The phantom menace in representation theory

TL;DR

This paper introduces the concept of phantoms in the representation theory of finite-dimensional algebras, highlighting their role in understanding subcategory embeddings and module interactions.

Contribution

It provides a comprehensive overview of phantoms, including their definitions, properties, and applications, especially over string algebras, advancing the understanding of subcategory embeddings.

Findings

Phantoms are key to understanding subcategory embeddings.

Contravariant finiteness plays a crucial role in module theory.

Applications over string algebras demonstrate the practical relevance.

Abstract

Our principal goal in this overview is to explain and motivate the concept of a phantom in the representation theory of a finite dimensional algebra $\Lambda$. In particular, we exhibit the key role of phantoms towards understanding how a full subcategory $\cal A$ of the category $\Lambda\text{-mod}$ of all finitely generated left $\Lambda$-modules is embedded into $\Lambda\text{-mod}$, in terms of maps leaving or entering $\cal A$. Contents: 1. Introduction and prerequisites; 2. Contravariant finiteness and first examples; 3. Homological importance of contravariant finiteness and a model application of the theory; 4. Phantoms. Definitions, existence, and basic properties; 5. An application: Phantoms over string algebras.