Adjacent integrally closed ideals in dimension two

Noh, Sunsook

doi:10.1016/0022-4049(93)90051-t

Cited by 15 publications

(17 citation statements)

References 10 publications

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“…Note however that our result says more; and besides being valid in arbitrary two-dimensional regular local rings, it also shows that q β is actually the smallest complete ideal strictly containing ℘ β . (This has been proved previously by Noh [12,Thm. 3.1], at least when the residue field of α is algebraically closed.)…”

Section: Proposition (45)supporting

confidence: 63%

Proximity inequalities for complete ideals in two-dimensional regular local rings

Lipman¹

1994

Commutative Algebra: Syzygies, Multiplicities, and Birational Algebra

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Section: Proposition (45)supporting

confidence: 63%

Proximity inequalities for complete ideals in two-dimensional regular local rings

Lipman¹

1994

Commutative Algebra: Syzygies, Multiplicities, and Birational Algebra

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“…If P is a simple complete ideal, by [10] any power of P other than P has infinitely many adjacent over-ideals. There also exist at least n adjacent over-ideals of J = J 1 · · · J n if n > 1 and J i is not a w j -ideal for j = i, where R(J ) = {w j | j = 1, .…”

Section: Introductionmentioning

confidence: 99%

“…It was proved that the v-predecessor P v is the unique complete adjacent over-ideal of P (cf. [6][7][8], [9,Appendix], [10]). It was further shown that either P v is simple, in which case it equals P t−1 , or it is a product of two simple complete ideals P t−1 P i for some 0 i < t − 1 using proximity relations among prime divisors v j associated to the simple v-ideals P j for 0 j t. However, the v-successor I +1 of P is not adjacent to P from below as it was shown that λ A (I /I +1 ) = 2 [9].…”

Section: Introductionmentioning

confidence: 99%

Adjacent integrally closed ideals in 2-dimensional regular local rings

Noh

Watanabe

2006

Journal of Algebra

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Let (A, m) be a 2-dimensional regular local ring with algebraically closed residue field. Zariski's Unique Factorization Theorem asserts that every integrally closed (complete) m-primary ideal I is uniquely factored into a product of powers of simple complete ideals I = P a 1 1 P a 2 2 · · · P a n n , where P i is a simple complete ideal for a i 1 and n 1. In this paper, we give a new characterization for a simple complete ideal in terms of adjacent complete ideals. We also give a characterization for a complete ideal I to have finitely many adjacent complete m-primary over-ideals. Namely, we show that I is simple if and only if it has a unique adjacent over-ideal and that I = P a 1 1 P a 2 2 · · · P a n n has only finitely many complete adjacent over-ideals if and only if a i = 1 for every i and there are no proximity relations among P i .

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“…If α = α , then ν x − α y k+1 = 2k + 2 since P k+1 is also a ν-ideal and P k = x, y k+1 ⊃ x − α y k+1 , y k+2 are adjacent but x − α y k+1 , y k+2 is not a ν-ideal [9,Lemma 3.3]. Then,…”

Section: B = Even Casementioning

confidence: 99%

“…If S is proximate to T and P ⊃ Q are the simple integrally closed ideals associated to the m(T )-adic and m(S)-adic order valuations respectively, then we also say that Q is proximate to P . In the sequence (1.1), the v-predecessor I n−1 of I is the unique integrally closed ideal adjacent to I from above [6,Theorem 4.11], [9,Theorem 3.1]. Furthermore, I n−1 is the product of simple v-ideals P i s associated to R i s to which R t is proximate, and that there are at most two such R i s to which R t is proximate [6,Theorem 4.11].…”

Section: Introductionmentioning

confidence: 99%

Valuation ideals of order two in 2‐dimensional regular local rings

Noh

2003

Mathematische Nachrichten

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Key words Valuation ideal, value semigroup, order of ideal, rank of valuation, simple valuation ideal MSC (2000) 13A18, 13H05, 13B02, 13B22Let K be the quotient field of a 2-dimensional regular local ring (R, m) and let v be a prime divisor of R, i.e., a valuation of K birationally dominating R which is residually transcendental over R. Zariski showed that: such prime divisor v is uniquely associated to a simple m-primary integrally closed ideal I of R, there are only finitely many simple v-ideals including I, and all the other v-ideals can be uniquely factored into products of simple v-ideals. The number of nonmaximal simple v-ideals is called the rank of v or the rank of I as well. It is also known that such an m-primary ideal I is minimally generated by o(I) + 1 elements, where o(I) denotes the m-adic order of I. Given a simple valuation ideal of order two associated to a prime divisor v of arbitrary rank, in this paper we find minimal generating sets of all the simple v-ideals and the value semigroup v(R) in terms of its rank and the v-value difference of two elements in a regular system of parameters of R. We also obtain unique factorizations of all the composite v-ideals and describe the complete sequence of v-ideals as explicitly as possible.

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Adjacent integrally closed ideals in dimension two

Cited by 15 publications

References 10 publications

Proximity inequalities for complete ideals in two-dimensional regular local rings

Proximity inequalities for complete ideals in two-dimensional regular local rings

Adjacent integrally closed ideals in 2-dimensional regular local rings

Valuation ideals of order two in 2‐dimensional regular local rings

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