Krull-Schmidt categories and projective covers

TL;DR

This paper introduces Krull-Schmidt categories, emphasizing their decomposition properties, through a self-contained approach centered on the concept of projective covers, to clarify their structure and foundational aspects.

Contribution

It offers a self-contained introduction to Krull-Schmidt categories focusing on projective covers, enhancing understanding of their decomposition properties.

Findings

Krull-Schmidt categories allow unique decomposition into indecomposables.

Projective covers are central to understanding these categories.

The paper clarifies foundational concepts for further research.

Abstract

Krull-Schmidt categories are additive categories such that each object decomposes into a finite direct sum of indecomposable objects having local endomorphism rings. We provide a self-contained introduction which is based on the concept of a projective cover.