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).

Multiplicities in graded rings II: integral equivalence and the Buchsbaum–Rim multiplicity

1996 ◽  
Vol 119 (3) ◽  
pp. 425-445 ◽  
Author(s):  
D. Kirby ◽  
D. Rees
Keyword(s):  
Exact Sequence ◽  
Short Exact Sequence ◽  
Noetherian Ring ◽  
Free Module ◽  
Graded Rings ◽  
Finitely Generated ◽  
Graded Ring ◽  
The Subject ◽  
Integral Equivalence ◽  
Image Position

While this paper is principally a continuation of [5], with as its object the application of sections 6 and 7 of that paper to obtain results related to the Buchsbaum–Rim multiplicity, it also has connections with [8] which are the subject of the first of the four sections. These concern integral equivalence of finitely generated R-modules. where R is an arbitrary noetherian ring. We therefore introduce a finitely generated R-module M and relate to it a short exact sequence (s.e.s.),where F is a free module generated by m elements u1,…, um, and L is generated by elements yj, (j = 1, …, n), of F. We identify the elements u1, …, um with a set of indeterminates X1, …, Xm, and F with the R-module S1 of elements of degree 1 of the graded ring S = R[X1, …, Xm].

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.

Obstructions to Liftings in Commutative Squares

1971 ◽  
Vol 23 (6) ◽  
pp. 977-982 ◽  
Author(s):  
Irwin S. Pressman
Keyword(s):  
Exact Sequence ◽  
Short Exact Sequence ◽  
Abelian Category ◽  
One To One ◽  
Image Position ◽  
Zero Class

A commutative square (1) of morphisms is said to have a lifting if there is a morphism λ: B1 → A2 such that λϕ1 = α and ϕ2λ = β1Let us assume that we are working in a fixed abelian category . Therefore, ϕi will have a kernel “Ki” and a cokernel “Ci” for i = 1, 2. Let k : K1 → K2 and c: C1 → C2 denote the canonical morphisms induced by α and β.We shall construct a short exact sequence (s.e.s.)2using the data of (1). We shall prove that (1) has a lifting if and only if k = 0, c = 0, and (2) represents the zero class in Ext1(C1, K2). Furthermore, if (1) has one lifting, then the liftings will be in one-to-one correspondence with the elements of the set |Hom(G1, K2)|.

Preservation of Rees exact sequences

2019 ◽  
Vol 69 (6) ◽  
pp. 1293-1302
Author(s):  
Morteza Jafari ◽  
Akbar Golchin ◽  
Hossein Mohammadzadeh Saany
Keyword(s):  
Semigroup Forum ◽  
Exact Sequence ◽  
Short Exact Sequence ◽  
Left And Right ◽  
Exact Sequences

Abstract Yuqun Chen and K. P. Shum in [Rees short exact sequence of S-systems, Semigroup Forum 65 (2002), 141–148] introduced Rees short exact sequence of acts and considered conditions under which a Rees short exact sequence of acts is left and right split, respectively. To our knowledge, conditions under which the induced sequences by functors Hom(RLS, –), Hom(–, RLS) and AS ⊗ S– (where R, S are monoids) are exact, are unknown. This article addresses these conditions. Results are different from that of modules.

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].

The projective extensions of a finite group

1981 ◽  
Vol 90 (2) ◽  
pp. 251-257
Author(s):  
P. J. Webb
Keyword(s):  
Finite Group ◽  
Group Ring ◽  
Projective Module ◽  
Integral Group Ring ◽  
Augmentation Ideal ◽  
Integral Group ◽  
Definition Of ◽  
Image Position ◽  
A Finite Group ◽  
Exact Sequences

Let G be a finite group and let g be the augmentation ideal of the integral group ring G. Following Gruenberg(5) we let (g̱) denote the category whose objects are short exact sequences of zG-modules of the form and in which the morphisms are commutative diagramsIn this paper we describe the projective objects in this category. These are the objects which satisfy the usual categorical definition of projectivity, but they may also be characterized as the short exact sequencesin which P is a projective module.

scholarly journals Splittings of link concordance groups

2017 ◽  
Vol 26 (02) ◽  
pp. 1740007
Author(s):  
Taylor Martin ◽  
Carolyn Otto
Keyword(s):  
Exact Sequence ◽  
Short Exact Sequence ◽  
Link Concordance ◽  
Exact Sequences

We establish several results about two short exact sequences involving lower terms of the [Formula: see text]-solvable filtration, [Formula: see text] of the string link concordance group [Formula: see text]. We utilize the Thom–Pontryagin construction to show that the Sato–Levine invariants [Formula: see text] must vanish for 0.5-solvable links. Using this result, we show that the short exact sequence [Formula: see text] does not split for links of two or more components, in contrast to the fact that it splits for knots. Considering lower terms of the filtration [Formula: see text] in the short exact sequence [Formula: see text], we show that while the sequence does not split for [Formula: see text], it does indeed split for [Formula: see text]. This allows us to determine that the quotient [Formula: see text].

scholarly journals The (CO)homology of groups given by presentations in which each defining relator involves at most two types of generators

1992 ◽  
Vol 52 (2) ◽  
pp. 205-218 ◽  
Author(s):  
Stephen J. Pride
Keyword(s):  
Exact Sequence ◽  
Short Exact Sequence ◽  
Augmentation Ideal ◽  
Trivial Group ◽  
Vertex Set ◽  
Set Up ◽  
Image Position

AbstractOur set-up will consist of the following: (i) a graph with vertex set V and edge set E; (ii) for each vertex ∈ V a non-trivial group Gv given by a presentation (xν; rν); (iii) for each edge e = {u, ν} ∈ E a group Ge given by a presentation (xu, xv; re) where re consists of the elements of ru ∪ rv, together with some further words on xu ∪ xv. We let G = (x; r) where x = ∪v∈v xv, r = ∪e∈E re. Ouraim is to try to describe the structure of G in terms of the groups Gv, (v ∈ V), Ge (e ∈ E). Under suitable conditions the natural homomorphisms Gv, → G (ν ∈ V), Ge → Ge (e ε E) are injective; and there is a short exact sequence (where, for any group H, IH is the augmentation ideal). Some (co)homological consequences of these resultsare derived.

scholarly journals THE FIRST U-EXTENSION MODULE AS CLASSES OF SHORT U-EXACT SEQUENCES

2021 ◽  
Vol 10 (4) ◽  
pp. 553
Author(s):  
Yudi Mahatma
Keyword(s):  
Exact Sequence ◽  
Short Exact Sequence ◽  
Projective Resolution ◽  
Equivalence Classes ◽  
Exact Sequences

Inspired by the notions of the U-exact sequence introduced by Davvaz and Parnian-Garamaleky in 1999, and of the chain U-complex introduced by Davvaz and Shabani-Solt in 2002, Mahatma and Muchtadi-Alamsyah in 2017 developed the concept of the U-projective resolution and the U-extension module, which are the generalizations of the concept of the projective resolution and the concept of extension module, respectively. It is already known that every element of a first extension module can be identified as a short exact sequence. To the simple, there is a relation between the first extension module and the short exact sequence. It is proper to expect the relation to be provided in the U-version. In this paper, we aim to construct a one-one correspondence between the first U-extension module and the set consisting of equivalence classes of short U-exact sequence.Keywords: Chain U-complex, U-projective resolution, U-extension module

scholarly journals Relation modules of amalgamated free products and HNN extensions

1989 ◽  
Vol 31 (3) ◽  
pp. 263-270 ◽  
Author(s):  
Torsten Hannebauer
Keyword(s):  
Normal Subgroup ◽  
Exact Sequence ◽  
Short Exact Sequence ◽  
Free Products ◽  
Relation Module ◽  
Amalgamated Free Products ◽  
Hnn Extensions ◽  
Image Position

Let G be a group anda free presentation of G, i.e. a short exact sequence of groups with F free. Conjugation in F induces on = R/R', the abelianized normal subgroup R, the structure of a right G-module (if r∈ R, x∈ F then (r)(xπ) = x-1rxR'). The G-module is called the relation module determined by the presentation (1). For a detailed discussion of this subject we refer to Gruenberg [3].

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