Categories of lattices, and their global structure in terms of almost split sequences

Abstract

A major part of Iyama's characterization of Auslander-Reiten quivers of representation-finite orders \(\Lambda\) consists of an induction via rejective subcategories of \(\Lambda\)-lattices, which amounts to a resolution of \(\Lambda\) as an isolated singularity. Despite of its useful applications (proof of Solomon's second conjecture and the finiteness of representation dimension of any artinian algebra), rejective induction cannot be generalized to higher dimensional Cohen-Macaulay orders \(\Lambda\). Our previous characterization of finite Auslander-Reiten quivers of \(\Lambda\) in terms of additive functions [22] was proved by means of L-functors, but we still had to rely on rejective induction. In the present article, this dependence will be eliminated.