scholarly journals Divisor class groups and graded canonical modules of multisection rings

2013 ◽  
Vol 212 ◽  
pp. 139-157 ◽  
Author(s):  
Kazuhiko Kurano
Keyword(s):  
Mild Condition ◽  
Projective Variety ◽  
Class Group ◽  
Class Groups ◽  
Divisor Class Group ◽  
Divisor Class ◽  
Canonical Module ◽  
Krull Domain

AbstractWe describe the divisor class group and the graded canonical module of the multisection ring T (X;D1,…, Ds) for a normal projective variety X and Weil divisors D1,…, Ds on X under a mild condition. In the proof, we use the theory of Krull domain and the equivariant twisted inverse functor.

scholarly journals Divisor class groups and graded canonical modules of multisection rings

2013 ◽  
Vol 212 ◽  
pp. 139-157
Author(s):  
Kazuhiko Kurano
Keyword(s):  
Mild Condition ◽  
Projective Variety ◽  
Class Group ◽  
Class Groups ◽  
Divisor Class Group ◽  
Divisor Class ◽  
Canonical Module ◽  
Krull Domain

AbstractWe describe the divisor class group and the graded canonical module of the multisection ringT(X;D1,…,Ds) for a normal projective varietyXand Weil divisorsD1,…,DsonXunder a mild condition. In the proof, we use the theory of Krull domain and the equivariant twisted inverse functor.

Krull Semigroups and Divisor Class Groups

1981 ◽  
Vol 33 (6) ◽  
pp. 1459-1468 ◽  
Author(s):  
Leo G. Chouinard II
Keyword(s):  
Abelian Group ◽  
Class Group ◽  
Chain Condition ◽  
Divisor Class Group ◽  
Divisor Class ◽  
Cyclic Subgroups ◽  
Krull Domain ◽  
Coefficient Ring ◽  
Ascending Chain Condition

R. Matsuda has shown that a group ring is a Krull domain if and only if the coefficient ring is a Krull domain and the group is a torsion-free abelian group satisfying the ascending chain condition (ace) on cyclic subgroups [6]. D. F. Anderson has used this to obtain a partial determination of when a semigroup ring is a Krull domain, and under certain circumstances to describe the divisor class group of such a ring ([1], [2]). Using some of Anderson's techniques, but taking a different approach, we arrive at a complete answer of a different nature to these questions. We call a semigroup satisfying the major new conditions arising a Krull semigroup, and define its divisor class group.In particular, every abelian group is the divisor class group of such a ring, and it follows that every abelian group is the divisor class group of a quasi-local ring, which seems to be a new result.

The divisor class group of splittable Zariski surfaces

2018 ◽  
Vol 17 (08) ◽  
pp. 1850150 ◽  
Author(s):  
Jeffrey Lang
Keyword(s):  
Class Group ◽  
Closed Field ◽  
Class Groups ◽  
Divisor Class Group ◽  
Divisor Class ◽  
Algebraically Closed Field ◽  
Normal Variety

Let [Formula: see text] be an algebraically closed field of characteristic [Formula: see text] and [Formula: see text]. Let [Formula: see text] be a normal variety defined by the equation [Formula: see text]. If [Formula: see text] is a product of [Formula: see text] linear factors in [Formula: see text], not necessarily homogeneous, where [Formula: see text] equals the degree of [Formula: see text], we say that [Formula: see text] is splittable. In this paper, we calculate the group of Weil divisors of a splittable [Formula: see text] for a generic [Formula: see text] when the characteristic of [Formula: see text] equals [Formula: see text]. In the process, we describe a general approach to studying class groups of splittable [Formula: see text] that we believe should yield results when [Formula: see text].

scholarly journals Some factorization properties of Krull domains with infinite cyclic divisor class group

1994 ◽  
Vol 96 (2) ◽  
pp. 97-112 ◽  
Author(s):  
David F. Anderson ◽  
Scott T. Chapman ◽  
William W. Smith
Keyword(s):  
Class Group ◽  
Divisor Class Group ◽  
Divisor Class ◽  
Krull Domains

scholarly journals The divisor class group of a Quot scheme

2010 ◽  
Vol 3 (0) ◽  
pp. 1-14
Author(s):  
Rafael Hernández ◽  
Daniel Ortega
Keyword(s):  
Class Group ◽  
Divisor Class Group ◽  
Divisor Class ◽  
Quot Scheme

On Central Ω-Krull Rings and their Class Groups

1984 ◽  
Vol 36 (2) ◽  
pp. 206-239 ◽  
Author(s):  
E. Jespers ◽  
P. Wauters
Keyword(s):  
Polynomial Ring ◽  
Class Group ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Semi Group ◽  
Class Groups ◽  
Krull Domain ◽  
Direct Limits ◽  
Commutative Theory ◽  
Necessary And Sufficient

The aim of this note is to study the class group of a central Ω-Krull ring and to determine in some cases whether a twisted (semi) group ring is a central Ω-Krull ring. In [8] we defined an Ω-Krull ring as a generalization of a commutative Krull domain. In the commutative theory, the class group plays an important role. In the second and third section, we generalize some results to the noncommutative case, in particular the relation between the class group of a central Ω-Krull ring and the class group of a localization. Some results are obtained in case the ring is graded. Theorem 3.2 establishes the relation between the class group and the graded class group. In particular, in the P.I. case we obtain that the class group is equal to the graded class group. As a consequence of a result on direct limits of Ω-Krull rings, we are able to derive a necessary and sufficient condition in order that a polynomial ring R[(Xi)i∊I] (I may be infinite) is a central Ω-Krull ring.

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