Integral Closure Of Ideals Research Articles

Integral closure and bounds for quotients of multiplicities of monomial ideals

Given a pair of monomial ideals I and J of finite colength of the ring of analytic function germs (Cn,0)→C, we prove that some power of I admits a reduction formed by homogeneous polynomials with respect to the Newton filtration induced by J if and only if the quotient of multiplicities e(I)/e(J) attains a suitable upper bound expressed in terms of the Newton polyhedra of I and J. We also explore other connections between mixed multiplicities, Newton filtrations and the integral closure of ideals.

$t$-reductions and $t$-integral closure of ideals

Let $R$ be an integral domain and $I$ a non\-zero ideal of $R$. An ideal $J\subseteq I$ is a $t$-reduction of~$I$ if $(JI^{n})_{t}=(I^{n+1})_{t}$ for some integer $n\geq 0$. An ele\-ment $x\in R$ is $t$-integral over $I$ if there is an equation $x^{n}+a_{1}x^{n-1}+ \cdots +a_{n-1}x+a_{n}=0$ with $a_{i}\in (I^{i})_{t}$ for $i=1,\ldots ,n$. The set of all elements that are $t$-integral over $I$ is called the $t$-integral closure of $I$. This paper investigates the $t$-reductions and $t$-integral closure of ideals. Our objective is to establish satisfactory $t$-analogues of well known results in the literature, on the integral closure of ideals and~its corr\-el\-ation with reductions, namely, Section 2 identifies basic properties of $t$-reductions of ideals and features explicit examples discriminating between the notions of reduction~and $t$-reduction. Section~3 investigates the concept of $t$-integral closure of ideals, including its correlation with $t$-reductions. Section~4 studies the persistence and contraction of $t$-integral closure of id\-eals under ring homomorphisms. Throughout the paper, the main results are illustrated with original examples.

The Special Closure of Polynomial Maps and Global Non-degeneracy

Let \(F:\mathbb C^n\rightarrow \mathbb C^n\) be a polynomial map such that \(F^{-1}(0)\) is finite. We analyze the connections between the multiplicity of F, the Newton polyhedron of F and the set of special monomials with respect to F, which is a notion motivated by the integral closure of ideals in the ring of analytic function germs \((\mathbb C^n,0)\rightarrow \mathbb C\). In particular, we characterize the polynomial maps whose set of special monomials is maximal.

Specialization and integral closure

We prove that the integral closedness of any ideal of height at least two is compatible with specialization by a generic element. This opens the possibility for proofs using induction on the height of an ideal. Also, with additional assumptions, we show that an element is integral over a module if it is integral modulo a generic element of the module. This turns questions about integral closures of modules into problems about integral closures of ideals, by means of a construction known as Bourbaki ideal.

INTEGRAL AND COMPLETE INTEGRAL CLOSURES OF IDEALS IN INTEGRAL DOMAINS

This paper studies the integral and complete integral closures of an ideal in an integral domain. By definition, the integral closure of an ideal I of a domain R is the ideal given by I′ ≔ {x ∈ R | x satisfies an equation of the form xr + a1xr-1 + ⋯ + ar = 0, where ai ∈ Ii for each i ∈ {1, …, r}}, and the complete integral closure of I is the ideal Ī ≔ {x ∈ R | there exists 0 ≠ = c ∈ R such that cxn ∈ In for all n ≥ 1}. An ideal I is said to be integrally closed or complete (respectively, completely integrally closed) if I = I′ (respectively, I = Ī). We investigate the integral and complete integral closures of ideals in many different classes of integral domains and we give a new characterization of almost Dedekind domains via the complete integral closure of ideals.

Weak subintegral closure of ideals

We describe some basic facts about the weak subintegral closure of ideals in both the algebraic and complex-analytic settings. We focus on the analogy between results on the integral closure of ideals and modules and the weak subintegral closure of an ideal. We start by giving a new geometric interpretation of the Reid–Roberts–Singh criterion for when an element is weakly subintegral over a subring. We give new characterizations of the weak subintegral closure of an ideal. We associate with an ideal I of a ring A an ideal I > , which consists of all elements of A such that v ( a ) > v ( I ) , for all Rees valuations v of I . The ideal I > plays an important role in conditions from stratification theory such as Whitney's condition A and Thom's condition A f and is contained in every reduction of I . We close with a valuative criterion for when an element is in the weak subintegral closure of an ideal. For this, we introduce a new closure operation for a pair of modules, which we call relative weak closure. We illustrate the usefulness of our valuative criterion.

Weakly G K -Perfect and Integral Closure of Ideals

Let R be a commutative Noetherian ring, K a nonzero finitely generated suitable R-module, and I an ideal of R. It is shown that if (R, 𝔪) is local, then 𝔪 is G K -perfect if and only if K is a canonical module for R. Furthermore, if I is integrally closed and G K − dim R I < ∞, then K 𝔭 is a canonical R 𝔭-module for every 𝔭 ∈ Ass R R/I whenever K satisfies Serre's condition (S 1) or grade K I > 0. Finally, it is shown that if CM − dim R I < ∞, then R 𝔭 is Cohen–Macaulay for every 𝔭 ∈ Ass R R/I.

Resolution of curve and surface singularities in characteristic zero

Preface. Note to the Reader. Terminology. I: Valuation Theory. 1. Marot Rings. 2. Manis Valuation Rings. 3. Valuation Rings and Valuations. 4. The Approximate Theorem for Independent Valuations. 5. Extensions of Valuations. 6. Extending Valuations to Algebraic Overfields. 7. Extensions of Discrete Valuations. 8. Ramification Theory of Valuations. 9. Extending Valuations to Non-Algebraic Overfields. 10. Valuations of Algebraic Function Fields. 11. Valuations Dominating a Local Domain. II: One-Dimensional Semilocal Cohen-Macaulay Rings. 1. Transversal Elements. 2. Integral Closure of One-Dimensional Semilocal Cohen-Macaulay Rings. 3. One-Dimensional Analytically Unramified and Analytically Irreducible CM-Rings. 4. Blowing up Ideals. 5. Infinitely Near Rings. III: Differential Modules and Ramification. 1. Introduction. 2. Norms and Traces. 3. Formally Unramified and Ramified Extensions. 4. Unramified Extensions and Discriminants. 5. Ramification for Quasilocal Rings. 6. Integral Closure and Completion. IV: Formal and Convergent Power Series Rings. 1. Formal Power Series Rings. 2. Convergent Power Series Rings. 3. Weierstrass Preparation Theorem. 4. The Category of Formal and Analytic Algebras. 5. Extensions of Formal and Analytic Algebras. V: Quasiordinary Singularities. 1. Fractionary Power Series. 2. The Jung-Abhyankar Theorem: Formal Case. 3. The Jung-Abhyankar Theorem: Analytic Case. 4. Quasiordinary Power Series.5. A Generalized Newton Algorithm. 6. Strictly Generated Semigroups. VI: The Singularity Zq = XYp. 1. Hirzebruch-Jung Singularities. 2. Semigroups and Semigroup Rings. 3. Continued Factions. 4. Two-Dimensional Cones. 5. Resolution of Singularities. VII: Two-Dimensional Regular Local Rings. 1. Ideal Transform. 2. Quadratic Transforms and Ideal Transforms. 3. Complete Ideals. 4. Factorization of Complete Ideals. 5. The Predecessors of a Simple Ideal. 6. Uniformization. 7. Resolution of Surface Singularities II: Blowing up and Normalizing. Appendices. A: Results from Classical Algebraic Geometry. 1. Generalities. 2. Affine and Finite Morphisms. 3. Products. 4. Proper Morphisms. 5. Algebraic Cones and Projective Varieties. 6. Regular and Singular points. 7. Normalization of a Variety. 8. Desingularization of a Variety. 9. Dimension of Fibres. 10. Quasifinite Morphisms and Ramification. 11. Divisors. 12. Some Results on Projections. 13. Blowing up. 14. Blowing up: the Local Rings. B: Miscellaneous Results. 1. Ordered Abelian Groups. 2. Localization. 3. Integral Extensions. 4. Some Results on Graded Rings and Modules. 5. Properties of the Rees Ring. 6. Integral Closure of Ideals. 7. Decomposition Group and Inertia Group. 8. Decomposable Rings. 9.

Łojasiewicz exponents, the integral closure of ideals and Newton polyhedra

We give an upper estimate for the Łojasiewicz exponent ℓ(J,I) of an ideal J⊆A(Kn) with respect to another ideal I in the ring A(Kn) of germs analytic functions f : (Kn,0)→K, where K=C or R, using Newton polyhedrons. In particular, we give a method to estimate the Łojasiewicz exponent α0(f) of a germ f∈A(Kn) that can be applied when f is Newton degenerate with respect to its Newton polyhedron.

The Integral Closure of Ideals in ℂ{x, y}

We give some techniques to determine the ideal K I generated by the monomials x k 1 y k 2 belonging to the integral closure Ī of an ideal I ⊆ ℂ{x, y}. We also give a sufficient condition for a weighted homogeneous ideal I ⊆ ℂ{x, y} to satisfy the relation Ī = I + K I .

LINEAR GROWTH OF PRIMARY DECOMPOSITIONS OF MODULES AND INTEGRAL CLOSURES

Let be a commutative Noetherian ring with identity. I.Swanson proved in[10] that every ideal in every commutative Noetherian ring has linear growth of primary decompositions. Later, in[8], R.Y. Sharp generalized this result to finitely generated modules over . In the first section we present another (may be simpler) proof of this generalized result. Sharp also proved in[7] that every proper ideal in has linear growth of primary decompositions for integral closures of ideals. We extend, in the last section, this result to finitely generated modules over .

F-rational rings and the integral closures of ideals

This is a preprint of an article published in Michigan Mathematical Journal, vol 49 (2001), pp. 3-11.

Derivations and the integral closure of ideals

Let (R,m) be a complete local domain containing the rationals. Then there exists an integer I such that for any ideal I C R, if f E m, f In, then there exists a derivation 6 of R with S(f) 5 In+l. This note grew out of a question raised by C. Huneke some years ago: Suppose (R, m) is a complete local domain with perfect residue class field k which it contains. Call an element x E R derivationally constant for R/k if 6(x) = 0 for all k-derivations 6 E Derk(R), and denote by C(R/k) the subring of derivationally constant elements of R/k. In this context Craig Huneke asked: (1) If I C R is an ideal, does there exist a constant I = l(R, I) E N with the following property: If x E R with 6(x) E In+l for all 6 E Derk(R), then there exists a c E C(R/k) with x c E In? (2) Is it possible to bound I in (1) in some uniform way? Here (2) is meant to cover the situation arising via the techniques of tight closure and reduction mod p: If I arises via reduction mod p from an ideal 3 given in an algebra 91 containing the integers, then Huneke asks for a bound I = 1(3) working for the reductions modulo all but finitely many primes p. Originally this question arose in the study of rational and F-rational singularities and the relations between them (cf. [F]), and interest in it has been revived by recent attempts to prove Kodaira vanishing with tight closure techniques (cf. [HS]). In [FHH] a positive answer to (1) was given in case char(k) = 0 and I C R is an m-primary and one-fibred ideal. Some partial results in positive characteristics are available, for instance in the graded case ([F]) and for regular local rings. In this note we will prove the following generalization of [FHH], (1.6): Theorem. Let k be a field, char(k) = 0 and let (R, m)/k be a local domain such that the universally finite derivation dR/k : R -/k exists. Then there exists a constant I = I(R) with the following property: If I C R is an ideal and r E m is an element with 6(r) E In+l for all 6 E Derk(R), then r E In. This completely answers (1), and it also gives a uniform bound (as asked for in (2)), depending on the ring only. A first attempt to solve this problem would be Received by the editors November 20, 1997 and, in revised form, February 24, 1998. 1991 Mathematics Subject Classification. Primary 13N05, 13J10.

Asymptotic behaviour of integral closures of ideals

Asymptotic behaviour of integral closures of ideals

On the integral closure of ideals

Among the several types of closures of an idealI that have been defined and studied in the past decades, the integral closureĪ has a central place being one of the earliest and most relevant. Despite this role, it is often a difficult challenge to describe it concretely once the generators ofI are known. Our aim in this note is to show that in a broad class of ideals their radicals play a fundamental role in testing for integral closedness, and in caseI ≠Ī, ✓I is still helpful in finding some fresh new elements inĪ/I. Among the classes of ideals under consideration are: complete intersection ideals of codimension two, generic complete intersection ideals, and generically Gorenstein ideals.

Symbolic squares in regular local rings

In this paper we prove for several classes of ideals in regular local rings of equicharacteristic 0 that the symbolic square of the ideal contains no minimal generator of the ideal. Our techniques come from residual intersections, integral closures of ideals, and differentials.

On the integral closure of ideals

On the integral closure of ideals

Integral Closure of Ideals in Excellent Local Rings

In [1] Briancon and Skoda proved, using analytic methods, that if I is an ideal in the convergent power series ring C{x1, . . . , xn} then In, the integral closure of I, is contained in I. Extensive work has been done in the direction of proving “Briancon-Skoda type theorems”, that is, statements about I t being contained in (I t−k)#, where k is a constant independent of t, and # is a closure operation on ideals (cf. [5], [6], [8], [11]). In this paper we study the following related problem: given an ideal I of a Noetherian local ring (R, m), find a “linear” integer-valued function f(n) such that I + mn ⊆ I +m for all n or for all sufficiently large n. An element x of R is said to be in the integral closure J of an ideal J if it satisfies a relation of the form x+α1x n−1+α2xn−2+. . .+αn = 0, with αt ∈ J t for all t. We first observe that if (R, m) is a noetherian local ring which is complete in the m-adic topology, then there exists an integer-valued function f(n), with limn→∞ f(n) = ∞, such that I + mn ⊆ I + m. To see this, we use the fact that I = ∩ V φ V −1(IV ), where the intersection is over all discrete valuation domains V which are R algebras via φ V and whose maximal ideal contracts to m. With this it is easily shown that ∩ n I + mn = I. By Chevalley’s theorem ([12, p. 270]) (for R/I and the descending sequence of ideals {I + mn / I}n) then I + mn ⊆ I + m for some function f(n) such that limn→∞ f(n) = ∞. Chevalley’s theorem does not help in determining the order of growth of f(n), and, in fact, it cannot because it takes into account only the topology determined by a given descending sequence of ideals. By also considering the algebraic properties of the sequence we prove a stronger statement:

The integral closure of ideals and Whitney equisingularity of germs of hypersurfaces

The integral closure of ideals and Whitney equisingularity of germs of hypersurfaces

Integral closures of ideals in the Rees ring

Integral closures of ideals in the Rees ring