Decomposition properties of strict Mittag-Leffler modules

Asensio, Pedro A. Guil; Torrecillas, Blas

doi:10.1016/j.jalgebra.2006.10.006

Cited by 4 publications

(10 citation statements)

References 18 publications

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“…The preceding results stress the close connection that seems to exist between these modules and the property that some λ-directed colimits of λ-presentable objects of λ-accessibly embedded subcategories of R-Mod are trivial. This connection had also been observed in [20,38,37] for (ℵ 0 -)strict Mittag-Leffler modules (see also [32,33,34]). …”

Section: Notation 53 Letsupporting

confidence: 63%

“…Finally, an arbitrary module M has perfect decompositions if and only if each totally ordered directed colimit (that is, a directed colimit in which the set of indices is totally ordered) of modules in Add M belongs to Add M . On the other hand, in [20] we establish the connection between perfect decompositions and the existence of modules satisfying certain Mittag-Leffler conditions, which are implied by the notion of separability. In our next result we study the relationship between totally ordered λ-directed colimits in λ-accessibly embedded subcategories of R-Mod and λ-separable modules (λ an infinite regular cardinal).…”

Section: Notation 53 Letmentioning

confidence: 99%

“…Following [8,20,22], we will call objects satisfying this property κ-strict Mittag-Leffler (or κ-locally pure-projective). The preceding results stress the close connection that seems to exist between these modules and the property that some λ-directed colimits of λ-presentable objects of λ-accessibly embedded subcategories of R-Mod are trivial.…”

Section: Notation 53 Letmentioning

confidence: 99%

“…From this point of view, they stress the close relation existing between the pure-semisimplicity of certain accessible subcategories of modules, (λ-)separable modules and Mittag-Leffler conditions (see e.g. [14,20,32,33,34,38]). …”

Section: Introductionmentioning

confidence: 97%

“…The results included in this paper are mainly inspired by an extension of the interesting construction made by Eklof in [12] (that is developed in Section 4), as well as by the study of the decomposition properties of strict Mittag-Leffler modules made in [20]. They also extend several previous constructions of separable modules in module categories given, for instance, by Angeleri-Hügel [5], Corner [10], Eklof and Shelah [14], Griffith [21], Hill [23] or Huisgen-Zimmermann [37,38].…”

Section: Introductionmentioning

confidence: 99%

See 4 more Smart Citations

Accessible subcategories of modules and pathological objects

Asensio¹,

Torrecillas²

2010

Forum Mathematicum

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Let λ be an infinite regular cardinal. It is proved, under the assumption of the Generalized Continuum Hypothesis, that any λ-accessible and λ-accessibly embedded subcategory K of a category of modules closed under direct sums gives rise to non-trivial κ-separable modules, for arbitrarily large regular cardinals κ ≥ λ, when some modules of K are not direct sum of λ-presentable modules (namely, those modules which are totally ordered λ-directed colimits of λ-presentable modules). In particular, it is shown that any ring R that is not left pure-semisimple has non-trivial λ-separable left modules for arbitrarily large regular cardinals λ. These results extend previous constructions by Corner, Griffith, Hill, Eklof, Shelah and Huisgen-Zimmermann. We point up that κ-separable modules satisfy certain generalized Mittag-Leffler conditions.

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