Cluster combinatorics of d-cluster categories

TL;DR

This paper explores the combinatorial structure of d-cluster tilting objects in d-cluster categories, extending known results from cluster categories to higher dimensions and applying these to generalized cluster complexes in finite and infinite root systems.

Contribution

It establishes equivalences among d-cluster tilting, maximal rigid, and complete rigid objects, and computes extension groups, extending classical cluster category results to d-cluster categories.

Findings

Any almost complete d-cluster tilting object has exactly d+1 complements.

Computed extension groups between complements.

Studied middle terms of the d+1 triangles in d-cluster categories.

Abstract

We study the cluster combinatorics of $d-$cluster tilting objects in $d-$cluster categories. By using mutations of maximal rigid objects in $d-$cluster categories which are defined similarly for $d-$cluster tilting objects, we prove the equivalences between $d-$cluster tilting objects, maximal rigid objects and complete rigid objects. Using the chain of $d+1$ triangles of $d-$cluster tilting objects in [IY], we prove that any almost complete $d-$cluster tilting object has exactly $d+1$ complements, compute the extension groups between these complements, and study the middle terms of these $d+1$ triangles. All results are the extensions of corresponding results on cluster tilting objects in cluster categories established in [BMRRT] to $d-$cluster categories. They are applied to the Fomin-Reading's generalized cluster complexes of finite root systems defined and studied in [FR2] [Th] [BaM1-2], and to that of infinite root systems [Zh3].