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 THE GRADED CENTER OF A TRIANGULATED CATEGORY

2016 ◽  
Vol 102 (1) ◽  
pp. 74-95
Author(s):  
JON F. CARLSON ◽  
PETER WEBB
Keyword(s):  
Group Algebra ◽  
Special Kind ◽  
Module Category ◽  
Triangulated Category ◽  
Triangulated Categories ◽  
Stable Module Category ◽  
Natural Transformations ◽  
Stable Module ◽  
Serre Functor

With applications in mind to the representations and cohomology of block algebras, we examine elements of the graded center of a triangulated category when the category has a Serre functor. These are natural transformations from the identity functor to powers of the shift functor that commute with the shift functor. We show that such natural transformations that have support in a single shift orbit of indecomposable objects are necessarily of a kind previously constructed by Linckelmann. Under further conditions, when the support is contained in only finitely many shift orbits, sums of transformations of this special kind account for all possibilities. Allowing infinitely many shift orbits in the support, we construct elements of the graded center of the stable module category of a tame group algebra of a kind that cannot occur with wild block algebras. We use functorial methods extensively in the proof, developing some of this theory in the context of triangulated categories.

scholarly journals Tropical Duality in $$(d+2)$$-Angulated Categories

2021 ◽  
Author(s):  
Joseph Reid
Keyword(s):  
Higher Dimensions ◽  
Module Category ◽  
Triangulated Category ◽  
Triangulated Categories ◽  
Basic Cluster ◽  
Grothendieck Groups ◽  
Cluster Tilting ◽  
Tilting Objects

AbstractLet $$\mathscr {C}$$ C be a 2-Calabi–Yau triangulated category with two cluster tilting subcategories $$\mathscr {T}$$ T and $$\mathscr {U}$$ U . A result from Jørgensen and Yakimov (Sel Math (NS) 26:71–90, 2020) and Demonet et al. (Int Math Res Not 2019:852–892, 2017) known as tropical duality says that the index with respect to $$\mathscr {T}$$ T provides an isomorphism between the split Grothendieck groups of $$\mathscr {U}$$ U and $$\mathscr {T}$$ T . We also have the notion of c-vectors, which using tropical duality have been proven to have sign coherence, and to be recoverable as dimension vectors of modules in a module category. The notion of triangulated categories extends to the notion of $$(d+2)$$ ( d + 2 ) -angulated categories. Using a higher analogue of cluster tilting objects, this paper generalises tropical duality to higher dimensions. This implies that these basic cluster tilting objects have the same number of indecomposable summands. It also proves that under conditions of mutability, c-vectors in the $$(d+2)$$ ( d + 2 ) -angulated case have sign coherence, and shows formulae for their computation. Finally, it proves that under the condition of mutability, the c-vectors are recoverable as dimension vectors of modules in a module category.

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.

scholarly journals Completing perfect complexes

2020 ◽  
Vol 296 (3-4) ◽  
pp. 1387-1427 ◽  
Author(s):  
Henning Krause
Keyword(s):  
Derived Categories ◽  
Derived Category ◽  
Triangulated Category ◽  
Triangulated Categories ◽  
Direct Construction ◽  
Coherent Sheaves ◽  
Perfect Complexes ◽  
Coherent Ring ◽  
Abelian Categories ◽  
Finitely Presented

Abstract This note proposes a new method to complete a triangulated category, which is based on the notion of a Cauchy sequence. We apply this to categories of perfect complexes. It is shown that the bounded derived category of finitely presented modules over a right coherent ring is the completion of the category of perfect complexes. The result extends to non-affine noetherian schemes and gives rise to a direct construction of the singularity category. The parallel theory of completion for abelian categories is compatible with the completion of derived categories. There are three appendices. The first one by Tobias Barthel discusses the completion of perfect complexes for ring spectra. The second one by Tobias Barthel and Henning Krause refines for a separated noetherian scheme the description of the bounded derived category of coherent sheaves as a completion. The final appendix by Bernhard Keller introduces the concept of a morphic enhancement for triangulated categories and provides a foundation for completing a triangulated category.

scholarly journals A local-global principle for small triangulated categories

2015 ◽  
Vol 158 (3) ◽  
pp. 451-476 ◽  
Author(s):  
DAVE BENSON ◽  
SRIKANTH B. IYENGAR ◽  
HENNING KRAUSE
Keyword(s):  
Noetherian Ring ◽  
Local Cohomology ◽  
Triangulated Category ◽  
Triangulated Categories ◽  
Commutative Noetherian Ring ◽  
Global Principle ◽  
Compactly Generated

AbstractLocal cohomology functors are constructed for the category of cohomological functors on an essentially small triangulated category ⊺ equipped with an action of a commutative noetherian ring. This is used to establish a local-global principle and to develop a notion of stratification, for ⊺ and the cohomological functors on it, analogous to such concepts for compactly generated triangulated categories.

Cohomologically triangulated categories II

2008 ◽  
Vol 3 (1) ◽  
pp. 1-52 ◽  
Author(s):  
H.-J. Baues ◽  
F. Muro
Keyword(s):  
Cohomology Class ◽  
Triangulated Category ◽  
Triangulated Categories ◽  
Additive Category

AbstractA cohomologically triangulated category is an additive categoryAtogether with a translation functortand a cohomology class Δ ∈H3(A,t) such that any good translation track category representing Δ is a triangulated track category. In this paper we give purely cohomological conditions on Δ which imply that (A,t,Δ) is a cohomologically triangulated category, avoiding the use of track categories. This yields a purely cohomological characterization of triangulated cohomology classes.

Homotopical algebra and triangulated categories

1995 ◽  
Vol 118 (2) ◽  
pp. 259-285 ◽  
Author(s):  
Marco Grandis
Keyword(s):  
Triangulated Category ◽  
Triangulated Categories ◽  
Homotopical Algebra ◽  
Abstract Setting

AbstractWe study here the connections between the well known Puppe-Verdier notion of triangulated category and an abstract setting for homotopical algebra, based on homotopy kernels and cokernels, which was expounded by the author in [11, 13[.We show that a right-homotopical category A (having well-behaved homotopy cokernels, i.e. mapping cones) has a sort of weak triangulated structure with regard to the suspension endofunctor σ, called σ-homotopical category. If A is homotopical and h-stable (in a sense related to the suspension-loop adjunction), this structure is also h-stable, i.e. satisfies ‘up to homotopy’ the axioms of Verdier[29[ for a triangulated category, excepting the octahedral one which depends on some further elementary conditions on the cone endofunctor of A. Every σ-homotopical category can be stabilized, by two universal procedures, respectively initial and terminal.

scholarly journals Auslander-Reiten triangles in subcategories

2008 ◽  
Vol 3 (3) ◽  
pp. 583-601 ◽  
Author(s):  
Peter Jørgensen
Keyword(s):  
Triangulated Category ◽  
Triangulated Categories ◽  
Approximation Properties ◽  
Finite Dimensional ◽  
Image Position ◽  
Finite Dimensional Algebras

AbstractThis paper studies Auslander-Reiten triangles in subcategories of triangulated categories. The main theorem shows that the Auslander-Reiten triangles in a subcategory are closely connected with the approximation properties of the subcategory. Namely, let C be an object in the subcategory C of the triangulated category T, and letbe an Auslander-Reiten triangle in T. Then under suitable assumptions, there is an Auslander-Reiten trianglein C if and only if there is a minimal right-C-approximation of the form.The theory is used to give a new proof of the existence of Auslander-Reiten sequences over finite dimensional algebras.

Right ADA algebras

2017 ◽  
Vol 16 (11) ◽  
pp. 1750210
Author(s):  
Edson Ribeiro Alvares ◽  
Ibrahim Assem ◽  
Diane Castonguay ◽  
Rosana R. S. Vargas
Keyword(s):  
Projective Module ◽  
Module Category ◽  
Artin Algebra ◽  
Indecomposable Projective Module ◽  
The Right

We introduce and study the class of right ADA algebras. An artin algebra is right ADA if every indecomposable projective module lies in the left or in the right part of its module category. We study the Auslander–Reiten components of a right ADA algebra which is not quasi-tilted and prove that they are of three types: components of the left and of the right support, and transitional components each containing a right section.

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