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arXiv · 1803.07186

Fabric idempotents and homological dimensions

Jordan McMahon
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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.

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Jordan McMahon. 2018-03-19. Fabric idempotents and homological dimensions. https://arxiv.org/abs/1803.07186

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