Silting mutation in triangulated categories

In representation theory of algebras the notion of ‘mutation’ often plays important roles, and two cases are well known, that is, ‘cluster tilting mutation’ and ‘exceptional mutation’. In this paper we focus on ‘tilting mutation’, which has a disadvantage that it is often impossible, that is, some of summands of a tilting object cannot be replaced to get a new tilting object. The aim of this paper is to take away this disadvantage by introducing ‘silting mutation’ for silting objects as a generalization of ‘tilting mutation’. We shall develop a basic theory of silting mutation. In particular, we introduce a partial order on the set of silting objects and establish the relationship with ‘silting mutation’ by generalizing the theory of Riedtmann–Schofield and Happel–Unger. We show that iterated silting mutations act transitively on the set of silting objects for local, hereditary or canonical algebras. Finally, we give a bijection between silting subcategories and certain t‐structures.