Decompositions of locally finite endo-length modules over a skeletally small category

TL;DR

This paper proves that every locally finite endo-length module over a skeletally small category can be decomposed into indecomposable modules with local endomorphism algebras, extending the understanding of module structure in category theory.

Contribution

It establishes a decomposition theorem for locally finite endo-length modules over skeletally small categories, identifying the structure of indecomposable components.

Findings

Any such module decomposes into indecomposables.

Indecomposable modules have local endomorphism algebras.

Provides a structural insight into modules over small categories.

Abstract

Given a skeletally small category $\mathcal{C}$, we show that any locally finite endo-length $\mathcal{C}$-module is the direct sum of indecomposable $\mathcal{C}$-modules, whose endomorphism algebra is local.