Locally quasi-unmixed Noetherian rings and ideals of the principal class

Ratliff, Louis J.

doi:10.2140/pjm.1974.52.185

Cited by 52 publications

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References 12 publications

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“…Using this last result, McAdam proved, in [3,Theorem 3], the very interesting result that if / C P are ideals in a Noetherian domain R that satisfies the altitude formula and if P is prime, then P is a prime divisor of the integral closures of all large powers of /" if and only if (height P) equals the analytic spread of IRP. Our main results, Theorems 1 and 2, were suggested by McAdam's result and the related result [6,Theorem 2.29], and they characterize Noetherian domains that satisfy the altitude formula in terms of these asymptotic prime divisors and analytic spreads. Then, after proving several corollaries of these theorems, we close by showing certain additional rings (besides those in [3,Theorem 6]) have the property that a prime ideal is a prime divisor of all large powers of an ideal / if and only if it is a prime divisor of the integral closures of all large powers of /.…”

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confidence: 84%

Note on Asymptotic Prime Divisors, Analytic Spreads and the Altitude Formula

Ratliff¹

1981

Proceedings of the American Mathematical Society

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Abstract.The main result gives several characterizations of Noethenan domains that satisfy the altitude formula in terms of analytic spreads and asymptotic prime divisors.1. Introduction. In [7], I proved several results concerning the prime divisors ( = associated primes) of /" and the integral closure of /", where / is an ideal in a Noetherian ring R and n is large. These results were considerably sharpened in [1] and [2]. In particular, in [1], Brodmann showed that all large powers of /" have the same prime divisors, and in [7, Theorem 2.5], I showed that the integral closures of all large powers of /" have the same prime divisors. Using this last result, McAdam proved,in [3, Theorem 3], the very interesting result that if / C P are ideals in a Noetherian domain R that satisfies the altitude formula and if P is prime, then P is a prime divisor of the integral closures of all large powers of /" if and only if (height P) equals the analytic spread of IRP. Our main results, Theorems 1 and 2, were suggested by McAdam's result and the related result [6, Theorem 2.29], and they characterize Noetherian domains that satisfy the altitude formula in terms of these asymptotic prime divisors and analytic spreads. Then, after proving several corollaries of these theorems, we close by showing certain additional rings (besides those in [3, Theorem 6]) have the property that a prime ideal is a prime divisor of all large powers of an ideal / if and only if it is a prime divisor of the integral

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confidence: 84%

Note on Asymptotic Prime Divisors, Analytic Spreads and the Altitude Formula

Ratliff¹

1981

Proceedings of the American Mathematical Society

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“…It follows that, for all positive integers n, u n R = {u n R q ∩ R | q ∈ Ass(R/(uR))} (by [9, (12.6)]) and that each u n R q ∩ R is integrally closed, so u n R = (u n R) a , by [11,Lemma 4]. Therefore I n = u n R ∩ R = (u n R) a ∩ R = I n a (by [12,Lemma 2.5]) for all positive integers n, so it follows that I is a normal ideal. 2…”

Section: Examples Of Ideals With Some Rees Integer Equal To Onementioning

confidence: 96%

Projectively full ideals in Noetherian rings (II)

et al. 2006

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Let R be a Noetherian commutative ring with unit 1 = 0, and let I be a regular proper ideal of R. The set P(I ) of integrally closed ideals projectively equivalent to I is linearly ordered by inclusion and discrete. There is naturally associated to I and to P(I ) a numerical semigroup S(I ); we have S(I ) = N if and only if every element of P(I ) is the integral closure of a power of the largest element K of P(I ). If this holds, the ideal K and the set P(I ) are said to be projectively full. A special case of the main result in this paper shows that if R contains the rational number field Q, then there exists a finite free integral extension ring A of R such that P(I A) is projectively full. If R is an integral domain, then the integral extension A has the property that P((I A + z * )/z * ) is projectively full for all minimal prime ideals z * in A. Therefore in the case where R is an integral domain there exists a finite integral extension domain B = A/z * of R such that P(I B) is projectively full.

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“…p is a regular local ring, p h R ′ p is integrally closed, and as R ′ is finitely generated over a locally formally equidimensional (regular) ring, R ′ q is locally formally equidimensional for every prime ideal q containing p. By a theorem of Ratliff, from [14], since p is generated by a regular sequence, the integral closure of p h R ′ q has no embedded prime ideals. It follows that the integral closure of…”

Section: Integral Closures Of (General) Monomial Idealsmentioning

confidence: 99%

Adjoints of ideals

Hübl¹,

Swanson²

2008

Michigan Math. J.

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We characterize ideals whose adjoints are determined by their Rees valuations. We generalize the notion of a regular system of parameters, and prove that for ideals generated by monomials in such elements, the integral closure and adjoints are generated by monomials. We prove that the adjoints of such ideals and of all ideals in twodimensional regular local rings are determined by their Rees valuations. We prove special cases of subadditivity of adjoints.Adjoint ideals and multiplier ideals have recently emerged as a fundamental tool in commutative algebra and algebraic geometry. In characteristic 0 they may be defined using resolution of singularities. In all characteristics, even mixed, Lipman gave the following definition:Definition 0.1: Let R be a regular domain, I an ideal in R. The adjoint adj I of I is defined as follows:where the intersection varies over all valuations v on the field of fractions K of R that are non-negative on R and for which the corresponding valuation ring R v is a localization of a finitely generated R-algebra. The symbol J R v /R denotes the Jacobian ideal of R v over R.By the assumption on v, each valuation in the definition of adj I is Noetherian. Many valuations v have the same valuation ring R v ; any two such valuations are positive real multiples of each other, and are called equivalent. In the definition of adj I above, one need only use one v from each equivalence class. In the sequel, we will always choose normalized valuations, that is, the integer-valued valuation v such that for all r ∈ R, v(r) equals that non-negative integer n which satisfies that rR v equals the nth power of the maximal ideal of R v .Lipman proved that for any ideal I in R and any x ∈ R, adj(xI) = x adj(I). In particular, adj(xR) = (x).A crucial and very powerful property is the subadditivity of adjoints: adj(IJ) ⊆ adj(I) adj(J). This was proved in characteristic zero by Demailly, Ein and Lazarsfeld [2], and is unknown in general. We prove it for generalized monomial ideals in Section 4, and for ideals in two-dimensional regular domains in Section 5. The case of subadditivity of adjoints for ordinary monomial ideals can be deduced from Howald's work [3] using toric resolutions, and the two-dimensional case has been proved by Takagi and Watanabe [17] using multiplier ideals. The case for generalized monomial ideals proved here is new.

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Locally quasi-unmixed Noetherian rings and ideals of the principal class

Cited by 52 publications

References 12 publications

Note on Asymptotic Prime Divisors, Analytic Spreads and the Altitude Formula

Note on Asymptotic Prime Divisors, Analytic Spreads and the Altitude Formula

Projectively full ideals in Noetherian rings (II)

Adjoints of ideals

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