Triangular matrix coalgebras and applications

Abstract

Abstract. We formally introduce and study generalized comatrix coalgebras and upper triangular comatrix coalgebras, which are not only a dualization but also an extension of classical generalized matrix algebras. We use these to answer several open questions on Noetherian and Artinian type notions in the theory of coalgebras, and to give complete connections between these. We also solve completely the so called finite splitting problem for coalgebras: we show that C is a coalgebra such that the rational part of every left finitely generated C * -module splits off if and only if C = D M 0 E is an upper triangular matrix coalgebra, for a serial coalgebra D whose Ext-quiver is a finite union of cycles, a finite dimensional coalgebra E and a finite dimensional bicomodule M .

Hereditary triangular matrix comonads

“…We recover thus the construction of a bipartite K-coalgebra from [21, p. 91] (called triangular matrix coalgebra in [17]). …”

“…be a triangular matrix comonad of order 2, with the comonad structure given by a sextuple of natural transformations as in (17) satisfying the conditions (18) and (19). If the equalizer of every pair of arrows of the form V C…”

“…Section 4 illustrates our general results by giving a characterization of the bipartite coalgebras (see [21,17]) that are right hereditary. Our methods allow us to work over a general commutative ring.…”