Silting mutation in triangulated categories

Abstract

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 to take away this disadvantage by introducing `silting mutation' for silting objects as a generalization of `tilting mutation'. We shall develope 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 mutation 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.

29 pages

Cited by in corpus (50)

• $\tau$-tilting finite algebras, bricks and $g$-vectors
• Reduction of $\tau$-tilting modules and torsion pairs
• Intermediate co-t-structures, two-term silting objects, tau-tilting modules, and torsion classes
• A reduction theorem for $\tau$-rigid modules
• Torsion pairs in silting theory
• Topological aspects of completely integrable foliations
• Lattice structure of torsion classes for path algebras
• Classification of two-term tilting complexes over Brauer graph algebras
• Non-kissing complexes and tau-tilting for gentle algebras
• Classifying tilting complexes over preprojective algebras of Dynkin type
• Silting objects
• Discreteness of silting objects and t-structures in triangulated categories
• Definability and approximations in triangulated categories
• Introduction to $\tau$-tilting theory
• Classifying $\tau$-tilting modules over the Auslander algebra of $K[x]/(x^n)$
• Contractible stability spaces and faithful braid group actions
• $t$-Structures with Grothendieck hearts via functor categories
• C-vectors via $\tau$-tilting theory
• Torsion pairs and simple-minded systems in triangulated categories
• Tilting modules over Auslander-Gorenstein Algebras
• On the Finiteness of the Derived Equivalence Classes of some Stable Endomorphism Rings
• Report on the finiteness of silting objects
• Derived equivalence classification of Brauer graph algebras
• The strong global dimension of piecewise hereditary algebras
• Lifting of recollements and gluing of partial silting sets
• Tilting theory for Gorenstein rings in dimension one
• Tame block algebras of Hecke algebras of classical type
• Derived equivalences for hereditary Artin algebras
• Complete gentle and special biserial algebras are $g$-tame
• Cluster categories and rational curves
• Discrete triangulated categories
• Tilting pairs in extriangulated categories
• $\mathcal{G}$-systems
• A resolution theorem for extriangulated categories with applications to the index
• Tilting complexes and codimension functions over commutative noetherian rings
• Gluing simple-minded collections in triangulated categories
• Some examples of $t$-structures for finite-dimensional algebras
• Representation type of cyclotomic quiver Hecke algebras of type $A_{\ell }^{(1)}$
• Mutation and torsion pairs
• $\tau$-tilting finiteness of two-point algebras II
• A new characterization of silting subcategories in the stable category of a Frobenius extriangulated category
• Classification of co-slicings and co-t-structures for the Kronecker algebra
• Fishing for complements
• On triangle equivalences of stable categories
• Approximations, ghosts and derived equivalences
• Relative left Bongartz completions and their compatibility with mutations
• Derived projective covers and Koszul duality of simple-minded and silting collections
• A note on gluing cosilting objects
• The non-decreasing condition on g-vectors
• Representation type of higher level cyclotomic quiver Hecke algebras in affine type C