Auslander-Reiten theory via Brown representability
Under mild assumptions, a triangulated category is known to satisfy Brown representability. That is, a covariant functor into abelian groups is representable if and only if it is exact and sends arbitrary coproducts to products. The paper under review develops some Auslander-Reiten theory for triangulated categories by using Brown representability as a basis. For example, Brown representability yields the existence of objects playing the role of Auslander-Reiten translates of given objects. It is shown that Auslander-Reiten triangles exist for compact objects. (Recall that over a finite-dimensional algebra, a compact complex is a bounded complex of finitely generated projective modules.) Moreover, pure-injective objects are discussed as well as morphisms determined by objects, and defects of triangles.
scientific article; zbMATH DE number 1563904