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.

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).

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 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 An Exact Sequence Associated with a Generalized Crossed Product

1973 ◽  
Vol 49 ◽  
pp. 21-51 ◽  
Author(s):  
Yôichi Miyashita
Keyword(s):  
Exact Sequence ◽  
Crossed Product ◽  
Group Homomorphism ◽  
Ring Extension ◽  
Exact Sequences

The purpose of this paper is to generalize the seven terms exact sequence given by Chase, Harrison and Rosenberg [8]. Our work was motivated by Kanzaki [16] and, of course, [8], [9]. The main theorem holds for any generalized crossed product, which is a more general one than that in Kanzaki [16]. In §1, we define a group P(A/B) for any ring extension A/B, and prove some preliminary exact sequences. In §2, we fix a group homomorphism J from a group G to the group of all invertible two-sided B-submodules of A.

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.

scholarly journals Group cohomology with coefficients in a crossed module

2010 ◽  
Vol 10 (2) ◽  
pp. 359-404 ◽  
Author(s):  
Behrang Noohi
Keyword(s):  
Exact Sequence ◽  
Short Exact Sequence ◽  
Geometric Approach ◽  
Group Cohomology ◽  
Theoretic Approach ◽  
Crossed Module ◽  
Explicit Approach ◽  
Crossed Modules ◽  
Cohomology Sequence

AbstractWe compare three different ways of defining group cohomology with coefficients in a crossed module: (1) explicit approach via cocycles; (2) geometric approach via gerbes; (3) group theoretic approach via butterflies. We discuss the case where the crossed module is braided and the case where the braiding is symmetric. We prove the functoriality of the cohomologies with respect to weak morphisms of crossed modules and also prove the ‘long’ exact cohomology sequence associated to a short exact sequence of crossed modules and weak morphisms.

Cellular covers of ℵ1-free abelian groups

2015 ◽  
Vol 14 (10) ◽  
pp. 1550139 ◽  
Author(s):  
José L. Rodríguez ◽  
Lutz Strüngmann
Keyword(s):  
Abelian Group ◽  
Exact Sequence ◽  
Free Abelian Group ◽  
Abelian Groups ◽  
Countable Rank ◽  
Free Abelian Groups ◽  
Rank 2 ◽  
Exact Sequences ◽  
Torsion Free ◽  
Cellular Cover

In this paper, we first show that for every natural number n and every countable reduced cotorsion-free group K there is a short exact sequence [Formula: see text] such that the map G → H is a cellular cover over H and the rank of H is exactly n. In particular, the free abelian group of infinite countable rank is the kernel of a cellular exact sequence of co-rank 2 which answers an open problem from Rodríguez–Strüngmann [J. L. Rodríguez and L. Strüngmann, Mediterr. J. Math.6 (2010) 139–150]. Moreover, we give a new method to construct cellular exact sequences with prescribed torsion free kernels and cokernels. In particular we apply this method to the class of ℵ1-free abelian groups in order to complement results from the cited work and Göbel–Rodríguez–Strüngmann [R. Göbel, J. L. Rodríguez and L. Strüngmann, Fund. Math.217 (2012) 211–231].

Gelfand-Kirillov Dimension is Exact for Noetherian PI Algebras

1984 ◽  
Vol 27 (2) ◽  
pp. 247-250 ◽  
Author(s):  
T. H. Lenagan
Keyword(s):  
Exact Sequence ◽  
Short Exact Sequence ◽  
Finitely Generated ◽  
Finitely Generated Modules ◽  
Kirillov Dimension

AbstractIf O → A → C → B → O is a short exact sequence of finitely generated modules over a Noetherian Pi-algebra then we show that GK(C) = max{GK(A), GK(B)}.

The Nature of Short Exact Sequence

2020 ◽  
Vol 10 (08) ◽  
pp. 719-725
Author(s):  
宏涛 范
Keyword(s):  
Exact Sequence ◽  
Short Exact Sequence

scholarly journals Equivariant splitting of the Hodge–de Rham exact sequence

2021 ◽  
Author(s):  
Jędrzej Garnek
Keyword(s):  
Finite Group ◽  
Exact Sequence ◽  
Short Exact Sequence ◽  
Algebraic Curve ◽  
Hyperelliptic Curves ◽  
Witt Vectors ◽  
De Rham ◽  
A Finite Group

AbstractLet X be an algebraic curve with a faithful action of a finite group G over a field k. We show that if the Hodge–de Rham short exact sequence of X splits G-equivariantly then the action of G on X is weakly ramified. In particular, this generalizes the result of Köck and Tait for hyperelliptic curves. We discuss also converse statements and tie this problem to lifting coverings of curves to the ring of Witt vectors of length 2.

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