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.

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.

scholarly journals Complete slices and homological properties of tilted algebras

1994 ◽  
Vol 36 (3) ◽  
pp. 347-354 ◽  
Author(s):  
Ibrahim Assem ◽  
Flávio Ulhoa Coelho
Keyword(s):  
Representation Theory ◽  
Global Dimension ◽  
Indecomposable Modules ◽  
Tilted Algebra ◽  
Finite Dimensional ◽  
Tilted Algebras ◽  
Homological Dimensions ◽  
The Subject ◽  
Left And Right ◽  
Almost All

It is reasonable to expect that the representation theory of an algebra (finite dimensional over a field, basic and connected) can be used to study its homological properties. In particular, much is known about the structure of the Auslander-Reiten quiver of an algebra, which records most of the information we have on its module category. We ask whether one can predict the homological dimensions of a module from its position in the Auslander-Reiten quiver. We are particularly interested in the case where the algebra is a tilted algebra. This class of algebras of global dimension two, introduced by Happel and Ringel in [7], has since then been the subject of many investigations, and its representation theory is well understood by now (see, for instance, [1], [7], [8], [9], [11], [13]).In this case, the most striking feature of the Auslander-Reiten quiver is the existence of complete slices, which reproduce the quiver of the hereditary algebra from which the tilted algebra arises. It follows from well-known results that any indecomposable successor (or predecessor) of a complete slice has injective (or projective, respectively) dimension at most one, from which one deduces that a tilted algebra is representation-finite if and only if both the projective and the injective dimensions of almost all (that is, all but at most finitely many non-isomorphic) indecomposable modules equal two (see (3.1) and (3.2)). On the other hand, the authors have shown in [2, (3.4)] that a representation-infinite algebra is concealed if and only if both the projective and the injective dimensions of almost all indecomposable modules equal one (see also [14]). This leads us to consider, for tilted algebras which are not concealed, the case when the projective (or injective) dimension of almost all indecomposable successors (or predecessors, respectively) of a complete slice equal two. In order to answer this question, we define the notions of left and right type of a tilted algebra, then those of reduced left and right types (see (2.2) and (3.4) for the definitions).

scholarly journals Reduction of triangulated categories and maximal modification algebras for cAn singularities

2018 ◽  
Vol 2018 (738) ◽  
pp. 149-202 ◽  
Author(s):  
Osamu Iyama ◽  
Michael Wemyss
Keyword(s):  
Complete Classification ◽  
Global Dimension ◽  
Triangulated Categories ◽  
Endomorphism Rings ◽  
Ground Field ◽  
Finite Global Dimension ◽  
Base Ring ◽  
Crepant Resolutions ◽  
Cluster Tilting ◽  
Stable Categories

Abstract In this paper we define and study triangulated categories in which the Hom-spaces have Krull dimension at most one over some base ring (hence they have a natural 2-step filtration), and each factor of the filtration satisfies some Calabi–Yau type property. If \mathcal{C} is such a category, we say that \mathcal{C} is Calabi–Yau with \dim\mathcal{C}\leq 1 . We extend the notion of Calabi–Yau reduction to this setting, and prove general results which are an analogue of known results in cluster theory. Such categories appear naturally in the setting of Gorenstein singularities in dimension three as the stable categories \mathop{\underline{\textup{CM}}}R of Cohen–Macaulay modules. We explain the connection between Calabi–Yau reduction of \mathop{\underline{\textup{CM}}}R and both partial crepant resolutions and \mathbb{Q} -factorial terminalizations of \operatorname{Spec}R , and we show under quite general assumptions that Calabi–Yau reductions exist. In the remainder of the paper we focus on complete local cA_{n} singularities R. By using a purely algebraic argument based on Calabi–Yau reduction of \mathop{\underline{\textup{CM}}}R , we give a complete classification of maximal modifying modules in terms of the symmetric group, generalizing and strengthening results in [I. Burban, O. Iyama, B. Keller and I. Reiten, Cluster tilting for one-dimensional hypersurface singularities, Adv. Math. 217 2008, 6, 2443–2484], [H. Dao and C. Huneke, Vanishing of Ext, cluster tilting and finite global dimension of endomorphism rings, Amer. J. Math. 135 2013, 2, 561–578], where we do not need any restriction on the ground field. We also describe the mutation of modifying modules at an arbitrary (not necessarily indecomposable) direct summand. As a corollary when k=\mathbb{C} we obtain many autoequivalences of the derived category of the \mathbb{Q} -factorial terminalizations of \operatorname{Spec}R .

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.

scholarly journals d-Auslander–Reiten sequences in subcategories

2020 ◽  
Vol 63 (2) ◽  
pp. 342-373
Author(s):  
Francesca Fedele
Keyword(s):  
Abelian Category ◽  
Dimensional Algebra ◽  
Finite Dimensional ◽  
Higher Dimensional ◽  
Image Position ◽  
Cluster Tilting ◽  
Cluster Tilting Subcategory ◽  
Exact Sequences ◽  
Algebra Over A Field

AbstractLet Φ be a finite-dimensional algebra over a field k. Kleiner described the Auslander–Reiten sequences in a precovering extension closed subcategory ${\rm {\cal X}}\subseteq {\rm mod }\,\Phi $. If $X\in \mathcal {X}$ is an indecomposable such that ${\rm Ext}_\Phi ^1 (X,{\rm {\cal X}})\ne 0$ and $\zeta X$ is the unique indecomposable direct summand of the $\mathcal {X}$-cover $g:Y\to D\,{\rm Tr}\,X$ such that ${\rm Ext}_\Phi ^1 (X,\zeta X)\ne 0$, then there is an Auslander–Reiten sequence in $\mathcal {X}$ of the form $${\rm \epsilon }:0\to \zeta X\to {X}^{\prime}\to X\to 0.$$Moreover, when ${\rm En}{\rm d}_\Phi (X)$ modulo the morphisms factoring through a projective is a division ring, Kleiner proved that each non-split short exact sequence of the form $$\delta :0\to Y\to {Y}^{\prime}\buildrel \eta \over \longrightarrow X\to 0$$is such that η is right almost split in $\mathcal {X}$, and the pushout of δ along g gives an Auslander–Reiten sequence in ${\rm mod}\,\Phi $ ending at X.In this paper, we give higher-dimensional generalizations of this. Let $d\geq 1$ be an integer. A d-cluster tilting subcategory ${\rm {\cal F}}\subseteq {\rm mod}\,\Phi $ plays the role of a higher ${\rm mod}\,\Phi $. Such an $\mathcal {F}$ is a d-abelian category, where kernels and cokernels are replaced by complexes of d objects and short exact sequences by complexes of d + 2 objects. We give higher versions of the above results for an additive ‘d-extension closed’ subcategory $\mathcal {X}$ of $\mathcal {F}$.

Sign in / Sign up

Export Citation Format

Share Document