideal and we apply our algorithm to some families of complete ideals.

An effective characterization of complete monomial ideals in two variables

We give a criterion for a monomial ideal in two variables to be complete (i.e. integrally closed) in terms of the exponents of the generators. This gives a positive answer to a question raised recently by P. Gimenez, A. Simis, W. Vasconcelos and R. Villarreal, On complete monomial ideals, J. Commut. Algebra, to appear, arXiv:1310.7793.

The new operations on complete ideals

Abstract We introduce the notion of K-ideals associated with Kuratowski partitions. Using new operations on complete ideals we show connections between K-ideals and precipitous ideals and prove that every complete ideal can be represented by some K-ideal.

“$${\mathbb {P}}^1$$P1-gluing” for local complete intersections

We prove an analogue of the Affine Horrocks’ Theorem for local complete intersection ideals of height n in R[T], where R is a regular domain of dimension d, which is essentially of finite type over an infinite perfect field of characteristic unequal to 2, and $$2n\ge d+3$$.

The arithmetical rank of k-complete ideals

We introduce the notions of algebraic and arithmetic derivation. As an application, we use the combinatorial decomposition of an ideal to provide a constructive method to find the algebraic invariants, as the arithmetical rank, for a family of squarefree monomial ideals, the $k$--complete ideals $I_k^n,$ also known as squarefree Veronese ideals of degree $k$.

Finitely supported ⁎-simple complete ideals and multiplicities in a regular local ring

Let (R,m) and (S,n) be regular local rings of dim⁡(S)=dim⁡(R)≥2 such that S birationally dominates R, and let V be the order valuation ring of S with corresponding valuation ν:=ordS. Assume that IS≠S and ν∈ReesSIS. Let u:=αt with IS=αIS, where α∈S. Then V=W∩Q(R) with W=(R[It]‾)Q=(S[ISu]‾)Q′, where Q∈Min(mR[It]‾) and Q′∈Min(nS[ISu]‾). Let P,P′ be the center of W on R[It] and S[Isu], respectively. We prove that if [Sn:Rm]=1, then R[It]P=S[Isu]P′. Let I be a finitely supported complete m-primary ideal of a regular local ring (R,m) of dimension d≥2. Let T be a terminal base point of I and V be the mT-adic order valuation of T with corresponding valuation v:=ordT. Let n≥1 be an integer. Assume that IT=mTn and [TmT:Rm]=1. Let P∈Min(mR[It]) such that P=Q∩R[It] with V=(R[It]‾)Q∩Q(R), where Q∈Min(mR[It]‾). We prove that the quotient ring R[It]P is d-dimensional normal Cohen–Macaulay standard graded domain over k with the multiplicity nd−1. In particular, R[It]P is regular if and only if n=1. We prove that k:=Rm is relatively algebraically closed in kv:=VmV. Also we determine the multiplicity of R[It]P, and we prove that if IT=mT, then R[It]P is regular if and only if [TmT:Rm]=1.

LOCAL COHOMOLOGY OF MULTI-REES ALGEBRAS, JOINT REDUCTION NUMBERS AND PRODUCT OF COMPLETE IDEALS

We find conditions on the local cohomology modules of multi-Rees algebras of admissible filtrations which enable us to predict joint reduction numbers. As a consequence, we are able to prove a generalization of a result of Reid, Roberts and Vitulli in the setting of analytically unramified local rings for completeness of power products of complete ideals.

The Stanley regularity of complete intersections and ideals of mixed products

Let [Formula: see text] be a polynomial ring over a field [Formula: see text] and [Formula: see text] a monomial ideal. We give some inequalities on Stanley regularity of monomial ideals. As consequences, we prove that [Formula: see text] and [Formula: see text] hold in the following cases: (1) [Formula: see text] is a complete intersection; (2) [Formula: see text] is an ideal of mixed products.

Blowing up finitely supported complete ideals in a regular local ring

Let I be a finitely supported complete m-primary ideal of a regular local ring (R,m). We consider singularities of the projective models ProjR[It] and ProjR[It]‾ over SpecR, where R[It]‾ denotes the integral closure of the Rees algebra R[It]. A theorem of Lipman implies that the ideal I has a unique factorization as a ⁎-product of special ⁎-simple complete ideals with possibly negative exponents for some of the factors. If ProjR[It]‾ is regular, we prove that ProjR[It]‾ is the regular model obtained by blowing up the finite set of base points of I. Extending work of Lipman and Huneke–Sally in dimension 2, we prove that every local ring S on ProjR[It]‾ that is a unique factorization domain is regular. Moreover, if dim⁡S≥2 and S dominates R, then S is an infinitely near point to R, that is, S is obtained from R by a finite sequence of local quadratic transforms.

Weak Rudin–Keisler reductions on projective ideals

We consider a slightly modified form of the standard Rudin-Keisler order on ideals and demonstrate the existence of complete (with respect to this order) ideals in various projective classes. Using our methods, we obtain a simple proof of Hjorth's theorem on the existence of a complete $\mathbf \Pi^1_1$ equivalence relation. Our proof of Hjorth's theorem enables us (under PD) to generalize his result to the classes of $\mathbf \Pi^1_{2n+1}$ equivalence relations.

Iterated elementary embeddings and the model theory of infinitary logic

We use iterations of elementary embeddings derived from countably complete ideals on ω1 to provide a uniform proof of some classical results connecting the number of models of cardinality ℵ1 in various infinitary logics to the number of syntactic types over the empty set. We introduce the notion of an analytically presented abstract elementary class (AEC), which allows the formulation and proof of generalizations of these results to refer to Galois types rather than syntactic types. We prove (Theorem 0.4) the equivalence of ℵ0-presented classes and analytically presented classes and, using this, generalize (Theorem 0.5) Keisler's theorem on few models in ℵ1 to bound the number of Galois types rather than the number of syntactic types. Theorem 0.6 gives a new proof (cf. [5]) for analytically presented AEC's of the absoluteness of ℵ1-categoricity from amalgamation in ℵ0 and almost Galois ω-stability.

Tensor products of semilattices and fuzzy ideals

We study (complete) fuzzy ideals on semilattices from the point of view of tensor products of semilattices. We show that the lattice of all complete fuzzy ideals on a semilattice is an extension of tensor product. We define the notion of semicomplete bi-ideals, and show that the lattice of all fuzzy ideals on a distributive lattice is isomorphic to the lattice of all semicomplete bi-ideals. Moreover, we show that the lattice of all (complete) fuzzy ideals on a distributive (algebraic) lattice is distributive.

Local Cohomology of Multi-Rees Algebras with Applications to joint Reductions and Complete Ideals

Let \((R,\mathfrak {m})\) be a Cohen-Macaulay local ring of dimension d and I=(I1,…,Id) be \(\mathfrak {m}-\)primary ideals in R. We prove that \({\lambda }_{R}([H^{d}_{(x_{ii}t_{i}:1\leq i \leq d)}({\mathcal {R}}^{\prime }({\mathcal {F}})]_{\mathbf {n}})\)\(< \infty \), for all \(\mathbf {n} \in \mathbb {N}^{d}\), where \({\mathcal {F}}=\{{\mathcal {F}}(\mathbf {n}):\mathbf {n}\in \mathbb {Z}^{d}\}\) is an I−admissible filtration and (xij) is a strict complete reduction of \({\mathcal {F}}\) and \({\mathcal {R}}^{\prime }({\mathcal {F}})\) is the extended multi-Rees algebra of \({\mathcal {F}}.\) As a consequence, we prove that the normal joint reduction number of I,J,K is zero in an analytically unramified Cohen-Macaulay local ring of dimension 3 if and only if \(\overline {e}_{3}(IJK)-[\overline {e}_{3}(IJ)+\overline {e}_{3}(IK)\)\(+\overline {e}_{3}(JK)]+\overline {e}_{3}(I)+ \overline {e}_{3}(J)+\overline {e}_{3}(K)=0\). This generalizes a theorem of Rees on joint reduction number zero in dimension 2. We apply this theorem to generalize a theorem of M. A. Vitulli in dimension 3.

The Rees Valuations of Complete Ideals in a Regular Local Ring

Let I be a complete m-primary ideal of a regular local ring (R, m) of dimension d ≥ 2. In the case of dimension two, the beautiful theory developed by Zariski implies that I factors uniquely as a product of powers of simple complete ideals and each of the simple complete factors of I has a unique Rees valuation. In the higher dimensional case, a simple complete ideal of R often has more than one Rees valuation, and a complete m-primary ideal I may have finitely many or infinitely many base points. For the ideals having finitely many base points Lipman proves a unique factorization involving special *-simple complete ideals and possibly negative exponents of the factors. Let T be an infinitely near point to R with dim R = dim T and R/m = T/m T . We prove that the special *-simple complete ideal P RT has a unique Rees valuation if and only if either dim R = 2 or there is no change of direction in the unique finite sequence of local quadratic transformations from R to T. We also examine conditions for a complete ideal to be projectively full.

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.

Finitely supported ⁎-simple complete ideals in a regular local ring

Let I be a finitely supported complete m-primary ideal of a regular local ring (R,m). A theorem of Lipman implies that I has a unique factorization as a ⁎-product of special ⁎-simple complete ideals with possibly negative exponents for some of the factors. The existence of negative exponents occurs if dimR⩾3 because of the existence of finitely supported ⁎-simple ideals that are not special. We consider properties of special ⁎-simple complete ideals such as their Rees valuations and point basis. Let (R,m) be a d-dimensional equicharacteristic regular local ring with m=(x1,…,xd)R. We define monomial quadratic transforms of R and consider transforms and inverse transforms of monomial ideals. For a large class of monomial ideals I that includes complete inverse transforms, we prove that the minimal number of generators of I is completely determined by the order of I. We give necessary and sufficient conditions for the complete inverse transform of a ⁎-product of monomial ideals to be the ⁎-product of the complete inverse transforms of the factors. This yields examples of finitely supported ⁎-simple monomial ideals that are not special. We prove that a finitely supported ⁎-simple monomial ideal with linearly ordered base points is special ⁎-simple.