Coalgebras with a radical rational functor. (Q2386053)

Let \(C\) be a coalgebra locally projective over a commutative (usually also Noetherian) ring \(K\) with dual \(C^*\) an algebra. The rational functor Rat from \(C^*\)-Mod, the category of left \(C^*\)-modules, to Comod-\(C\), the category of right \(C\)-comodules, is defined by taking \(M\) to \(\text{Rat}(M)\) where \(\text{Rat}(M)\) is the set of rational elements in \(M\). This paper studies the question of when the rational functor, which is a hereditary preradical, is a radical functor, or equivalently when a filter of left ideals, \(\mathcal L\), is a Gabriel filter, for a class of coalgebras defined using a strong semilattice of semigroups. The coalgebras studied in this paper are of the form \(C=K[S]\) where \(S\) is a semigroup constructed as follows. Let \(Y\) be an infinite, residually finite semilattice and let \(S'=\sum\{S_\alpha\mid\alpha\in Y\}\) be a strong semilattice sum of finite semigroups (the definition of strong semilattice of semigroups is given in the introduction) with \(S_\alpha=S_\alpha^2\) and such that each map \(\varphi\colon S_\alpha\to S_\beta\) is onto. Then \(S\) is the semigroup obtained by adjoining an identity element. A left ideal \(I\) of \(C^*\) containing the generalized power series ring \(K[\![\bigcup_{\alpha\in X}S_\alpha]\!]\) where \(Y-X\) is finite lies in the filter \(\mathcal L\). The authors show that all ideals in \(\mathcal L\) have this form and show that if \(Y\cup\{e\}\) is a mu-semilattice, then \(\mathcal L\) is a Gabriel filter. They give an example to show \(\mathcal L\) a Gabriel filter does not imply that \(\mathcal L\) has a cofinal subset of finitely generated left ideals, thus answering in the negative a conjecture of Cuadra, Năstăsescu and Van Oystaeyen. Finally the authors consider some new examples of left semiperfect coalgebras, i.e., coalgebras such that every simple right comodule has a projective cover in Comod-\(C\), by showing that if \(K\) is QF and each \(S_\alpha\) is a monoid, then \(C\) is right semiperfect.