In this note we discuss three notions of dimension for triangulated categories: Rouquier dimension, diagonal and Serre dimension. We prove some basic properties these dimensions, compare them open problems.

We study determinant functors which are defined on a triangulated category and take values in a Picard category. The two main results are the existence of a universal determinant functor for every small triangulated category, and a comparison theorem for determinant functors on a triangulated category with a non-degenerate bounded t-structure and determinant functors on its heart. For a small t...

In representation theory of algebras the notion of ‘mutation’ often plays important roles, and two cases are well known, i.e. ‘cluster tilting mutation’ and ‘exceptional mutation’. In this paper we focus on ‘tilting mutation’, which has a disadvantage that it is often impossible, i.e. some of summands of a tilting object can not be replaced to get a new tilting object. The aim of this paper is ...