Properties of triangulated and quotient categories arising from $n$-Calabi-Yau triples

Abstract

Let k be a field, n ≥ 3 an integer and T a k-linear triangulated category with a triangulated subcategory T f d and a subcategory M = add(M ) such that (T , T f d , M) is an n-Calabi-Yau triple. For every integer m and every object X in T , there is a unique, up to isomorphism, truncation triangle of the formIn this paper, we prove some properties of the triangulated categories T and T T f d . Our first result gives a relation between the Hom-spaces in these categories, using limits and colimits. Our second result is a Gap Theorem in T , showing when the truncation triangles split. Moreover, we apply our two theorems to present an alternative proof to a result by Guo, originally stated in a more specific setup of dg k-algebras A and subcategories of the derived category of dg A-modules. This proves that T T f d is Hom-finite and (n − 1)-Calabi-Yau, its object M is (n − 1)-cluster tilting and the endomorphism algebras of M over T and over T T f d are isomorphic. Note that these properties make T T f d a generalisation of the cluster category.