The phantom menace in representation theory

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.