Combinatorial Dimensions: Indecomposability on Certain Local Finite Dimensional Trivial Extension Algebras

TL;DR

This paper investigates the indecomposability of modules over specific local finite dimensional trivial extension algebras using combinatorial and categorical methods, providing new criteria for modules of various ranks.

Contribution

It introduces combinatorial and categorical tools to determine module indecomposability over certain algebras, extending to infinite rank modules.

Findings

Criteria for indecomposability of finite rank modules

Categorical framework for infinite rank modules

Combinatorial invariants for module analysis

Abstract

We study problems related to indecomposability of modules over certain local finite dimensional trivial extension algebras. We do this by purely combinatorial methods. We introduce the concepts of graph of cyclic modules, of combinatorial dimension, and of fundamental combinatorial dimension of a module. We use these concepts to establish, under favorable conditions, criteria for the indecomposability of a module. We present categorifed versions of these constructions and we use this categorical framework to establish criteria for the indecomposability of modules of infinite rank.