Generalization of a result of Hickel, Ito and Izumi about a Diophantine inequality

Beddani, Charef; Spivakovsky, Mark

doi:10.1016/j.jpaa.2014.07.008

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Remark 222

An Equality Between the Number Of Minimal Primes In The Comp1

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2015

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“…The following theorem is a generalization of a result proved in [4] to the case of unibranch local domain. Precisely, we replace the strong condition "R is analytically unramified" of Theorem 1 of [4] by the condition "R is finite over R".…”

Section: Remark 22mentioning

confidence: 75%

“…Precisely, we replace the strong condition "R is analytically unramified" of Theorem 1 of [4] by the condition "R is finite over R". The proof given in this paper is different from that in [4]. Proof.…”

Section: Remark 22mentioning

confidence: 99%

“…As the formal fibers of R are geometrically normal, the formal fibers of the normal domain S are also geometrically normal [4,Lemma 1.7]. Hence S is integrally closed [10, p. 185].…”

Section: An Equality Between the Number Of Minimal Primes In The Compmentioning

confidence: 99%

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Some results on quasi-unmixed local domains

Beddani

Spivakovsky

2015

Journal of Algebra

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Available online xxxx Communicated by Steven Dale Cutkosky MSC: 13B35 13B22 13A18 Keywords: Divisorial valuations Quasi-unmixed local domainsThis paper shows that if (R, m) is a Noetherian unibranch local domain with field of fractions K, then the integral closure S of R in K is analytically irreducible and finite over R if and only if R is analytically irreducible. We also prove the equality between the number of minimal prime ideals in R and the number of maximal ideals in S in the case when R is a Noetherian quasi-unmixed local domain such that S is finite over R and S n has only one minimal prime for all the maximal ideals n in S, where S n is the n-adic completion of S n .

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Section: Remark 22mentioning

confidence: 75%

Section: Remark 22mentioning

confidence: 99%