A Study on Adams Completion and Cocompletion

Many algebraic and geometrical constructions from different field of mathematics such as Algebra, Analysis, Topology, Algebraic Topology, Differential Topology, Differentiable Manifolds and so on can be obtained as Adams completion or cocompletion with respect to chosen sets of morphisms in suitable categories. Cayley’s Theorem, ascending central series and descending central series are well known facts in the area of group theory. It is shown how these concepts are identified with Adams completion. We obtain a Whitehead-like tower of a module by considering a suitable set of morphisms in the corresponding homotopy category (that is, category of right modules and homotopy module homomorphisms) whose different stages are the Adams cocompletion of the module. Indeed, the work is carried out in a general framework by considering a Serre class of abelian groups. The minimal model of a simply connected differential graded algebra is obtained as the Adams cocompletion with respect to the suitably chosen set of morphisms in the category of 1-connected differential graded algebras over Q and differential graded algebra homomorphisms. Also with the help of Kopylov and Timofeev result, the relationship between a graph and Adams cocompletion is established

On Adams Completion and Cocompletion

The minimal model of a 1-connected differential graded Lie algebra is obtained as the Adams cocompletion of the differential graded Lie algebra with respect to a chosen set of morphisms in the category of 1-connected differential graded Lie algebras (d.g.l.a.’s)over the field of rationals and d.g.l.a.-homomorphisms. The Postnikov-like approximation of a module is obtained as the Adams completions of the space with the help of a suitable set of morphisms in the category of some specific modules and module homomorphisms. The Cartan-Whitehead decomposition of topological G-module is obtained as the Adams cocompletion of the space with respect to suitable sets of morphisms. Postnikov-like approximation is obtained for a topological G-module, in terms of Adams completion with respect to a suitable sets of morphisms, using cohomology theory of topological G-modules.The ring of fractions of the algebra of all bounded linear operators on a separable infinite dimensional Banach space is isomorphic to the Adams completion of the algebra with respect to a carefully chosen set of morphisms in the category of separable infinite dimensional Banach spaces and bounded linear norm preserving operators of norms at most 1. The nth tensor algebra and symmetric algebra are each isomorphic to the Adams completions of the algebras. The exterior algebra and Clifford algebra are each isomorphic to the Adams completions of the algebra with respect to a chosen set of morphisms in the category of modules and module homomorphisms