Silting mutation in triangulated categories

By Takuma Aihara, Osamu Iyama. Journal of the London Mathematical Society, 2010

406 citations recorded by the source corpus.

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.

Subfields: Algebra and Number Theory, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/37da7005d12bc82de538fa742d26a17f441fb1d1

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Citations in the corpus

Silting Objects (2018, background): “This quiver is known to be connected - that is, iterated silting mutation is transitive - whenever A is a local, or a hereditary, or a canonical algebra [7], or belongs to the class of silting-discrete algebras which will be discussed in Section 7.”
Silting objects, simple-minded collections, $t$-structures and co-$t$-structures for finite-dimensional algebras (2012, background): “A subcategory M of C is called a silting subcategory [33, 1] if it is stable for taking direct summands and generates C (i.”
Torsion pairs in silting theory (2016, positive): “Its proof is inspired by the proof of [Aihara and Iyama 2012, Theorem 4.3].”
Silting reduction and Calabi--Yau reduction of triangulated categories (2014, background): “We call U the silting reduction of T with respect to P (see [2]).”
Realisation functors in tilting theory (2015, background): “For any triangulated category T, we denote by [1] its suspension functor.”
Semibricks (2016, background): “, PnA} of indecomposable direct summands of P gives a Z-basis of the Grothendieck group K0(K b(projA)) [3].”
Tilting-Connected Symmetric Algebras (2010, methodology): “[AI] Let T be a basic silting object in T .”
Discreteness of silting objects and t ‐structures in triangulated categories (2017, background): “In recent years the connection between silting theory and the theory of t-structures (and co-t-structures) has received much attention [4, 7, 14, 16, 38, 55, 62], and silting theory has been playing an increasingly important role in the study of triangulated categories of algebraic origin, due to su”
Ordered Exchange Graphs (2013, methodology): “Silting mutation was defined and studied by Buan, Reiten and Thomas in [18] for bounded derived categories of finitedimensional hereditary algebras and independently by Aihara and Iyama in [4] for the general case.”
On compactly generated torsion pairs and the classification of co--structures for commutative noetherian rings (2012, background): “However, the following result from [1], which generalizes both [11, Theorem III.”
A reduction theorem for τ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tau $$\end{document}-rigid modules (2016, background): “This is why Aihara and Iyama introduced the concept of silting mutation [6], where one observes quite the opposite behavior: any summand of a silting object can be replaced to get (infinitely) many new silting objects, and among all of those possibilities one is distinguished as the “right mutation””
Silting theory in triangulated categories with coproducts (2015, methodology): “We will then put ?[0] = 1D and ?[k] will denote the k-th power of ?[1], for each integer k.”
Intermediate co-t-structures, two-term silting objects, tau-tilting modules, and torsion classes (2013, background): “Applying the functor F gives a projective presentation P 1 π → P 0 → U → 0 (3) in modS, and U is a presilting subcategory ⇔ the class { π | u ∈ U } has Property (S).”
A complete derived invariant for gentle algebras via winding numbers and Arf invariants (2019, background): “The shift of a graded curve (γ, f ) is the graded curve (γ, f [1]), where f [1] = f −1.”