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)|.

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.

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].

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 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].

scholarly journals A note on free products with a normal amalgamation

1968 ◽  
Vol 8 (3) ◽  
pp. 631-637 ◽  
Author(s):  
R. A. Bryce
Keyword(s):  
Free Products ◽  
One To One ◽  
Image Position ◽  
Subgroup Theorem ◽  
Infinite Cycles

It is a consequence of the Kurosh subgroup theorem for free products that if a group has two decompositions where each Ai and each Bj is indecomposable, then I and J can be placed in one-to-one correspondence so that corresponding groups if not conjugate are infinite cycles. We prove here a corresponding result for free products with a normal amalgamation.

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.

Behavior of Coefficients of Inverses of α-Spiral Functions

1986 ◽  
Vol 38 (6) ◽  
pp. 1329-1337 ◽  
Author(s):  
Richard J. Libera ◽  
Eligiusz J. Złotkiewicz
Keyword(s):  
Unit Disk ◽  
Series Expansion ◽  
Open Unit ◽  
Open Unit Disk ◽  
Maclaurin Series ◽  
Koebe Function ◽  
One To One ◽  
Image Position

If f(z) is univalent (regular and one-to-one) in the open unit disk Δ, Δ = {z ∊ C:│z│ < 1}, and has a Maclaurin series expansion of the form(1.1)then, as de Branges has shown, │ak│ = k, for k = 2, 3, … and the Koebe function.(1.1)serves to show that these bounds are the best ones possible (see [3]). The functions defined above are generally said to constitute the class .

scholarly journals Continuation homomorphism in Rabinowitz Floer homology for symplectic deformations

2011 ◽  
Vol 151 (3) ◽  
pp. 471-502 ◽  
Author(s):  
YOUNGJIN BAE ◽  
URS FRAUENFELDER
Keyword(s):  
Exact Sequence ◽  
Isoperimetric Inequality ◽  
Short Exact Sequence ◽  
Loop Space ◽  
Floer Homology ◽  
Critical Value ◽  
Alternative Proof ◽  
Rabinowitz Floer Homology ◽  
Space Homology ◽  
Loop Space Homology

AbstractWill J. Merry computed Rabinowitz Floer homology above Mañé's critical value in terms of loop space homology in [14] by establishing an Abbondandolo–Schwarz short exact sequence. The purpose of this paper is to provide an alternative proof of Merry's result. We construct a continuation homomorphism for symplectic deformations which enables us to reduce the computation to the untwisted case. Our construction takes advantage of a special version of the isoperimetric inequality which above Mañé's critical value holds true.

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