Exact Sequences Of Abelian Groups Research Articles

A method for obtaining proper classes of short exact sequences of Abelian groups

A method for obtaining proper classes of short exact sequences of Abelian groups

Distality in valued fields and related structures

We investigate distality and existence of distal expansions in valued fields and related structures. In particular, we characterize distality in a large class of ordered abelian groups, provide an Ax-Kochen-Eršov-style characterization for henselian valued fields, and demonstrate that certain expansions of fields, e.g., the differential field of logarithmic-exponential transseries, are distal. As a new tool for analyzing valued fields we employ a relative quantifier elimination for pure short exact sequences of abelian groups.

Characteristic classes as complete obstructions

In the first part of this paper, we propose a uniform interpretation of characteristic classes as obstructions to the reduction of the structure group and to the existence of an equivariant extension of a certain homomorphism defined a priori only on a single fiber of the bundle. Afterwards, we define a family of invariants of principal bundles that detect the number of group reductions that a principal bundle admits. We prove that they fit into a long exact sequence of abelian groups, together with the cohomology of the base space and the cohomology of the classifying space of the structure group.

Realization of long exact sequences of abelian groups

Given a long exact sequence of abelian groups [formula disponible al document original] a short exact sequence of complexes of free abelian groups is constructed whose cohomology long exact sequence is precisely L. In this sense, L is realized . Two techniques which are introduced to reduce or replace lengthy diagram chasing arguments may be of interest to some readers. One is an arithmetic of bicartesian squares; the other is the use of the fact that categories of morphisms of abelian categories are themselves abelian.

Gluing endo-permutation modules

In this paper, I show that if p is an odd prime, and if P is a finite p-group, then there exists an exact sequence of abelian groups 0→ T (P )→ D(P )→ lim ←− 1<Q≤P D ( NP (Q)/Q )→ H1A≥2(P ),Z )(P ) , where D(P ) is the Dade group of P and T (P ) is the subgroup of endo-trivial modules. Here lim ←− 1<Q≤P D ( NP (Q)/Q ) is the group of sequences of compatible elements in the Dade groups D ( NP (Q)/Q ) for non trivial subgroups Q of P . The poset A≥2(P ) is the set of elementary abelian subgroups of rank at least 2 of P , ordered by inclusion. The group H A≥2(P ),Z )(P ) is the subgroup of H1A≥2(P ),Z ) consisting of classes of P -invariant 1-cocycles. A key result to prove that the above sequence is exact is a characterization of elements of 2D(P ) by sequences of integers, indexed by sections (T, S) of P such that T/S ∼= (Z/pZ), fulfilling certain conditions associated to subquotients of P which are either elementary abelian of rank 3, or extraspecial of order p and exponent p. AMS Subject classification : 20C20

Homological Dimensions of Extensions of Artinian Rings by Right Projective Bimodules

ABSTRACT Let Λ be an Artinian ring and let B be a right projective Λ-Λ-bimodule. Recall that a ring Γ is an extension of Λ by B if we have a short exact sequence of Abelian groups , where f is a homomorphism of rings. Recall that a Λ-module M is called liftable to Γ if there exists a Γ-module X such that X/BX≅ M and Tor Γ n (Λ,X) = 0 for all n > 0. The finitistic dimension of Γ is the supremum of the projective dimensions of finitely generated Γ-modules of finite projective dimension. We study Γ-modules of finite projective dimension and we give upper bounds on the finitistic dimension of Γ in terms of the finitistic dimension of liftable Λ-modules.

Prebalanced and Balanced Sequences of Modules over Domains

Balanced and prebalanced exact sequences of abelian groups are intimately tied to the structure of Butler groups and have been used a great deal in this regard. Butler modules as well as torsion-free modules over integral domains receive a lot of attention and we feel it is necessary to explore the notions of balanced and prebalanced in a more general setting. As a consequence of our results, we are able to provide a new characterization of Butler modules over certain integral domains; examples of which include the subrings of algebraic number fields.

A Picard–Brauer exact sequence of categorical groups

A categorical group is a monoidal groupoid in which each object has a tensorial inverse. Two main examples are the Picard categorical group of a monoidal category and the Brauer categorical group of a braided monoidal category with stable coequalizers. After discussing the notions of kernel, cokernel and exact sequence for categorical groups, we show that, given a suitable monoidal functor between two symmetric monoidal categories with stable coequalizers, it is possible to build up a five-term Picard–Brauer exact sequence of categorical groups. The usual Units-Picard and Picard–Brauer exact sequences of abelian groups follow from this exact sequence of categorical groups. We also discuss the direct sum decomposition of the Brauer–Long group.

Algorithmic Methods for Finitely Generated Abelian Groups

We describe algorithmic tools to compute with exact sequences of Abelian groups. Although simple in nature, these are essential for a number of applications such as the determination of the structure of (ZK/m)*for an ideal m of a number field K, of ray class groups of number fields, and of conductors of the corresponding Abelian extensions.

Computing ray class groups, conductors and discriminants

We use the algorithmic computation of exact sequences of Abelian groups to compute the complete structure of ( Z K / m ) ∗ (\mathbb {Z}_{K}/\mathfrak {m})^{*} for an ideal m \mathfrak {m} of a number field K K , as well as ray class groups of number fields, and conductors and discriminants of the corresponding Abelian extensions. As an application we give several number fields with discriminants less than previously known ones.

Generation of proper classes of short exact sequences

The generation of proper classes of short exact sequences of modules by subclasses is considered. The class generated by two proper classes is studied by means of some operations between these classes. These operations are investigated in details for classes of short quasi‐splitting, torsion‐splitting and pure exact sequences of abelian groups.

Realization of long exact sequences of abelian groups

Realization of long exact sequences of abelian groups

ON THE HOMOTOPY AND HOMOLOGY REALIZATION OF EXACT SEQUENCES OF GROUPS

In this article we prove the existence of homology and homotopy realizations of exact sequences of abelian groups. For exact sequences ending in a nonabelian group, we give a sufficient condition for the existence of a realization.Bibliography: 2 items.

Extensions of Endomorphisms from the Higher Centres

If 0 → A → C → B → 0 is an exact sequence of abelian groups, if ƒ is a 2-cocyle for this extension, if α ∈ End A, and if β ∈ End B, then a necessary and sufficient condition that α extend to an endomorphism γ of C which induces β is that (M) αƒ and ƒβ be cohomologous ; see Montgomery (2). We shall extend this result to the case where 1 → A → G → B → 1 is an exact sequence of groups and A is abelian.

Imbedding of abelian categories

As a consequence of this theorem, every object A of (i has elementsnamely, the elements of the image A' of A under the imbedding-and all the usual propositions and constructions performed by means of diagram chasing may be carried out in an arbitrary abelian category precisely as in the category of abelian groups. In fact, if we identify (a with its image Wt' under the imbedding, then a sequence is exact in (t if and only if it is an exact sequence of abelian groups. The kernel, cokernel, image, and coimage of a map f of et are the kernel, cokernel, image, and coimage of f in the category of abelian groups; the map f is an epimorphism, monomorphism, or isomorphism if and only if it has the corresponding property considered as a map of abelian groups. The direct product of finitely many objects of et is their direct product as abelian groups. If A (c, then every subobject [4] of A is a subgroup of A, and the intersection (or sum) of finitely many subobjects of A is their ordinary intersection (or sum); the direct (or inverse) image of a subobject of A by a map of Q is the usual set-theoretic direct (or inverse) image. Moreover, if