# Triangular matrix coalgebras and applications

## TL;DR

This paper explores generalized and upper triangular comatrix coalgebras, establishing their properties, connections to classical algebra, and solving the finite splitting problem for coalgebras with specific structural conditions.

## Contribution

It introduces and analyzes generalized comatrix coalgebras, extends classical algebra concepts to coalgebras, and fully characterizes coalgebras with the finite splitting property.

## Key findings

- Characterization of coalgebras as upper triangular matrix coalgebras.
- Complete solution to the finite splitting problem for specific coalgebras.
- Connections established between Noetherian, Artinian, and coalgebra structures.

## Abstract

We 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 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 a coalgebra $C$ has the property that the rational part of every finitely generated left $C^*$-module splits off if and only if $C$ has the form $C=\left(\begin{array}{cc} D & M \\ 0 & E \end{array}\right)$, 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 $D$-$E$-bicomodule $M$.