When does the rational submodule split off?

Cuadra, Juan

doi:10.1007/bf02824836

Cited by 5 publications

(1 citation statement)

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“…Many investigations in coalgebras are in fact connected to the relation between rational modules and arbitrary modules over C * . For example, one type of problem studied in this setting is when does the rational part of every module or of every finitely generated module over C * split off [5,15,16,17,23]. One problem of a particular interest is deciding when is this subcategory of rational modules closed under extensions.…”

Section: Introductionmentioning

confidence: 99%

Complete Path Algebras and Rational Modules

Iovanov

2015

Preprint

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Section: Introductionmentioning

confidence: 99%

Complete Path Algebras and Rational Modules

Iovanov

2015

Preprint

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When Does the Rational Torsion Split Off for Finitely Generated Modules

Iovanov

2009

Algebr Represent Theor

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It is well known that the torsion part of any finitely generated module over the formal power series ring K [[X]] is a direct summand. In fact, K [[X]] is an algebra dual to the divided power coalgebra over K and the torsion part of any K[[X]]-module actually identifies with the rational part of that module. More generally, for a certain general enough class of coalgebras-those having only finite dimensional subcomodules-we see that the above phenomenon is preserved: the set of torsion elements of any C * -module is exactly the rational submodule. With this starting point in mind, given a coalgebra C we investigate when the rational submodule of any finitely generated left C * -module is a direct summand. We prove various properties of coalgebras C having this splitting property. Just like in the K[[X]] case, we see that standard examples of coalgebras with this property are the chain coalgebras which are coalgebras whose lattice of left (or equivalently, right, two-sided) coideals form a chain. We give some representation theoretic characterizations of chain coalgebras, which turn out to make a left-right symmetric concept. In fact, in the main result of this paper we characterize the colocal coalgebras where this splitting property holds non-trivially (i.e. infinite dimensional coalgebras) as being exactly Presented by Stefaan Caenepeel and Alain Verschoren. Dedicated to Fred Van Oystaeyen for his sixtieth birthday. 288 M.C. Iovanov the chain coalgebras. This characterizes the cocommutative coalgebras of this kind. Furthermore, we give characterizations of chain coalgebras in particular cases and construct various and general classes of examples of coalgebras with this splitting property.

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