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In this paper, the (Co)homology theory of pure algebra will be studied. We aim to study the homological theory of cyclic homology and its properties. The triviality property of cohomology groups of algebras will be obtained. We will study the Mayer-Vietortis of cyclic homology. The long exact sequence of cyclic homology and the relation between cyclic homology and dihedral homology will be proved. We will introduce the proof of isomorphism HCn(A×A′)≅HCn(A)⊕HCn(A′) and the definition of Morita invariance for cyclic homology.

The objective of this study is to calculate the K-Cohomology invariants of certain classification spaces of homotopy groups in short exact sequences, where the computation is done using properties of the K-theoretical features. For these packages we use various applications including K-theory Algebra, which transformed the theory of operator algebra from a smaller branch of functional analysis into a topic with deep and wide application areas in all algebra and topology in mathematics.

We prove a number of results on the survival of the type-I property under extensions of locally compact groups: (a) that given a closed normal embedding \mathbb {N}\trianglelefteq \mathbb {E} of locally compact groups and a twisted action (\alpha,\tau) thereof on a (post)liminal C^*-algebra A the twisted crossed product A\rtimes _{\alpha,\tau }\mathbb {E} is again (post)liminal and (b) a number of converses to the effect that under various conditions a normal, closed, cocompact subgroup \mathbb {N}\trianglelefteq \mathbb {E} is type-I as soon as \mathbb {E} is. This happens for instance if \mathbb {N} is discrete and \mathbb {E} is Lie, or if \mathbb {N} is finitely-generated discrete (with no further restrictions except cocompactness). Examples show that there is not much scope for dropping these conditions. In the same spirit, call a locally compact group \mathbb {G} type-I-preserving if all semidirect products \mathbb {N}\rtimes \mathbb {G} are type-I as soon as \mathbb {N} is, and linearly type-I-preserving if the same conclusion holds for semidirect products V\rtimes \mathbb {G} arising from finite-dimensional \mathbb {G}-representations. We characterize the (linearly) type-I-preserving groups that are (1) discrete-by-compact-Lie, (2) nilpotent, or (3) solvable Lie. [ABSTRACT FROM AUTHOR]

The notion of a U-exact sequence is a generalization of the exact sequence. In this paper, we introduce a rough U-exact sequence in a rough group in an approximation space. Furthermore, we provide the properties of the rough U-exact sequence in a rough group.

Bousfield introduced an algebraic category of modules that reflects the structure detected by p-localized complex topological K-theory. He constructed, for any module M in this category, a natural 4-term exact sequence 0 → M → U M → U M → M ⊗ Q → 0 , where U denotes the co-free functor, right adjoint to the forgetful functor to Z (p) -modules. Clarke et al. identified the objects of Bousfield's category as the 'discrete' modules for a certain topological ring A, obtained as a completion of the polynomial ring Z (p) [ x ] , and simplified the construction of the Bousfield sequence in this context. We introduce the notion of 'cobasically discrete' R-modules as a clarification of the Clarke et al. modules, noting that these correspond to comodules over the coalgebra that R is dual to. We study analogues of the Bousfield sequence for other polynomial completion rings, noting a variety of behaviour in the last term of the sequence. [ABSTRACT FROM AUTHOR]

In this paper we calculate cohomology of a classical Lie algebra of type A 2 over an algebraically field k of characteristic p = 3 with coefficients in simple modules. To describe their structure we will consider them as modules over an algebraic group SL 3( k ) . In the case of characteristic p = 3 , there are only two peculiar simple modules: a simple that module isomorphic to the quotient module of the adjoint module by the center, and a one-dimensional trivial module. The results on the cohomology of simple nontrivial module are used for calculating the cohomology of the adjoint module. We also calculate cohomology of the simple quotient algebra Lie of A 2 by the center.