scholarly journals Discreteness of silting objects and t-structures in triangulated categories

2018 ◽  
Vol 118 (1) ◽  
pp. 1-42 ◽  
Author(s):  
Takahide Adachi ◽  
Yuya Mizuno ◽  
Dong Yang
Keyword(s):  
Triangulated Categories ◽  
Silting Objects

scholarly journals Report on the finiteness of silting objects

2021 ◽  
pp. 1-17
Author(s):  
Takuma Aihara ◽  
Takahiro Honma ◽  
Kengo Miyamoto ◽  
Qi Wang
Keyword(s):  
Finite Algebra ◽  
Triangulated Categories ◽  
Finite Algebras ◽  
Silting Objects ◽  
Silting Object

Abstract We discuss the finiteness of (two-term) silting objects. First, we investigate new triangulated categories without silting object. Second, we study two classes of $\tau$ -tilting-finite algebras and give the numbers of their two-term silting objects. Finally, we explore when $\tau$ -tilting-finiteness implies representation-finiteness and obtain several classes of algebras in which a $\tau$ -tilting-finite algebra is representation-finite.

scholarly journals Lifting of recollements and gluing of partial silting sets

2021 ◽  
pp. 1-49
Author(s):  
Manuel Saorín ◽  
Alexandra Zvonareva
Keyword(s):  
Finite Length ◽  
Explicit Construction ◽  
Derived Categories ◽  
Triangulated Categories ◽  
Artin Algebras ◽  
Stable Categories ◽  
Silting Objects ◽  
Torsion Free

This paper focuses on recollements and silting theory in triangulated categories. It consists of two main parts. In the first part a criterion for a recollement of triangulated subcategories to lift to a torsion torsion-free triple (TTF triple) of ambient triangulated categories with coproducts is proved. As a consequence, lifting of TTF triples is possible for recollements of stable categories of repetitive algebras or self-injective finite length algebras and recollements of bounded derived categories of separated Noetherian schemes. When, in addition, the outer subcategories in the recollement are derived categories of small linear categories the conditions from the criterion are sufficient to lift the recollement to a recollement of ambient triangulated categories up to equivalence. In the second part we use these results to study the problem of constructing silting sets in the central category of a recollement generating the t-structure glued from the silting t-structures in the outer categories. In the case of a recollement of bounded derived categories of Artin algebras we provide an explicit construction for gluing classical silting objects.

scholarly journals Recollements, comma categories and morphic enhancements

2021 ◽  
pp. 1-25
Author(s):  
Xiao-Wu Chen ◽  
Jue Le
Keyword(s):  
Module Category ◽  
Triangulated Category ◽  
Triangulated Categories ◽  
Comma Category ◽  
The Right ◽  
Projective Lines ◽  
Triangle Functor

For each recollement of triangulated categories, there is an epivalence between the middle category and the comma category associated with a triangle functor from the category on the right to the category on the left. For a morphic enhancement of a triangulated category $\mathcal {T}$ , there are three explicit ideals of the enhancing category, whose corresponding factor categories are all equivalent to the module category over $\mathcal {T}$ . Examples related to inflation categories and weighted projective lines are discussed.

scholarly journals MODEL STRUCTURES ON TRIANGULATED CATEGORIES

2014 ◽  
Vol 57 (2) ◽  
pp. 263-284 ◽  
Author(s):  
XIAOYAN YANG
Keyword(s):  
Proper Class ◽  
Triangulated Category ◽  
Triangulated Categories ◽  
General Study ◽  
Proper Class Of Triangles ◽  
Cotorsion Pairs ◽  
One To One ◽  
The Relationship ◽  
Model Structures

AbstractWe define model structures on a triangulated category with respect to some proper classes of triangles and give a general study of triangulated model structures. We look at the relationship between these model structures and cotorsion pairs with respect to a proper class of triangles on the triangulated category. In particular, we get Hovey's one-to-one correspondence between triangulated model structures and complete cotorsion pairs with respect to a proper class of triangles. Some applications are given.

New model structures and projective (injective) cotorsion pairs

2021 ◽  
Author(s):  
Aimin Xu
Keyword(s):  
Positive Integer ◽  
Triangulated Categories ◽  
Model Structure ◽  
Cotorsion Pair ◽  
Flat Dimension ◽  
Gorenstein Projective Module ◽  
Chain Complexes ◽  
Cotorsion Pairs ◽  
Gorenstein Injective ◽  
Model Structures

Let [Formula: see text] be either the category of [Formula: see text]-modules or the category of chain complexes of [Formula: see text]-modules and [Formula: see text] a cofibrantly generated hereditary abelian model structure on [Formula: see text]. First, we get a new cofibrantly generated model structure on [Formula: see text] related to [Formula: see text] for any positive integer [Formula: see text], and hence, one can get new algebraic triangulated categories. Second, it is shown that any [Formula: see text]-strongly Gorenstein projective module gives rise to a projective cotorsion pair cogenerated by a set. Finally, let [Formula: see text] be an [Formula: see text]-module with finite flat dimension and [Formula: see text] a positive integer, if [Formula: see text] is an exact sequence of [Formula: see text]-modules with every [Formula: see text] Gorenstein injective, then [Formula: see text] is injective.

scholarly journals On graded centres and block cohomology

2009 ◽  
Vol 52 (2) ◽  
pp. 489-514 ◽  
Author(s):  
Markus Linckelmann
Keyword(s):  
Triangulated Categories ◽  
Nilpotent Ideal ◽  
Stable Elements ◽  
Block Algebra

AbstractWe extend the group theoretic notions of transfer and stable elements to graded centres of triangulated categories. When applied to the centre Z*(Db(B) of the derived bounded category of a block algebra B we show that the block cohomology H*(B) is isomorphic to a quotient of a certain subalgebra of stable elements of Z*(Db(B)) by some nilpotent ideal, and that a quotient of Z*(Db(B)) by some nilpotent ideal is Noetherian over H*(B).

scholarly journals Homological Systems in Triangulated Categories

2014 ◽  
Vol 24 (1) ◽  
pp. 1-35 ◽  
Author(s):  
O. Mendoza ◽  
V. Santiago
Keyword(s):  
Triangulated Categories

scholarly journals Relative rigid objects in triangulated categories

2019 ◽  
Vol 520 ◽  
pp. 171-185 ◽  
Author(s):  
Changjian Fu ◽  
Shengfei Geng ◽  
Pin Liu
Keyword(s):  
Triangulated Categories ◽  
Rigid Objects
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