Fabric idempotents and higher Auslander-Reiten theory

Abstract

Over a finite-dimensional algebra A, simple A-modules that have projective dimension one have special properties. For example, Geigle-Lenzing studied them in connection to homological epimorphisms of rings and they have also appeared in work concerning the finitistic dimension conjecture. If we however work in a d-cluster-tilting subcategory, then not all simples are contained in the subcategory. In this context, a replacement might be to work with idempotents instead, and utilise the theory of Auslander-Platzeck-Todorov. We introduce the notion of a fabric idempotent as an analogue of the localisable objects studied by Chen-Krause, and show that they provide rich combinatorial properties to illustrate the theory. In particular, we use fabric idempotents to extend the classification of singularity categories of Nakayama algebras by Chen-Ye to higher Nakayama algebras.