Class groups and Brauer groups

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].

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Brauer groups and class groups for Krull domains: a K-theoretic approach

Journal of Pure and Applied Algebra ◽

10.1016/0022-4049(84)90007-0 ◽

1984 ◽

Vol 33 (2) ◽

pp. 219-224 ◽

Cited By ~ 6

Author(s):

A. Verschoren

Keyword(s):

Theoretic Approach ◽

Brauer Groups ◽

Class Groups ◽

Krull Domains

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

Canadian Journal of Mathematics ◽

10.4153/cjm-1982-072-4 ◽

1982 ◽

Vol 34 (4) ◽

pp. 996-1010 ◽

Cited By ~ 9

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].

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The Nielsen-Thurston Classification

10.23943/princeton/9780691147949.003.0014 ◽

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.

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

Journal für die reine und angewandte Mathematik (Crelles Journal) ◽

10.1515/crelle-2020-0034 ◽

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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Social Class and Code Elaboration in Oral Communication

Journal of Speech and Hearing Research ◽

10.1044/jshr.1402.421 ◽

1971 ◽

Vol 14 (2) ◽

pp. 421-427 ◽

Cited By ~ 5

Author(s):

Millicent E. Poole ◽

T. W. Field

Keyword(s):

Social Class ◽

Middle Class ◽

Working Class ◽

Oral Communication ◽

Class Groups ◽

Parental Occupation ◽

The Social ◽

Linguistic Coding

The Bernstein thesis of elaborated and restricted coding orientation in oral communication was explored at an Australian tertiary institute. A working-class/middle-class dichotomy was established on the basis of parental occupation and education, and differences in overall coding orientation were found to be associated with social class. This study differed from others in the area in that the social class groups were contrasted in the totality of their coding orientation on the elaborated/restricted continuum, rather than on discrete indices of linguistic coding.

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