Brauer groups and class groups for a Krull domain

1982 ◽  
pp. 66-90 ◽  
Author(s):  
Morris Orzech
Keyword(s):  
Brauer Groups ◽  
Class Groups ◽  
Krull Domain

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.

scholarly journals Brauer groups and Néron class groups

2020 ◽  
Vol 16 (10) ◽  
pp. 2275-2292
Author(s):  
Cristian D. González-Avilés
Keyword(s):  
Exact Sequence ◽  
Abelian Variety ◽  
Duality Theorem ◽  
Class Group ◽  
Global Field ◽  
Brauer Group ◽  
Jacobian Variety ◽  
Brauer Groups ◽  
Class Groups ◽  
Finite Set

Let [Formula: see text] be a global field and let [Formula: see text] be a finite set of primes of [Formula: see text] containing the Archimedean primes. We generalize the duality theorem for the Néron [Formula: see text]-class group of an abelian variety [Formula: see text] over [Formula: see text] established previously by removing the requirement that the Tate–Shafarevich group of [Formula: see text] be finite. We also derive an exact sequence that relates the indicated group associated to the Jacobian variety of a proper, smooth and geometrically connected curve [Formula: see text] over [Formula: see text] to a certain finite subquotient of the Brauer group of [Formula: see text].

Class groups and Brauer groups

1979 ◽  
Vol 34 (1-2) ◽  
pp. 97-105 ◽  
Author(s):  
Jack Sonn
Keyword(s):  
Brauer Groups ◽  
Class Groups

Brauer Groups, Class Groups and Maximal Orders for a Krull Scheme

1982 ◽  
Vol 34 (4) ◽  
pp. 996-1010 ◽  
Author(s):  
Heisook Lee ◽  
Morris Orzech
Keyword(s):  
Exact Sequence ◽  
Monoidal Categories ◽  
Brauer Groups ◽  
Class Groups ◽  
Maximal Orders ◽  
Image Position ◽  
Exact Sequences ◽  
Krull Domains

In a previous paper [13] one of us considered Brauer groups Br(C) and class groups Cl(C) attached to certain monoidal categories C of divisorial R-lattices. That paper showed that the groups arising for a suitable pair of categories C1 ⊆ C2 could be related by a tidy exact sequenceIt was shown that this exact sequence specializes to a number of exact sequences which had formerly been handled separately. At the same time the conventional setting of noetherian normal domains was replaced by that of Krull domains, thus generalizing previous results while also simplifying the proofs. This work was carried out in an affine setting, and one aim of the present paper is to carry these results over to Krull schemes. This will enable us to recover the non-affine version of an exact sequence obtained by Auslander [1, p. 261], as well as to introduce a new, non-affine version of a different sequence derived by the same author [2, Theorem 1].

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.

Involutive Brauer groups of a Krull domain

1995 ◽  
Vol 23 (14) ◽  
pp. 5269-5295
Author(s):  
M. V. Reyes Sanchez ◽  
A. Verschoren
Keyword(s):  
Brauer Groups ◽  
Krull Domain

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.

The Nielsen-Thurston Classification

2017 ◽  
Author(s):  
Benson Farb ◽  
Dan Margalit
Keyword(s):  
Hyperbolic Geometry ◽  
Mapping Class Groups ◽  
Classification Theorem ◽  
Mapping Class ◽  
Fundamental Result ◽  
Class Groups ◽  
Mapping Torus ◽  
Higher Genus ◽  
Manifold Theory

This chapter explains and proves the Nielsen–Thurston classification of elements of Mod(S), one of the central theorems in the study of mapping class groups. It first considers the classification of elements for the torus of Mod(T² before discussing higher-genus analogues for each of the three types of elements of Mod(T². It then states the Nielsen–Thurston classification theorem in various forms, as well as a connection to 3-manifold theory, along with Thurston's geometric classification of mapping torus. The rest of the chapter is devoted to Bers' proof of the Nielsen–Thurston classification. The collar lemma is highlighted as a new ingredient, as it is also a fundamental result in the hyperbolic geometry of surfaces.

Pointwise bound for ℓ-torsion in class groups: Elementary abelian extensions

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Jiuya Wang
Keyword(s):  
Galois Group ◽  
Prime Number ◽  
Finite Groups ◽  
Upper Bound ◽  
Abelian Groups ◽  
Class Groups ◽  
Abelian Extensions ◽  
Pointwise Bound ◽  
First Case

AbstractElementary abelian groups are finite groups in the form of {A=(\mathbb{Z}/p\mathbb{Z})^{r}} for a prime number p. For every integer {\ell>1} and {r>1}, we prove a non-trivial upper bound on the {\ell}-torsion in class groups of every A-extension. Our results are pointwise and unconditional. This establishes the first case where for some Galois group G, the {\ell}-torsion in class groups are bounded non-trivially for every G-extension and every integer {\ell>1}. When r is large enough, the unconditional pointwise bound we obtain also breaks the previously best known bound shown by Ellenberg and Venkatesh under GRH.

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