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Homological Algebra in Locally Compact Abelian Groups

1967; American Mathematical Society; Volume: 127; Issue: 3 Linguagem: Inglês

10.2307/1994421

1088-6850

Martin Moskowitz,

Homotopy and Cohomology in Algebraic Topology

I. Introduction.This paper is concerned with homological algebra in the category S? of locally compact abelian topological groups.The morphisms of £f are the continuous homomorphisms.However, only certain exact sequences, resolutions etc., are admissible.These are sequences whose continuous homomorphisms are open onto their respective ranges.Such maps are called proper and the corresponding sequences, proper exact.Concomitant with this is the fact that although Sf is an additive category it is not abelian.The material in §11 may be regarded as preparatory, although there may be some independent interest here.Various structural facts are proven (some of which are well known) and basic properties of important dual subcategories of £f are investigated.In §111 the projectives and injeptives of ¿C are computed.It turns out that subgroups of projectives are projective and quotient groups of injectives are injective.Vector groups are characterized by the fact that they are both projective and injective.Finally, necessary and sufficient conditions are given for the existence of proper resolutions.In § §IV and V continuous versions of the functors Horn and (x) (via dualization) are defined on certain subcategories of =5f and their functorial properties, including exactness, are investigated.These extend the usual notions for discrete groups.In order to do this it is necessary to study topological groups of continuous multilinear functions.Sufficient, and in a sense, necessary conditions are given for the various groups to be locally compact.Under these conditions the appropriate functors are shown to be isomorphic.Finally, more or less explicit computations are made for Horn, (x), etc., sharpening some of the earlier results, and their geometric and structural significance is investigated.In §VI Tor and Ext are defined by resolutions as derived functors of (g) and Horn, and their functorial properties are studied.It turns out that Torn and Extn vanish for n g 2, and that Torx and Ex^ are computable.Knowledge of Extj in turn gives information about certain group extensions in ££.Th ,hout this paper, complete duality of all concepts and theorems is obtained, in order to accomplish this some concessions have to be made to the topology at various stages.For this reason the functors Horn, (x), Tor, and Ext are not defined on .£? but only on certain subcategories.