Semiperfect and coreflexive coalgebras. (Q494963)

The authors study non-counital coalgebras \(C\) and non-unital algebras \(A\) over a field. Much of the development parallels the theory of (counital) coalgebras and (unital) algebras. For example, \(C\)-comodules are locally finite, and \(C^*\) is an algebra. A finite dual \(A^0\) is constructed, and characterized as in the unital case. Examples are given involving quivers, and also incidence algebras and coalgebras. A (counital) coalgebra \(C\) has been called coreflexive if the injective evaluation map of \(C\) into \((C^*)^0\) is surjective. In this case (\(C\) counital), the authors define a map from \(C\) into \((\mathrm{Rat}(_{C^*})C^*))^0\), and call \(C\) left coreflexive if it is a bijection. \(\mathrm{Rat}(_{C^*})C^*\) is the left rational \((C^*)\)-module of \(C^*\). As an ideal in \(C^*\), it can be regarded as a non-unital algebra, and has a finite dual. The authors show that if \(C\) is left coreflexive, then \(C\) is right semiperfect, i.e., its injective indecomposable left comodules are finite-dimensional. Conversely, they show that if \(C\) is right semiperfect and the coradical of \(C\) is coreflexive, then \(C\) is left coreflexive. A concluding section gives some applications to Hopf algebras.