Relative Cohomology

1957 ◽  
Vol 9 ◽  
pp. 19-34 ◽  
Author(s):  
D. G. Higman
Keyword(s):  
Cohomology Theory ◽  
Cohomology Groups ◽  
Dual Pairs ◽  
Relative Cohomology

It is our purpose in this paper to present certain aspects of a cohomology theory of a ring R relative to a subring S, basing the theory on the notions of induced and produced pairs of our earlier paper (2), but making the paper self-contained except for references to a few specific results of (2). The cohomology groups introduced occur in dual pairs. Generic cocycles are defined, and the groups are related to the protractions and retractions of R-modules.

scholarly journals A new description of equivariant cohomology for totally disconnected groups

2008 ◽  
Vol 1 (3) ◽  
pp. 431-472 ◽  
Author(s):  
Christian Voigt
Keyword(s):  
Equivariant Cohomology ◽  
Simplicial Complexes ◽  
Cyclic Homology ◽  
Cohomology Theory ◽  
Cohomology Groups ◽  
Natural Class ◽  
New Approach ◽  
Totally Disconnected

AbstractWe consider smooth actions of totally disconnected groups on simplicial complexes and compare different equivariant cohomology groups associated to such actions. Our main result is that the bivariant equivariant cohomology theory introduced by Baum and Schneider can be described using equivariant periodic cyclic homology. This provides a new approach to the construction of Baum and Schneider as well as a computation of equivariant periodic cyclic homology for a natural class of examples. In addition we discuss the relation between cosheaf homology and equivariant Bredon homology. Since the theory of Baum and Schneider generalizes cosheaf homology we finally see that all these approaches to equivariant cohomology for totally disconnected groups are closely related.

scholarly journals Note on the Cohomology Groups of Associative Algebras

1953 ◽  
Vol 6 ◽  
pp. 85-92 ◽  
Author(s):  
Hirosi Nagao
Keyword(s):  
Detailed Analysis ◽  
Structural Characterization ◽  
Cohomology Theory ◽  
Cohomology Groups ◽  
Associative Algebras ◽  
3 Dimensional

The cohomology theory of associative algebras has been developed by G. liochschild [1], [2], [3], and the 1-, 2-, and 3-dimensional cohomology groups have been interpreted with reference to classical notions of structure in his papers. Recently M. Ikeda has obtained, by a detailed analysis of Hochschild’s modules, an interesting structural characterization of the class of algebras whose 2-dimensional cohomology groups are all zero [5].

scholarly journals Cohomology Theory for Non-Normal Subgroups and Non-Normal Fields

1954 ◽  
Vol 2 (2) ◽  
pp. 66-76 ◽  
Author(s):  
Iain T. Adamson
Keyword(s):  
Finite Group ◽  
Exact Sequence ◽  
Cohomology Group ◽  
Cohomology Theory ◽  
Normal Subgroups ◽  
Cohomology Groups ◽  
Coset Space ◽  
Image Position ◽  
A Finite Group

Let G be a finite group, H an arbitrary subgroup (i.e., not necessarily normal); we decompose G as a union of left cosets modulo H:choosing fixed coset representatives v. In this paper we construct a “coset space complex” and assign cohomology groups; Hr([G: H], A), to it for all coefficient modules A and all dimensions, -∞<r<∞. We show that ifis an exact sequence of coefficient modules such that H1U, A')= 0 for all subgroups U of H, then a cohomology group sequencemay be defined and is exact for -∞<r<∞. We also provide a link between the cohomology groups Hr([G: H], A) and the cohomology groups of G and H; namely, we prove that if Hv(U, A)= 0 for all subgroups U of H and for v = 1, 2, …, n–1, then the sequenceis exact, where the homomorphisms of the sequence are those induced by injection, inflation and restriction respectively.

scholarly journals On the Cohomology of Relative Babach Algebras

2019 ◽  
Vol 13 (10) ◽  
pp. 1
Author(s):  
Alaa Hassan Noreldeen Mohamed
Keyword(s):  
Banach Algebra ◽  
Cohomology Theory ◽  
Basic Theorem ◽  
Relative Cohomology

We study the relative cohomology theory of Banach algebra and give some important basic theorem of it. More, we give discuss about some of properties which we require it in our investigation. At long last, we ponder the dihedral cohomology aggregate as definitions and hypotheses and we characterize the Banach S-relative dihedral cohomology gathering and some theorems.

scholarly journals Continuous cohomology of topological quandles

2019 ◽  
Vol 28 (06) ◽  
pp. 1950036 ◽  
Author(s):  
Mohamed Elhamdadi ◽  
Masahico Saito ◽  
Emanuele Zappala
Keyword(s):  
Inverse Limit ◽  
Cohomology Theory ◽  
Cohomology Groups ◽  
Algebraic Theories ◽  
Continuous Cohomology

A continuous cohomology theory for topological quandles is introduced, and compared to the algebraic theories. Extensions of topological quandles are studied with respect to continuous 2-cocycles, and used to show the differences in second cohomology groups for specific topological quandles. A method of computing the cohomology groups of the inverse limit is applied to quandles.

scholarly journals Algebras With Vanishing n-Cohomology Groups

1954 ◽  
Vol 7 ◽  
pp. 115-131 ◽  
Author(s):  
Masatoshi Ikeda ◽  
Hiroshi Nagao ◽  
Tadashi Nakayama
Keyword(s):  
Associative Algebra ◽  
Finite Rank ◽  
Unit Element ◽  
Cohomology Theory ◽  
Cohomology Groups ◽  
Associative Algebras ◽  
Semisimple Algebras

Cohomology theory for (associative) algebras was first established in general higher dimensionalities by G. Hochschild [3], [4], [5]. Algebras with vanishing 1-cohomology groups are separable semisimple algebras ([3], Theorem 4.1). On extending and refining our recent results [6], [8], [12], we establish in the present paper the following:Let n ≧ 2. Let A be an (associative) algebra (of finite rank) possessing a unit element 1 over a field Ω, and N be its radical.

Gorenstein cohomology of N-complexes

2019 ◽  
Vol 19 (09) ◽  
pp. 2050174
Author(s):  
Bo Lu ◽  
Zhenxing Di
Keyword(s):  
Projective Dimension ◽  
Complex Case ◽  
Binomial Coefficients ◽  
Cohomology Groups ◽  
Injective Dimension ◽  
Relative Cohomology ◽  
Gorenstein Injective Dimension ◽  
Gorenstein Injective ◽  
Gaussian Binomial Coefficients

Let [Formula: see text] and [Formula: see text] be [Formula: see text]-complexes with [Formula: see text] an integer such that [Formula: see text] has finite Gorenstein projective dimension and [Formula: see text] has finite Gorenstein injective dimension. We define the [Formula: see text]th Gorenstein cohomology groups [Formula: see text] [Formula: see text] via a strict Gorenstein precover [Formula: see text] of [Formula: see text] and a strict Gorenstein preenvelope [Formula: see text] of [Formula: see text]. Using Gaussian binomial coefficients we show that there exists an isomorphism [Formula: see text] which extends the balance result of Liu [Relative cohomology of complexes. J. Algebra 502 (2018) 79–97] to the [Formula: see text]-complex case.

scholarly journals Transporting cohomology in Lazard correspondence

2017 ◽  
Vol 16 (06) ◽  
pp. 1750119
Author(s):  
Oihana Garaialde Ocaña ◽  
Jon González-Sánchez
Keyword(s):  
Lie Algebras ◽  
Group Cohomology ◽  
Nilpotency Class ◽  
Cohomology Groups ◽  
Finitely Generated ◽  
Relative Cohomology ◽  
Category Of Modules ◽  
Lazard Correspondence

Lazard correspondence provides an isomorphism of categories between finitely generated nilpotent pro-[Formula: see text] groups of nilpotency class smaller than [Formula: see text] and finitely generated nilpotent [Formula: see text]-Lie algebras of nilpotency class smaller than [Formula: see text]. Denote by [Formula: see text] and [Formula: see text] the group cohomology functors and the Lie cohomology functors respectively. The aim of this paper is to show that for [Formula: see text], [Formula: see text] and [Formula: see text], and for a given category of modules the cohomology functors [Formula: see text] and [Formula: see text] are naturally equivalent. A similar result is proved for [Formula: see text] with the relative cohomology groups.

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