Combinatorial dimensions: Indecomposability on certain local finite-dimensional trivial extension algebras.

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 categorified versions of these constructions and we use this categorical framework to establish criteria for the indecomposability of modules of infinite rank. [ABSTRACT FROM AUTHOR]