On the existence of cluster tilting objects in triangulated categories

We show that in a triangulated category, the existence of a cluster tilting object often implies that the homomorphism groups are bounded in size. This holds for the stable module category of a selfinjective algebra, and as a corollary we recover a theorem of Erdmann and Holm. We then apply our result to Calabi–Yau triangulated categories, in particular stable categories of maximal Cohen–Macaulay modules over commutative local complete Gorenstein algebras with isolated singularities. We show that the existence of almost all kinds of cluster tilting objects can only occur if the algebra is a hypersurface.