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Silting mutation in triangulated categories
Takuma Aihara, Osamu Iyama
TL;DR
Tilting mutation often cannot replace some summands while preserving tilting, limiting the objects reachable by iteration. The paper introduces silting mutation as a generalization, develops its partial-order theory, and establishes transitivity for local, hereditary, and canonical algebras, alongside structural correspondences with t-structures.
Problem
Tilting mutation is often impossible for some summands, so iterated tilting mutation may not produce sufficiently many tilting objects.
Method
The paper introduces silting mutation for silting objects, a partial order related to mutation, silting reduction, and correspondences between silting subcategories and t-structures.
Results
Iterated irreducible silting mutation is transitive for local, hereditary, and canonical algebras, and silting subcategories correspond bijectively to t-structures under certain conditions.
Takeaways & Limitations
Silting mutation provides a broader mutation framework, while the paper’s structural results connect silting objects and subcategories with partial orders, quotient categories, and t-structures.
Takeaways & Limitations
Silting mutation of a tilting subcategory is not necessarily a tilting subcategory, and the detailed mutation discussion assumes a Krull-Schmidt triangulated category.
Abstract
from arXiv · showhide
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.
1. Introduction
The paper introduces silting mutation to overcome the frequent impossibility of tilting mutation and develops its basic theory. It relates mutation to a partial order and studies transitivity and correspondences with t-structures.
- Motivation: Silting mutation generalizes tilting mutation so summands can be replaced even when tilting mutation cannot produce a new tilting object.The paper presents this as a response to the limited supply of objects obtainable by iterated tilting mutation.
- Main question: The paper studies whether iterated irreducible silting mutation acts transitively on basic silting objects in Kb(proj-A).This is formulated as property (T) for finite dimensional algebras.
- Transitivity: If A is local, hereditary, or canonical, property (T) is satisfied; hereditary algebras also have transitivity for iterated irreducible silting mutation.A symmetric algebra is known where (T) fails, while no algebra is known where (T′) fails.
- Basic theory: A partial order on silting objects is introduced, and its relationship with silting mutation is established by generalizing the Riedtmann-Schofield and Happel-Unger theory.The paper identifies this relationship as a basic tool for studying silting objects.
- Structural results: Silting subcategories yield a Grothendieck-group basis, implying that all basic silting objects have the same number of indecomposable summands.Silting reduction also gives a bijection with silting subcategories in a quotient triangulated category.
- t-structures: Under arbitrary coproducts, the paper establishes a one-to-one correspondence between silting subcategories and t-structures satisfying certain conditions.The setting also provides a torsion pair for each set of compact objects.
2. Silting subcategories
The paper develops silting subcategories and objects in triangulated categories, defining their basic properties and relating them to partial orders, mutation, reduction, and tilting constructions.
- Definitions: A silting subcategory M satisfies Hom_T(M, M[> 0]) = 0 and generates the triangulated category as thick M; tilting requires vanishing for all nonzero shifts.Silting and tilting objects are defined when their additive subcategories have these properties.
- Basic properties: A silting subcategory imposes eventual positive-degree Hom-vanishing between every pair of objects in the triangulated category.For tilting subcategories, the corresponding vanishing holds for sufficiently large positive and negative degrees.
- Krull-Schmidt categories: For a Krull-Schmidt triangulated category, the indecomposable objects of a silting subcategory form a basis of K0(T), so all basic silting objects have the same number of indecomposable summands.Silting subcategories can therefore be viewed as isomorphism classes of basic silting objects when a silting object exists.
- Silting mutation: Every mutation of a silting subcategory is silting, while under the stated conditions irreducible mutation is equivalent to being a covering relation in the partial order.If M > N, an irreducible left mutation L exists with M > L ≥ N.
- Silting reduction and examples: Silting reduction gives a bijection between silting subcategories containing a suitable functorially finite thick subcategory and silting subcategories of the corresponding quotient category.The reduction functor also induces a bijection between the relevant indecomposable objects, and iterated irreducible mutation is transitive in the stated settings.
3. Transitivity for piecewise hereditary algebras
For hereditary and canonical algebras, the paper proves transitivity of iterated irreducible silting mutation by relating silting objects to full exceptional sequences.
- Theorem 3.1 proves that iterated irreducible silting mutation acts transitively on silting objects in Kb(proj-A) for hereditary or canonical algebras.
- The proof compares silting objects with exceptional sequences and uses braid-group and shift actions on full exceptional sequences.
- A full exceptional sequence with suitable shifts yields a silting object, with shifts satisfying ℓ_i + a ≤ ℓ_i+1.
- Silting objects arising from the same full exceptional sequence with ordered shifts are connected by iterated irreducible silting mutation.
- Any basic silting object can be reordered into a full exceptional sequence.
- Exceptional-sequence mutations can be translated into iterated irreducible silting mutations, providing the inductive step for transitivity.
4. Silting subcategories in triangulated categories with coproducts
The section develops silting subcategories in triangulated categories with arbitrary coproducts, derives torsion pairs and t-structures from compact-object sets, and establishes bijections with certain t-structures.
- Definitions: Silting subcategories are skeletally small, compact, generating subcategories whose positive shifts have no morphisms from the subcategory.This modifies the definition used earlier for triangulated categories with arbitrary coproducts.
- Torsion pairs: For any set C of compact objects, (⊥(C⊥), C⊥) forms a torsion pair.The construction uses arbitrary coproducts and homotopy-colimit arguments.
- T- and co-t-structures: If C[1] ⊂ C, the torsion pair yields a t-structure, while C ⊂ C[1] yields a co-t-structure after shifting.These specialize the general torsion-pair result to closure conditions on C.
- Silting applications: For a silting subcategory M, (M[< 0]⊥, M[> 0]⊥) is a t-structure.The result identifies the two halves using orthogonality to positive and negative shifts of M.
- Correspondence: Silting subcategories correspond bijectively to silting t-structures, and tilting subcategories correspond bijectively to tilting t-structures.The bijections are mutually inverse and distinguish tilting structures by containment in the heart.