$n\mathbb{Z}$-cluster tilting subcategories for Nakayama algebras

Abstract

$n\mathbb{Z}$-cluster tilting subcategories are an ideal setting for higher dimensional Auslander-Reiten theory. We give a complete classification of $n\mathbb{Z}$-cluster tilting subcategories of module categories of Nakayama algebras. In particular, we show that there are three kinds of Nakayama algebras that admit $n\mathbb{Z}$-cluster tilting subcategories: finite global dimension, selfinjective and non-Iwanaga-Gorenstein. Only the selfinjective ones can admit more than one $n\mathbb{Z}$-cluster tilting subcategory. It has been shown by the second author, that each such $n\mathbb{Z}$-cluster tilting subcategory induces an $n\mathbb{Z}$-cluster tilting subcategory of the corresponding singularity category. For each Nakayama algebra in our classification, we describe its singularity category, the canonical functor from its module category to its singularity category, and provide a complete comparison of $n\mathbb{Z}$-cluster tilting subcategories in the module category and the singularity category. This relies heavily of results by Shen, who described the singularity categories of all Nakayama algebras.