When Does the Rational Torsion Split Off for Finitely Generated Modules

Abstract

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.

Triangular matrix coalgebras and applications

“…We call a left C-comodule M almost finite if it has only finite dimensional proper subcomodules. We recall the following result which combines several results from [IO,Section 1 & 2].…”

“…We recall that a coalgebra is left Artinian if and only if C is Artinian as a right (respectively left) C * -module, equivalently, C * is a left (respectively, right) Noetherian algebra. We first note that we have an extended version of [IO,Proposition 2.5]. …”

“…By Proposition [IO,2.9], the coalgebra C I has the left finite Ratsplitting property as a subcoalgebra of C. So we now concentrate on characterizing coalgebras C with the left finite Rat-splitting, and such that all injective indecomposable left comodules are infinite dimensional.…”

“…The above mentioned embedding of C-comodules into C * -modules provides another natural framework for this problem. This was studied in several places [C,I1,IO,NT]; in [NT,I1] it is shown that if all C * -modules split, then C must be finite dimensional. The situation for which only finitely generated C * -modules split (the finite splitting property) is studied in [C, IO].…”

Semiartinian Profinite Algebras have Nilpotent Jacobson Radical

“…its largest semiartinian submodule) splits off. This is part of a general class of problems called splitting problems (relative to torsion theories), which have a long history of mathematical interest (see, for example, [G,K1,K2,Rot,T1,T2,T3]; also, [C,I1,IO,NT] for splitting problems for profinite and pseudocompact algebras). It was originally conjectured that if the Dickson torsion splits off in any left A-module for a ring A, then A is left semiartinian; this was proved to be false by a counterexample in [Co].…”