The valued Gabriel quiver of a wedge product and semiprime coalgebras

TL;DR

This paper explores the representation theory of wedge products of coalgebras through Gabriel quivers, introduces semiprime coalgebras, and characterizes their comodules, revealing structural properties and classifications.

Contribution

It provides the first analysis of the valued Gabriel quiver for wedge products and characterizes semiprime coalgebras using localization techniques.

Findings

Monomial semiprime fc-tame coalgebras are string if Gabriel quiver is locally finite.

Hereditary semiprime strictly quasi-finite coalgebras are serial.

Introduces a new approach to the representation theory of coalgebras via Gabriel quivers.

Abstract

We make a first approach to the representation theory of the wedge product of coalgebras by means of the description of its valued Gabriel quiver. Then we define semiprime coalgebras and study its category of comodules by the use of localization techniques. In particular, we prove that, whether its Gabriel quiver is locally finite, any monomial semiprime fc-tame coalgebra is string. We also prove a weaker version of Eisenbud-Griffith theorem, namely, any hereditary semiprime strictly quasi-finite coalgebra is serial.