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

Abstract $$n\mathbb {Z}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>n</mml:mi> <mml:mi>Z</mml:mi> </mml:mrow> </mml:math> -cluster tilting subcategories are an ideal setting for higher dimensional Auslander–Reiten theory. We give a complete classification of $$n\mathbb {Z}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>n</mml:mi> <mml:mi>Z</mml:mi> </mml:mrow> </mml:math> -cluster tilting subcategories of module categories of Nakayama algebras given by quivers with relations. In particular, we show that there are three kinds of Nakayama algebras that admit $$n\mathbb {Z}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>n</mml:mi> <mml:mi>Z</mml:mi> </mml:mrow> </mml:math> -cluster tilting subcategories: finite global dimension, selfinjective and non-Iwanaga–Gorenstein. Only the selfinjective ones can admit more than one $$n\mathbb {Z}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>n</mml:mi> <mml:mi>Z</mml:mi> </mml:mrow> </mml:math> -cluster tilting subcategory. It has been shown by the second author, that each such $$n\mathbb {Z}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>n</mml:mi> <mml:mi>Z</mml:mi> </mml:mrow> </mml:math> -cluster tilting subcategory induces an $$n\mathbb {Z}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>n</mml:mi> <mml:mi>Z</mml:mi> </mml:mrow> </mml:math> -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 $$n\mathbb {Z}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>n</mml:mi> <mml:mi>Z</mml:mi> </mml:mrow> </mml:math> -cluster tilting subcategories in the module category and the singularity category. This relies heavily on results by Shen, who described the singularity categories of all Nakayama algebras.