scholarly journals Almost split morphisms in subcategories of triangulated categories

2021 ◽  
pp. 2250239
Author(s):  
Francesca Fedele
Keyword(s):  
Full Description ◽  
Triangulated Category ◽  
Triangulated Categories ◽  
Cluster Category ◽  
Indecomposable Object ◽  
Type Formula ◽  
Dynkin Type ◽  
Serre Functor

For a suitable triangulated category [Formula: see text] with a Serre functor [Formula: see text] and a full precovering subcategory [Formula: see text] closed under summands and extensions, an indecomposable object [Formula: see text] in [Formula: see text] is called Ext-projective if Ext[Formula: see text]. Then there is no Auslander–Reiten triangle in [Formula: see text] with end term [Formula: see text]. In this paper, we show that if, for such an object [Formula: see text], there is a minimal right almost split morphism [Formula: see text] in [Formula: see text], then [Formula: see text] appears in something very similar to an Auslander–Reiten triangle in [Formula: see text]: an essentially unique triangle in [Formula: see text] of the form [Formula: see text] where [Formula: see text] is an indecomposable not in [Formula: see text] and [Formula: see text] is a [Formula: see text]-envelope of [Formula: see text]. Moreover, under some extra assumptions, we show that removing [Formula: see text] from [Formula: see text] and replacing it with [Formula: see text] produces a new subcategory of [Formula: see text] closed under extensions. We prove that this process coincides with the classic mutation of [Formula: see text] with respect to the rigid subcategory of [Formula: see text] generated by all the indecomposable Ext-projectives in [Formula: see text] apart from [Formula: see text]. When [Formula: see text] is the cluster category of Dynkin type [Formula: see text] and [Formula: see text] has the above properties, we give a full description of the triangles in [Formula: see text] of the form [Formula: see text] and show under which circumstances replacing [Formula: see text] by [Formula: see text] gives a new extension closed subcategory.

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 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.

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.

Relative n-rigid objects in (n + 2)-angulated categories

2020 ◽  
pp. 2150157
Author(s):  
Zongyang Xie ◽  
Zhongkui Liu ◽  
Zhenxing Di
Keyword(s):  
Closed Field ◽  
Indecomposable Object ◽  
Endomorphism Algebra ◽  
Rigid Object ◽  
Algebraically Closed Field ◽  
Basic Formula ◽  
Rigid Objects ◽  
Serre Functor

Let [Formula: see text] be an algebraically closed field, [Formula: see text] an integer, [Formula: see text] a [Formula: see text]-linear Hom-finite [Formula: see text]-angulated category with [Formula: see text]-suspension functor [Formula: see text], a Serre functor [Formula: see text], and split idempotents. Let [Formula: see text] be a basic [Formula: see text]-rigid object and [Formula: see text] the endomorphism algebra of [Formula: see text]. We introduce the notion of relative [Formula: see text]-rigid objects, i.e. [Formula: see text]-rigid objects of [Formula: see text]. Then we show that the basic maximal [Formula: see text]-rigid objects in [Formula: see text] are in bijection with basic maximal [Formula: see text]-rigid pairs of [Formula: see text]-modules when every indecomposable object in [Formula: see text] is [Formula: see text]-rigid. As an application, we recover a result in Jacobsen–Jørgensen [Maximal [Formula: see text]-rigid pairs, J. Algebra 546 (2020) 119–134].

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.

scholarly journals NEARLY MORITA EQUIVALENCES AND RIGID OBJECTS

2016 ◽  
Vol 225 ◽  
pp. 64-99 ◽  
Author(s):  
ROBERT J. MARSH ◽  
YANN PALU
Keyword(s):  
Simple Module ◽  
Triangulated Category ◽  
Triangulated Categories ◽  
Morita Equivalent ◽  
Rigid Objects ◽  
Endomorphism Algebras ◽  
Tilted Algebras ◽  
Calculus Of Fractions ◽  
Cluster Tilting ◽  
Maximal Rigid Object

If $T$ and $T^{\prime }$ are two cluster-tilting objects of an acyclic cluster category related by a mutation, their endomorphism algebras are nearly Morita equivalent (Buan et al., Cluster-tilted algebras, Trans. Amer. Math. Soc. 359(1) (2007), 323–332 (electronic)); that is, their module categories are equivalent “up to a simple module”. This result has been generalized by Yang, using a result of Plamondon, to any simple mutation of maximal rigid objects in a 2-Calabi–Yau triangulated category. In this paper, we investigate the more general case of any mutation of a (non-necessarily maximal) rigid object in a triangulated category with a Serre functor. In that setup, the endomorphism algebras might not be nearly Morita equivalent, and we obtain a weaker property that we call pseudo-Morita equivalence. Inspired by Buan and Marsh (From triangulated categories to module categories via localization II: calculus of fractions, J. Lond. Math. Soc. (2) 86(1) (2012), 152–170; From triangulated categories to module categories via localisation, Trans. Amer. Math. Soc. 365(6) (2013), 2845–2861), we also describe our result in terms of localizations.

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