Noetherian Hereditary Abelian Categories Satisfying Serre Duality

Notations and conventions 296 Introduction 296 I. Serre duality and almost split sequences 300 I.1. Preliminaries on Serre duality 300 I.2. Connection between Serre duality and Auslander–Reiten triangles 304 I.3. Serre functors on hereditary abelian categories 307 II. Hereditary noetherian abelian categories with non-zero projective objects 309 II.1. Hereditary abelian categories constructed from quivers 310 II.2. Hereditary abelian categories generated by preprojectives 314 II.3. Derived equivalences 318 II.4. The classification 324 III. Sources of hereditary abelian categories with no projectives or injectives 329 III.1. Hereditary abelian categories with Serre functor and all objects of finite length 329 III.2. Sheaves of hereditary orders and graded rings 331 III.3. Hereditary abelian categories associated to infinite Dynkin and tame quivers 333 IV. Hereditary noetherian abelian categories with no projectives or injectives 345 IV.1. Preliminaries 345 IV.2. Completion 349 IV.3. Description in terms of a pullback diagram 353 IV.4. The finite orbit case 356 IV.5. The infinite orbit case 358 V. Applications 360 V.1. Saturatedness 361