Fabric idempotents and homological dimensions

Abstract

Over a finite-dimensonal algbera $A$, 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 this subcategory. In this context, a replacement might be to work with idempotent ideals instead, and utilise the theory of Auslander-Platzeck-Todorov. We introduce the notion of a fabric idempotent as an analogue of the localising modules studied by Chen-Krause, and to illustrate the theory we show that they provide rich combinatorial properties. An application is to extend the classification of singularity categories of Nakayama algebras by Chen-Ye to higher Nakayama algebras.

19 pages. Examples of higher canonical algebras included, typos corrected

References in corpus (6)

• Higher Nakayama algebras I: Construction
• Gorenstein Homological Algebra of Artin Algebras
• Special tilting modules for algebras with positive dominant dimension
• Gendo-symmetric algebras, dominant dimensions and Gorenstein homological algebra
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Cited by in corpus (2)

• $d\mathbb{Z}$-cluster tilting subcategories of singularity categories
• Higher gentle algebras