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Torsion and localization

University of British Columbia

The purpose of this thesis is to develop the machinery of noncommutative localization as it is being used to date, along with some fundamental results and examples. We are not concerned with a search for a "true torsion theory" for R-modules, but rather with a unification of previous generalisations in a more natural categorical setting. In section 1, the generalisation of torsion for a ring R manifests itself as a kernel functor which is a left exact subfunctor of the identity functor on the category of R-modules. If a kernel functor ơ also has the property ơ(M/ơ(M)) = 0 for any R-module M, we say that ơ is idempotent. We treat the Gabriel correspondence which establishes a canonical bijection between kernel functors, filters of left ideals in R , and classes of R-modules closed under submodules, extensions, homomorphic images, and arbitrary direct sums. This result, which allows us to view torsion in several equivalent ways, is fundamental to the rest of the thesis. Section 2 presents some positive and negative observations on when a kernel functor is idempotent. In section 3 we begin by generalising the concept of injective module by defining ơ-injectivity relative to an idempotent kernel functor ơ. This yields a full coreflective subcategory of the category of R-modules. The localization functor relative to ơ is then constructed as the composite of the coreflector with the embedding of the subcategory. In section 4 we discuss the important "property T" which allows us to express the localization of an R-module as the module tensored with the localized ring, just as in the classical commutative case of localizing at a prime ideal. Finally in section 5 we see that every idempotent kernel functor can be represented by a finitely cogenerating injective R-module V and the relative localization of R by the double centralizer of V . Indications are that the generalised concept of torsion with its relative localization will prove itself increasingly valuable in the further study of rings and modules.

Text

2011-04-07

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http://hdl.handle.net/2429/33374

Mathematics

University of British Columbia

Graduate