Auslander-Reiten Sequences or Triangles Related to Rigid Subcategories

2016 ◽  
Vol 23 (01) ◽  
pp. 1-14
Author(s):  
Ming Lu
Keyword(s):  
Triangulated Category ◽  
Stable Category ◽  
Frobenius Category ◽  
Cluster Tilting ◽  
Cluster Tilting Subcategory

Let 𝒞 be a triangulated category which has Auslander-Reiten triangles, and ℛ a functorially finite rigid subcategory of 𝒞. It is well known that there exist Auslander-Reiten sequences in mod ℛ. In this paper, we give explicitly the relations between the Auslander-Reiten translations, sequences in mod ℛ and the Auslander-Reiten functors, triangles in 𝒞, respectively. Furthermore, if 𝒯 is a cluster-tilting subcategory of 𝒞 and mod 𝒯 is a Frobenius category, we also get the Auslander-Reiten functor and the translation functor of mod 𝒯 corresponding to the ones in 𝒞. As a consequence, we get that if the quotient of a d-Calabi-Yau triangulated category modulo a cluster tilting subcategory is Frobenius, then its stable category is (2d-1)-Calabi-Yau. This result was first proved by Keller and Reiten in the case d=2, and then by Dugas in the general case, using different methods.

Relative Quotient Triangulated Categories

2014 ◽  
Vol 21 (02) ◽  
pp. 195-206 ◽  
Author(s):  
Shengyong Pan
Keyword(s):  
Triangulated Category ◽  
Finite Dimensional Algebra ◽  
Dimensional Algebra ◽  
Stable Category ◽  
Gorenstein Algebra ◽  
Finite Dimensional ◽  
Frobenius Category ◽  
Relative Version ◽  
Relative Derived Category ◽  
Algebra Over A Field

Let A be a finite dimensional algebra over a field k. We consider a subfunctor F of [Formula: see text], which has enough projectives and injectives such that [Formula: see text] is of finite type, where [Formula: see text] denotes the set of F-projectives. One can get the relative derived category [Formula: see text] of A-mod. For an F-self-orthogonal module TF, we discuss the relation between the relative quotient triangulated category [Formula: see text] and the relative stable category of the Frobenius category of TF-Cohen-Macaulay modules. In particular, for an F-Gorenstein algebra A and an F-tilting A-module TF, we get a triangle equivalence between [Formula: see text] and the relative stable category of TF-Cohen-Macaulay modules. This gives the relative version of a result of Chen and Zhang.

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.

scholarly journals Cluster tilting vs. weak cluster tilting in Dynkin type A infinity

2015 ◽  
Vol 27 (2) ◽  
Author(s):  
Thorsten Holm ◽  
Peter Jørgensen
Keyword(s):  
Positive Integer ◽  
Type A ◽  
Triangulated Category ◽  
Tilting Theory ◽  
Dynkin Type ◽  
The One ◽  
Cluster Tilting

AbstractThis paper shows a new phenomenon in higher cluster tilting theory. For each positive integerOn the one hand, theThe category 𝖢 is the algebraic triangulated category generated by a (

scholarly journals -tilting theory

2013 ◽  
Vol 150 (3) ◽  
pp. 415-452 ◽  
Author(s):  
Takahide Adachi ◽  
Osamu Iyama ◽  
Idun Reiten
Keyword(s):  
Direct Summand ◽  
Triangulated Category ◽  
Finite Dimensional Algebra ◽  
Tilting Module ◽  
Tilting Modules ◽  
Dimensional Algebra ◽  
Tilting Theory ◽  
Finite Dimensional ◽  
Cluster Tilting ◽  
Tilting Objects

AbstractThe aim of this paper is to introduce $\tau $-tilting theory, which ‘completes’ (classical) tilting theory from the viewpoint of mutation. It is well known in tilting theory that an almost complete tilting module for any finite-dimensional algebra over a field $k$ is a direct summand of exactly one or two tilting modules. An important property in cluster-tilting theory is that an almost complete cluster-tilting object in a 2-CY triangulated category is a direct summand of exactly two cluster-tilting objects. Reformulated for path algebras $kQ$, this says that an almost complete support tilting module has exactly two complements. We generalize (support) tilting modules to what we call (support) $\tau $-tilting modules, and show that an almost complete support $\tau $-tilting module has exactly two complements for any finite-dimensional algebra. For a finite-dimensional $k$-algebra $\Lambda $, we establish bijections between functorially finite torsion classes in $ \mathsf{mod} \hspace{0.167em} \Lambda $, support $\tau $-tilting modules and two-term silting complexes in ${ \mathsf{K} }^{\mathrm{b} } ( \mathsf{proj} \hspace{0.167em} \Lambda )$. Moreover, these objects correspond bijectively to cluster-tilting objects in $ \mathcal{C} $ if $\Lambda $ is a 2-CY tilted algebra associated with a 2-CY triangulated category $ \mathcal{C} $. As an application, we show that the property of having two complements holds also for two-term silting complexes in ${ \mathsf{K} }^{\mathrm{b} } ( \mathsf{proj} \hspace{0.167em} \Lambda )$.

scholarly journals Tilting and Cluster Tilting for Preprojective Algebras and Coxeter Groups

2017 ◽  
Vol 2019 (18) ◽  
pp. 5597-5634 ◽  
Author(s):  
Yuta Kimura
Keyword(s):  
Coxeter Groups ◽  
Cluster Category ◽  
Preprojective Algebra ◽  
Endomorphism Algebra ◽  
Factor Algebra ◽  
Stable Category ◽  
Preprojective Algebras ◽  
Tilting Object ◽  
Cluster Tilting ◽  
Silting Object

AbstractWe study the stable category of the graded Cohen–Macaulay modules of the factor algebra of the preprojective algebra associated with an element $w$ of the Coxeter group of a quiver. We show that there exists a silting object $M(\boldsymbol{w})$ of this category associated with each reduced expression $\boldsymbol{w}$ of $w$ and give a sufficient condition on $\boldsymbol{w}$ such that $M(\boldsymbol{w})$ is a tilting object. In particular, the stable category is triangle equivalent to the derived category of the endomorphism algebra of $M(\boldsymbol{w})$. Moreover, we compare it with a triangle equivalence given by Amiot–Reiten–Todorov for a cluster category.

scholarly journals TILTING THEORY FOR GORENSTEIN RINGS IN DIMENSION ONE

2020 ◽  
Vol 8 ◽  
Author(s):  
RAGNAR-OLAF BUCHWEITZ ◽  
OSAMU IYAMA ◽  
KOTA YAMAURA
Keyword(s):  
Commutative Algebra ◽  
Prime Ideals ◽  
Gorenstein Ring ◽  
Triangulated Category ◽  
Stable Category ◽  
Tilting Theory ◽  
Exceptional Collection ◽  
Tilting Object ◽  
Dimension One ◽  
Silting Object

In representation theory, commutative algebra and algebraic geometry, it is an important problem to understand when the triangulated category $\mathsf{D}_{\operatorname{sg}}^{\mathbb{Z}}(R)=\text{}\underline{\mathsf{CM}}_{0}^{\mathbb{Z}}R$ admits a tilting (respectively, silting) object for a $\mathbb{Z}$ -graded commutative Gorenstein ring $R=\bigoplus _{i\geqslant 0}R_{i}$ . Here $\mathsf{D}_{\operatorname{sg}}^{\mathbb{Z}}(R)$ is the singularity category, and $\text{}\underline{\mathsf{CM}}_{0}^{\mathbb{Z}}R$ is the stable category of $\mathbb{Z}$ -graded Cohen–Macaulay (CM) $R$ -modules, which are locally free at all nonmaximal prime ideals of $R$ . In this paper, we give a complete answer to this problem in the case where $\dim R=1$ and $R_{0}$ is a field. We prove that $\text{}\underline{\mathsf{CM}}_{0}^{\mathbb{Z}}R$ always admits a silting object, and that $\text{}\underline{\mathsf{CM}}_{0}^{\mathbb{Z}}R$ admits a tilting object if and only if either $R$ is regular or the $a$ -invariant of $R$ is nonnegative. Our silting/tilting object will be given explicitly. We also show that if $R$ is reduced and nonregular, then its $a$ -invariant is nonnegative and the above tilting object gives a full strong exceptional collection in $\text{}\underline{\mathsf{CM}}_{0}^{\mathbb{Z}}R=\text{}\underline{\mathsf{CM}}^{\mathbb{Z}}R$ .

scholarly journals Gluing of n-Cluster Tilting Subcategories for Representation-directed Algebras

2020 ◽  
Author(s):  
Laertis Vaso
Keyword(s):  
Representation Theory ◽  
Global Dimension ◽  
Cluster Tilting ◽  
Cluster Tilting Subcategory

Abstract Given $n\leq d<\infty $ n ≤ d < ∞ , we investigate the existence of algebras of global dimension d which admit an n-cluster tilting subcategory. We construct many such examples using representation-directed algebras. First, given two representation-directed algebras A and B, a projective A-module P and an injective B-module I satisfying certain conditions, we show how we can construct a new representation-directed algebra "Image missing" in such a way that the representation theory of Λ is completely described by the representation theories of A and B. Next we introduce n-fractured subcategories which generalize n-cluster tilting subcategories for representation-directed algebras. We then show how one can construct an n-cluster tilting subcategory for Λ by using n-fractured subcategories of A and B. As an application of our construction, we show that if n is odd and d ≥ n then there exists an algebra admitting an n-cluster tilting subcategory and having global dimension d. We show the same result if n is even and d is odd or d ≥ 2n.

Buchweitz’s equivalences for Gorenstein flat modules with respect to semidualizing modules

2019 ◽  
pp. 2150006
Author(s):  
Jiangsheng Hu ◽  
Yuxian Geng ◽  
Jinyong Wu ◽  
Huanhuan Li
Keyword(s):  
Noetherian Ring ◽  
Full Subcategory ◽  
Stable Category ◽  
Commutative Noetherian Ring ◽  
Singularity Category ◽  
Flat Modules ◽  
Frobenius Category ◽  
Semidualizing Modules ◽  
Structure Formula ◽  
Gorenstein Flat

Let [Formula: see text] be a commutative Noetherian ring and [Formula: see text] a semidualizing [Formula: see text]-module. We obtain an exact structure [Formula: see text] and prove that the full subcategory [Formula: see text] of [Formula: see text] is a Frobenius category with [Formula: see text] the subcategory of projective and injective objects, where [Formula: see text] and [Formula: see text] (respectively, [Formula: see text]) is the subcategory of [Formula: see text]-Gorenstein flat (respectively, [Formula: see text]-flat [Formula: see text]-cotorsion) [Formula: see text]-modules. Then the stable category [Formula: see text] of [Formula: see text] and the singularity category [Formula: see text] of [Formula: see text] are also considered. As a consequence, we get that there is a Buchweitz’s equivalence [Formula: see text] if and only if [Formula: see text] is a part of some AB-context.

scholarly journals $$d\mathbb {Z}$$-Cluster tilting subcategories of singularity categories

2020 ◽  
Author(s):  
Sondre Kvamme
Keyword(s):  
Triangulated Categories ◽  
Finite Dimensional Algebra ◽  
Dimensional Algebra ◽  
Exact Category ◽  
Singularity Category ◽  
Finite Dimensional ◽  
Cluster Tilting ◽  
Cluster Tilting Subcategory

Abstract For an exact category $${{\mathcal {E}}}$$ E with enough projectives and with a $$d\mathbb {Z}$$ d Z -cluster tilting subcategory, we show that the singularity category of $${{\mathcal {E}}}$$ E admits a $$d\mathbb {Z}$$ d Z -cluster tilting subcategory. To do this we introduce cluster tilting subcategories of left triangulated categories, and we show that there is a correspondence between cluster tilting subcategories of $${{\mathcal {E}}}$$ E and $${\underline{{{\mathcal {E}}}}}$$ E ̲ . We also deduce that the Gorenstein projectives of $${{\mathcal {E}}}$$ E admit a $$d\mathbb {Z}$$ d Z -cluster tilting subcategory under some assumptions. Finally, we compute the $$d\mathbb {Z}$$ d Z -cluster tilting subcategory of the singularity category for a finite-dimensional algebra which is not Iwanaga–Gorenstein.

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