Slenderness in Abelian categories
Description
Our setting for the study of slenderness is quite general: we define slenderness in arbitrary abelian categories which have products and coproducts. An object S in such a category C is called slender if for every family {A(,n)}(,n(ELEM) ) of objects in C, the obvious monomorphism (DIAGRAM, TABLE OR GRAPHIC OMITTED...PLEASE SEE DAI) is an isomorphism of abelian groups This notion proves to be as useful as for abelian groups; in fact many results on slender abelian groups carry over mutatis mutandis to this general case. However, while in the case of groups, we could take advantage of very specific properties, those properties were not available here, thus a new approach was necessary. Namely the concept of a filtration A = V(,0) (R-HOOK) V(,1) (R-HOOK) ... (R-HOOK) V(,n) (R-HOOK) ... of an object A and its (Hausdorff) completion (DIAGRAM, TABLE OR GRAPHIC OMITTED...PLEASE SEE DAI) play a very important role since no complete object can be slender. The fundamental theorem characterizing slenderness is the following: An S (ELEM) obj C is slender if and only if (1) for every family {A(,n)}(,n(ELEM) ) of objects in C C( A(,n)/ A(,n),S) = 0, and (2) S does not have a subobject which is complete in a nondiscrete Hausdorff filtration In module categories, we obtain more results of topological nature, which allow us to further specify our description of slender objects: An R-Module M is slender if and only if for every family of cyclic modules {Ra(,n)}(,n(ELEM) ), we have (DIAGRAM, TABLE OR GRAPHIC OMITTED...PLEASE SEE DAI) is not a submodule of M, and (3) M does not contain any completion of Ra(,n) in a nondiscrete metrizable linear topology These seem to be the best possible results on slenderness in the general setting of abelian categories and arbitrary module categories, respectively. In order to get more informative results, we impose additional hypotheses as some restrictions on the cardinality of generating sets of the quotient field Q of a domain R. The distinction between countable and uncountable case is crucial