Serial Coalgebras and Their Valued Gabriel Quivers

We study serial coalgebras by means of their valued Gabriel quivers. In particular, Hom-computable and representation-directed coalgebras are characterized. The Auslander-Reiten quiver of a serial coalgebra is described. Finally, a version of Eisenbud-Griffith theorem is proved, namely, every subcoalgebra of a prime, hereditary and strictly quasi-finite coalgebra is serial. Introduction A systematic study of serial coalgebras was initiated in [4], where, in particular, it was shown that any serial indecomposable coalgebra over an algebraically closed field is Morita-Takeuchi equivalent to a subcoalgebra of a path coalgebra of a quiver which is either a cycle or a chain (finite or infinite) [4, Theorem 2.10]. In this paper, we take advantage of the valued Gabriel quivers associated to a coalgebra to characterize indecomposable serial coalgebras over any field (Theorem 1.5). In conjunction with localization techniques (see Section 2), this combinatorial tool allows to complete the study made in [4] in more remarkable aspects. Thus, in Section 3, we characterize Hom-computable serial coalgebras in the sense of [27] (Proposition 3.3), and representation-directed coalgebras (Proposition 3.4). Section 4 is devoted to describe the Auslander-Reiten quiver of the category of finite dimensional (right) comodules of a serial coalgebra. It was observed in [4] that a consequence of [7, Corollary 3.2] is that the finite dual coalgebra of a hereditary noetherian prime algebra over a field is serial. In Section 5 we reconsider this result of Eisenbud and Griffith from the coalgebraic point of view: we prove, using the results developed in the previous sections, that any subcoalgebra of a prime, hereditary and strictly quasi-finite coalgebra is serial (Corollary 5.3). Throughout we fix a field K and we assume C is a K-coalgebra. We refer the reader to the books [1], [17] and [29] for notions and notations about coalgebras. Unless otherwise stated, all C-comodules are right C-comodules. It is well-known that C has a decomposition, as right C-comodule,