AUSLANDER–REITEN TRIANGLES AND A THEOREM OF ZIMMERMANN

Abstract

A classical theorem of Zimmermann describes the relation between almost split sequences in the category of finitely presented modules and those in the category of all modules over some fixed ring. An analogue of Auslander–Reiten triangles in triangulated categories is proved in this paper. This is used to explain the relation between different existence results for Auslander–Reiten triangles, which are based either on Brown’s representability theorem, or on the existence of Serre functors. This paper is a continuation of [14], where Brown representability was introduced as a foundation for a general Auslander–Reiten theory in triangulated categories. More recently, Auslander–Reiten triangles have played an important role in work of Reiten and van den Bergh [18] and Jorgensen [11]. In [18], the existence of Auslander–Reiten triangles is translated into the existence of a Serre functor in the sense of Bondal and Kapranov [4]. This is used for a classification of the noetherian hereditary abelian categories satisfying Serre duality. In [11], the singular cochain differential graded algebra of a simply connected space is studied. The existence of