Categorical aspects of Banach modules and Fréchet modules: flatness, projectivity, and functorial isomorphisms
DOI:
https://doi.org/10.51867/ajernet.maths.7.2.84Palavras-chave:
Fréchet Modules, Banach Rings, Flatness, Ideals, Tensor ProductResumo
This paper presents a systematic categorical study of Banach modules over Banach rings and Fréchet modules over Fréchet algebras. We establish foundational results in the categories BanR and Ban_R^(≤1), including a characterization of projective objects, the fact that the tensor product of two projective modules is flat, and an isomorphism between the dual of a colimit and the limit of the duals. In the Fréchet setting, we prove that every essential module is naturally isomorphic to HomA(A, X), and that the strict topology makes this identification a homeomorphism under natural hypotheses. New contributions include a characterization of flatness via ideals, the existence of enough projectives, and the flatness of tensor products of flat modules. Applications to multiplier algebras and representation theory are discussed, and a list of open problems is provided.
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Direitos de Autor (c) 2026 Makila Patrick Wanjala
Este trabalho encontra-se publicado com a Licença Internacional Creative Commons Atribuição-NãoComercial 4.0.
