Cluster tilting vs. weak cluster tilting in Dynkin type : A infinity

Abstract

This paper shows a new phenomenon in higher cluster tilting theory. For each positive integer d, we exhibit a triangulated category C with the following properties. On the one hand, the d-cluster tilting subcategories of C have very simple mutation behaviour: Each indecomposable object has exactly d mutations. On the other hand, the weakly d-cluster tilting subcategories of C which lack functorial finiteness can have much more complicated mutation behaviour: For each 0 ≤ ℓ ≤ d-1, we show a weakly d-cluster tilting subcategory Tℓ which has an indecomposable object with precisely ℓ mutations. The category C is the algebraic triangulated category generated by a (d + 1)-spherical object and can be thought of as a higher cluster category of Dynkin type A∞. © 2015 by De Gruyter.