The Local Class Group of a Krull Domain

Bouvier, Alain

doi:10.4153/cmb-1983-003-1

Cited by 18 publications

(5 citation statements)

References 9 publications

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“…• Let D be a Krull domain and Div(D) its group of divisorial fractional ideals under the v-multiplication (Fossum, 1973, Proposition 3.4). The local class group G(D) of D is the factor group Div(D) modulo the subgroup of invertible fractional ideals (Bouvier and Zafrullah, 1988). By Fossum (1973, Corollary 18.15), a Krull domain has zero local class group if and only if it is locally factorial (hence it is a ⋆-domain).…”

Section: Ideal Theoretic Resultsmentioning

confidence: 99%

Some Applications of André–Quillen Homology to Classes of Arithmetic Rings

Dumitrescu

Ionescu

2008

Communications in Algebra

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Section: Ideal Theoretic Resultsmentioning

confidence: 99%

Some Applications of André–Quillen Homology to Classes of Arithmetic Rings

Dumitrescu

Ionescu

2008

Communications in Algebra

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“…In particular, t-invertibility has a key role for extending the notion of class group from Krull domains to general integral domains (cf. [8], [9], [10] and the survey paper [7]). An interesting chart of a large set of various t-, v-, d-invertibility based characterizations of several classes of integral domains can be found at the end of [4]; some motivations for introducing the w-invertibility and the first properties showing the "good" behaviour of this notion can be found in [47] (cf.…”

Section: Introduction and Background Resultsmentioning

confidence: 99%

Semistar Invertibility on Integral Domains

Fontana

Picozza

2005

Algebra Colloq.

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Abstract. After the introduction in 1994, by Okabe and Matsuda, of the notion of semistar operation, many authors have investigated different aspects of this general and powerful concept. A natural development of the recent work in this area leads to investigate the concept of invertibility in the semistar setting. In this paper, we will show the existence of a "theoretical obstruction" for extending many results, proved for star-invertibility, to the semistar case. For this reason, we will introduce two distinct notions of invertibility in the semistar setting (called ⋆-invertibility and quasi-⋆-invertibility), we will discuss the motivations of these "two levels" of invertibility and we will extend, accordingly, many classical results proved for the d-, v-, t-and w-invertibility. Among the main properties proved here, we mention the following: (a) several characterizations of ⋆-invertibility and quasi-⋆-invertibility and necessary and sufficient conditions for the equivalence of these two notions; (b) the relations between the ⋆-invertibility (or quasi-⋆-invertibility) and the invertibility (or quasi-invertibility) with respect to the semistar operation of finite type (denoted by ⋆ f ) and to the stable semistar operation of finite type (denoted by ⋆), canonically associated to ⋆; (c) a characterization of the H(⋆)-domains in terms of semistar-invertibility (note that the H(⋆)-domains generalize, in the semistar setting, the H-domains introduced by Glaz and Vasconcelos); (d) for a semistar operation of finite type a nonzero finitely generated (fractional) ideal I is ⋆-invertible (or, equivalently, quasi-⋆-invertible, in the stable semistar case) if and only if its extension to the Nagata semistar ring I Na(D, ⋆) is an invertible ideal of Na(D, ⋆).

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“…For our purposes the divisor class group being torsion means that for each proper divisorial ideal I there is some positive integer n such that (I n ) v is principal. The other concept to know is the local class group G(D) = Cl(D)/P ic(D) of a Krull domain D, introduced and studied by Bouvier in [10]. Now G(D) being torsion is equivalent to (I n ) v being invertible, for some integer n, for each proper divisorial ideal I.…”

Section: A Universal Restriction With Conditionsmentioning

confidence: 99%

Domains whose ideals meet a universal restriction

Zafrullah¹

2020

Preprint

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Let D be an integral domain, S(D) = I(D) (It(D)) the set of proper nonzero ideals (proper t-ideals) of D, M ax(D) (t-M ax(D) the set of maximal (t-) ideals of D, and let P be a predicate on S(D) with nonempty truth set Π S(D) ⊆ S(D), where P can be: "-is invertible" or "-is divisorial" etc.. We say S(D) meets P (S(D)We show that if S(D) ⊳ P, we have no control over dim D. We also show that I(D) ⊳ P does not imply I(R) ⊳ P, while It(D) ⊳ P implies It(R) ⊳ P, for most choices of P, when R = D[X] and have examples to show that generally S(D) ⊳ P does not extend to rings of fractions. We study restrictions that may control the dimension of D when S(D) ⊳ P. We also say S(D) ⊳ P with a twist (S(D) ⊳ t P ) if ∀s ∈ S(D) ∃π ∈ Π D (P )(s n ⊆ π for some n ∈ N ) and study S(D) ⊳ t P, along the same lines as S(D) ⊳ P and provide examples.

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The Local Class Group of a Krull Domain

Cited by 18 publications

References 9 publications

Some Applications of André–Quillen Homology to Classes of Arithmetic Rings

Some Applications of André–Quillen Homology to Classes of Arithmetic Rings

Semistar Invertibility on Integral Domains

Domains whose ideals meet a universal restriction

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