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 Non-abelian exterior products of groups and exact sequences in the homology of groups

1987 ◽  
Vol 29 (1) ◽  
pp. 13-19 ◽  
Author(s):  
G. J. Ellis
Keyword(s):  
Exact Sequence ◽  
Short Exact Sequence ◽  
Type Theorem ◽  
Exterior Product ◽  
Products Of Groups ◽  
Hopf Formula ◽  
Definition Of ◽  
Image Position ◽  
Exact Sequences

Various authors have obtained an eight term exact sequence in homologyfrom a short exact sequence of groups,the term V varying from author to author (see [7] and [2]; see also [5] for the simpler case where N is central in G, and [6] for the case where N is central and N ⊂ [G, G]). The most satisfying version of the sequence is obtained by Brown and Loday [2] (full details of [2] are in [3]) as a corollary to their van Kampen type theorem for squares of spaces: they give the term V as the kernel of a map G ∧ N → N from a “non-abelian exterior product” of G and N to the group N (the definition of G ∧ N, first published in [2], is recalled below). The two short exact sequencesandwhere F is free, together with the fact that H2(F) = 0 and H3(F) = 0, imply isomorphisms..The isomorphism (2) is essentially the description of H2(G) proved algebraically in [11]. As noted in [2], the isomorphism (3) is the analogue for H3(G) of the Hopf formula for H2(G).

Localization and class groups of module categories with exactness defects

1993 ◽  
Vol 113 (2) ◽  
pp. 233-251 ◽  
Author(s):  
D. Holland ◽  
S. M. J. Wilson
Keyword(s):  
Exact Sequence ◽  
Class Group ◽  
Grothendieck Group ◽  
Module Category ◽  
Class Groups ◽  
Definition Of ◽  
Free Modules ◽  
Exact Sequences ◽  
Torsion Free

AbstractWe present a new way of forming a grothendieck group with respect to exact sequences. A ‘defect’ is attached to each non-split sequence and the relation that would normally be derived from a collection of exact sequences is only effective if the (signed) sum of the corresponding defects is zero. The theory of the localization exact sequence and, in particular, of the relative group in this sequence is developed. The (‘locally free’) class group of a module category with exactness defect is defined and an idèlic formula for this is given. The role of torsion and of torsion-free modules is investigated. One aim of the work is to enhance the locally trivial, ‘class group’, invariants obtainable for a module while keeping to a minimum the local obstructions to the definition of such invariants.

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].

Quasi-Splitting Exact Sequence

1971 ◽  
Vol 23 (3) ◽  
pp. 503-506
Author(s):  
Hsiang-Dah Hou
Keyword(s):  
Exact Sequence ◽  
Full Subcategory ◽  
Abelian Category ◽  
Image Position

Let R be a ring with 1 ≠ 0 and α, β, γ R-endomorphisms of R-modules A, B, and C respectively. We shall denote by M(R) the category of R-modules, and by End(R) the category of R-endomorphisms. For objects α and β of End(R) a morphism λ: α → β is an R-homomorphism such that λα = β λ. We shall denote by Idm(R) the full subcategory of End(R) whose objects are idempotents. Idm(R) is an abelian category, ker, coker and im are constructed in the naive way and henceis exact in M(R) if and only ifis exact in Idm(R), where the domains of α,β, and γ are A, B, and C respectively. One sees that End (R) as well as Idm(R) is abelian.

Presentations of the Trefoil Group

1973 ◽  
Vol 16 (4) ◽  
pp. 517-520 ◽  
Author(s):  
M. J. Dunwoody ◽  
A. Pietrowski
Keyword(s):  
Exact Sequence ◽  
Free Group ◽  
Image Position

A presentation of a group G is an exact sequence of groupswhere F is a free group. Let l→S ⊆ F→G→1 be another presentation of G involving the same free group F.

The mod ℭ Suspension Theorem

1969 ◽  
Vol 21 ◽  
pp. 684-701 ◽  
Author(s):  
Benson Samuel Brown
Keyword(s):  
Finite Group ◽  
Exact Sequence ◽  
Abelian Groups ◽  
Finitely Generated ◽  
Finite Abelian Groups ◽  
Cw Complexes ◽  
Image Position ◽  
A Finite Group

Our aim in this paper is to prove the general mod ℭ suspension theorem: Suppose that X and Y are CW-complexes,ℭ is a class offinite abelian groups, and that(i) πi(Y) ∈ℭfor all i < n,(ii) H*(X; Z) is finitely generated,(iii) Hi(X;Z) ∈ℭfor all i > k.Then the suspension homomorphismis a(mod ℭ) monomorphism for 2 ≦ r ≦ 2n – k – 2 (when r= 1, ker E is a finite group of order d, where Zd∈ ℭ and is a (mod ℭ) epimorphism for 2 ≦ r ≦ 2n – k – 2The proof is basically the same as the proof of the regular suspension theorem. It depends essentially on (mod ℭ) versions of the Serre exact sequence and of the Whitehead theorem.

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