## n-abelian and n-exact categories (2014)

Citations: | 1 - 0 self |

### Citations

518 | Triangulated categories in the representation theory of finite dimensional algebras - Happel - 1988 |

509 |
Higher algebraic K-theory I
- Quillen
- 1973
(Show Context)
Citation Context ...nteresting additive categories are not abelian but still have good homological properties with respect with a restricted class of short exact sequences. Exact categories were introduced by Quillen in =-=[42]-=- from this perspective to axiomatize extension-closed subcategories of abelian categories. Derived categories play an important role in the study of the homological properties of abelian and exact cat... |

311 |
Sur quelques points d’Algèbre Homologique
- Grothendieck
- 1957
(Show Context)
Citation Context ...Throughout we use the comparative adjective “higher” in relation to the length of exact sequences and not in the sense of higher category theory. Abelian categories were introduced by Grothendieck in =-=[23]-=- to axiomatize the properties of the category of modules over a ring and of the category of sheaves over a scheme. It is often the case that interesting additive categories are not abelian but still h... |

249 |
Des catégories dérivées des catégories abéliennes
- Verdier
- 1996
(Show Context)
Citation Context ...mportant role in the study of the homological properties of abelian and exact categories. Their properties are captured by the notion of triangulated categories, introduced by Grothendieck-Verdier in =-=[44]-=-. By a result of Happel, the stable category of a Frobenius exact category has a natural structure of a triangulated category, see [24, Thm. I.2.6]. Triangulated categories arising in this way have be... |

199 |
Faisceaux Pervers”. Analysis and topology on singulas spaces
- Beilinson, Bernstein, et al.
- 1981
(Show Context)
Citation Context ...Λ be a finite dimensional algebra such that gl. dim.Λ = n. We say that Λ is n-representation-infinite if for all i ≥ 0 we have ν−in (Λ) ∈ modΛ. 54 G. JASSO Let T be a triangulated category. Following =-=[11]-=-, a t-structure on T is a pair (T≤0,T≥0) of strictly full subcategories of T which satisfies the following properties: (i) We have ΣT≤0 ⊆ T≤0 and Σ−1T≥0 ⊆ T≥0. (ii) For all X ∈ T≤0 and for all Y ∈ T≥0... |

188 | On differential graded categories
- Keller
- 2006
(Show Context)
Citation Context ...le category of a Frobenius exact category has a natural structure of a triangulated category, see [24, Thm. I.2.6]. Triangulated categories arising in this way have been called algebraic by Keller in =-=[36]-=-. Algebraic triangulated categories have a natural dg-enhancement in the sense of Bondal-Kapranov [15], thus are often considered as a more reasonable class than that of general triangulated categorie... |

187 |
Cohen-Macaulay modules over Cohen-Macaulay rings, London Mathematical Society Lecture Note Series 146
- Yoshino
- 1990
(Show Context)
Citation Context .... We remind the reader that R is an isolated singularity if R is not regular and for all non-maximal prime ideals p ⊂ R we have that Rp is a regular ring. In this case, CMR has almost-split sequences =-=[7, 46]-=-. Theorem 6.12. [29, Thm. 2.5] and [33, Cor. 8.2] Let K be an algebraically closed field of characteristic 0 and set S := KJx0, x1, . . . , xnK. Also, let G be a finite subgroup of SLn(K) acting freel... |

176 |
Noncommutative projective schemes
- Artin, Zhang
- 1994
(Show Context)
Citation Context ...modΛ) | ν−in (X) ∈ D ≥0 ∀i≫ 0 } . Then, the pair (X≤0,X≥0) is a t-structure in Db(modΛ). Moreover, the heart of this t-structure is equivalent to the non-commutative projective scheme qgrΠn+1(Λ), see =-=[2]-=- for the definition. The following result gives examples of n-exact categories. Theorem 6.8. [26] Let Λ be an n-representation infinite algebra such that Πn+1(Λ) is noetherian. Let (X≤0,X≥0) be the t-... |

176 | Quivers with potentials and their representations
- Derksen, Weyman, et al.
(Show Context)
Citation Context ...gebra Πn+1(Λ) := ⊕ d≥0 ExtnΛ(DΛ,Λ) ⊗Λd. The following is a structure theorem for 2-representation-finite algebras; it allows to produce examples of such algebras rather easily. We refer the reader to =-=[25, 18]-=- for details and definitions. Theorem 6.4. [25, Thm. 3.11] Let Λ be a finite dimensional algebra such that gl. dim.Λ = 2. Then, the following statements are equivalent: (i) The algebra Λ is 2-represen... |

150 |
Applications of contravariantly finite subcategories
- Auslander, Reiten
- 1991
(Show Context)
Citation Context ... A functorially finite subcategory of C is a subcategory which is both covariantly and contravariantly finite in C. For further information on functorially finite subcategories we refer the reader to =-=[6, 5]-=-. 2.2. n-cokernels, n-kernels, and n-exact sequences. Let C be an additive category and f : A → B a morphism in C. A weak cokernel of f is a morphism g : B → C such that for all C′ ∈ C the sequence of... |

149 | Cluster algebras I: Foundations
- Fomin, Zelevinsky
(Show Context)
Citation Context ...ategories were introduced by Buan-Marsh-Reiten-ReinekeTodorov in [16] as the key concept involved in the additive categorification of the mutation combinatorics of Fomin-Zelevinsky’s cluster algebras =-=[19]-=- via 2-CalabiYau triangulated categories. It was then observed by Iyama-Yoshino [33] that the notion of mutation can be extended to the class of n-cluster-tilting subcategories of triangulated categor... |

149 | An introduction to homological algebra, volume 38 of Cambridge Studies in Advanced Mathematics - Weibel - 1994 |

129 | Tilting theory and cluster combinatorics
- Buan, Marsh, et al.
- 2006
(Show Context)
Citation Context ...ral triangulated categories. Recently, a new class of additive categories appeared in representation theory. The 2-cluster-tilting subcategories were introduced by Buan-Marsh-Reiten-ReinekeTodorov in =-=[16]-=- as the key concept involved in the additive categorification of the mutation combinatorics of Fomin-Zelevinsky’s cluster algebras [19] via 2-CalabiYau triangulated categories. It was then observed by... |

118 |
Mutation in triangulated categories and rigid Cohen-Macaulay modules
- Iyama, Yoshino
(Show Context)
Citation Context ...cept involved in the additive categorification of the mutation combinatorics of Fomin-Zelevinsky’s cluster algebras [19] via 2-CalabiYau triangulated categories. It was then observed by Iyama-Yoshino =-=[33]-=- that the notion of mutation can be extended to the class of n-cluster-tilting subcategories of triangulated categories. From a different perspective, n-cluster-tilting subcategories of certain exact ... |

116 |
Coherent sheaves on P n and problems in linear algebra
- Beĭlinson
- 1978
(Show Context)
Citation Context ...g Theorem 6.8. Example 6.9. Let cohPnK be the category of coherent sheaves over the projective n-space over K, and let Λ be the endomorphism algebra of the tilting bundle O ⊕ O(1) ⊕ · · · ⊕ O(n), see =-=[10]-=-. It is known that Λ is an n-representation-infinite algebra and that Πn+1(Λ) is noetherian, see [27, Ex. 2.15]. Moreover, there is an equivalence of triangulated categories Db(modΛ) ∼= D b(cohPnK). n... |

101 | Chain complexes and stable categories - Keller - 1990 |

82 |
Almost split sequences in subcategories
- Auslander, Smalø
- 1981
(Show Context)
Citation Context ... A functorially finite subcategory of C is a subcategory which is both covariantly and contravariantly finite in C. For further information on functorially finite subcategories we refer the reader to =-=[6, 5]-=-. 2.2. n-cokernels, n-kernels, and n-exact sequences. Let C be an additive category and f : A → B a morphism in C. A weak cokernel of f is a morphism g : B → C such that for all C′ ∈ C the sequence of... |

80 | Higher dimensional Auslander-Reiten theory on maximal orthogonal subcategories
- Iyama
(Show Context)
Citation Context ...ended to the class of n-cluster-tilting subcategories of triangulated categories. From a different perspective, n-cluster-tilting subcategories of certain exact categories were introduced by Iyama in =-=[29]-=- and further investigated in [30, 28] from the viewpoint of higher Auslander-Reiten theory. In this theory, the notion of nalmost-split sequence, which are certain exact sequences with n+2 terms, play... |

79 |
Coherent functors
- Auslander
- 1965
(Show Context)
Citation Context ...ModC. Moreover, modC is closed under kernels in ModC if and only if C has weak kernels, in which case modC is an abelian category. For further information on coherent C-modules we refer the reader to =-=[3]-=-. Our aim is to prove the following theorem. Theorem 3.20. Let M be a small projectively generated n-abelian category. Let P be the category of projective objects in M and F : M → modP the functor def... |

43 |
Isolated singularities and the existence of almost split sequences
- Auslander
- 1986
(Show Context)
Citation Context .... We remind the reader that R is an isolated singularity if R is not regular and for all non-maximal prime ideals p ⊂ R we have that Rp is a regular ring. In this case, CMR has almost-split sequences =-=[7, 46]-=-. Theorem 6.12. [29, Thm. 2.5] and [33, Cor. 8.2] Let K be an algebraically closed field of characteristic 0 and set S := KJx0, x1, . . . , xnK. Also, let G be a finite subgroup of SLn(K) acting freel... |

41 |
The Derived Category of an Exact Category
- Neeman
- 1990
(Show Context)
Citation Context ...mplex if for each k ∈ Z the sequence Y k−1 Xk ։ Y k is an X-admissible exact sequence. If the class X is clear from the complex, then we may write “acyclic” instead of “X-acyclic”. Following Neeman =-=[41]-=-, the derived category D = D(E,X) is by definition the Verdier quotient H(E)/ thick(Ac(X)). Then, for all k ≥ 1 and for all E ∈ E we can define the functor ExtkX(E,−) : E→ ModZ by F 7→ HomD(E,F [k]). ... |

35 |
Deformed Calabi-Yau completions
- Keller
(Show Context)
Citation Context ...· · · τ ℓ2−1 n (I2), τ ℓ2 n (I2) ∼= Pσ(2) ... ... ... ... ... Id, τn(Id), τ 2 n(Id), · · · τ ℓd−1 n (Id), τ ℓd n (Id) ∼= Pσ(d) Let Λ be a finite dimensional algebra such that gl. dim.Λ ≤ n. Following =-=[38]-=-, the (n+ 1)-preprojective algebra of Λ is defined as the tensor algebra Πn+1(Λ) := ⊕ d≥0 ExtnΛ(DΛ,Λ) ⊗Λd. The following is a structure theorem for 2-representation-finite algebras; it allows to produ... |

34 |
I.: Stable equivalences of dualizing R-varieties
- Auslander, Reiten
- 1974
(Show Context)
Citation Context ...ster-tilting subcategory of modP. 26 G. JASSO Remark 3.21. The requirement in Theorem 3.20 of modP being injectively cogenerated is satisfied, for example, if P is a dualizing variety in the sense of =-=[4]-=-. In fact, instead of proving Theorem 3.20, we prove the following more general statement. Lemma 3.22. Let M be a small projectively generated n-abelian category. Let P be the category of projective o... |

34 | Acyclic Calabi-Yau categories
- Keller, Reiten
(Show Context)
Citation Context ...egories arising in this way are the following: • Let C = CQ be the cluster category associated with an acyclic quiver Q, see [16] for details. It is known that C is an algebraic triangulated category =-=[37]-=-. Moreover, a basic 2-cluster-tilting object T ∈ C satisfies Σ2T ∼= T if and only if C(T, T ) is a selfinjective algebra, see [32, Cor. 3.8]. All such algebras were classified by Ringel in [43]. In pa... |

30 | Cluster tilting for higher Auslander algebra
- Iyama
(Show Context)
Citation Context ...ilting subcategories of triangulated categories. From a different perspective, n-cluster-tilting subcategories of certain exact categories were introduced by Iyama in [29] and further investigated in =-=[30, 28]-=- from the viewpoint of higher Auslander-Reiten theory. In this theory, the notion of nalmost-split sequence, which are certain exact sequences with n+2 terms, plays an important role. With motivation ... |

23 | Exact categories
- Bühler
(Show Context)
Citation Context ...nces. Furthermore, if X0 X1 · · · Xn Xn+1 d0 d1 dn−1 dn is an X-admissible n-exact sequence, we say that d0 is an X-admissible monomorphism and that dn is an X-admissible epimorphism. In analogy with =-=[17]-=-, we depict X-admissible monomorphisms by and X-admissible epimorphisms by ։. A sequence → · · · →։ of n + 1 morphisms always denotes an X-admissible n-exact sequence. When the class X is clear fro... |

23 | Preprojective algebras and cluster algebras, preprint
- Geiss, Leclerc, et al.
- 2008
(Show Context)
Citation Context ... module category of a preprojective algebra of Dynkin type, which has infinite global dimension, are central in Geiß-Leclerc-Schröer’s categorification of cluster algebras arising in Lie theory, see =-=[22]-=- and the references therein. Further examples of n-cluster-tilting subcategories of abelian and exact categories have been constructed by Amiot-Iyama-Reiten in the category of CohenMacaulay modules ov... |

21 | n-representation-finite algebras and n-APR tilting
- Iyama, Oppermann
(Show Context)
Citation Context ...are one of the central objects of study of higher Auslander-Reiten theory. A distinguished class of such algebras, the socalled n-representation-finite algebras, were introduced by Iyama-Oppermann in =-=[31]-=- and have been studied in greater detail by Herschend-Iyama in the case n = 2, see [25]. In a parallel direction, 2-cluster-tilting subcategories of the module category of a preprojective algebra of D... |

21 | Stable categories of higher preprojective algebras - Iyama, Oppermann |

20 | The cluster category of a canonical algebra,
- Barot, Kussin, et al.
- 2010
(Show Context)
Citation Context ..., if T is a 2-cluster-tilting object in C such that Σ2T ∼= T , then addT ⊆ C is an algebraic 4-angulated category. • Let C = CX be the cluster category associated with a weighted projective line, see =-=[9]-=- for details. As in the previous case, a 2-cluster-tilting object T ∈ C satisfies Σ2T ∼= T if and only if C(T, T ) is a selfinjective algebra. All such algebras are classified in [34, Thm. 1.3]. Such ... |

18 | Stable categories of Cohen-Macaulay modules and cluster categories. arXiv:1104.3658
- Amiot, Iyama, et al.
(Show Context)
Citation Context .... Further examples of n-cluster-tilting subcategories of abelian and exact categories have been constructed by Amiot-Iyama-Reiten in the category of CohenMacaulay modules over an isolated singularity =-=[1]-=-. Finally, let us give a brief description of the contents of this article. In Section 2 we introduce the basic concepts behind the definitions of n-abelian and n-exact categories: n-cokernels, n-kern... |

15 | n-angulated categories,
- Geiss, Keller, et al.
- 2013
(Show Context)
Citation Context ...phisms for some k ∈ { 0, 1, . . . , n+ 1 }. Abusing the terminology, we say that two n-Σ-sequences are weakly isomorphic if they are connected by a finite zigzag of weak isomorphisms. Definition 5.1. =-=[20]-=- A pre-(n + 2)-angulated category is a triple (F,Σ, S) where F is an additive category, Σ: F → F is an automorphism1, and S is a class of nΣ-sequences (whose members we call (n + 2)-angles) which sati... |

13 | Selfinjective quivers with potential and 2-representation-finite algebras,
- Herschend, Iyama
- 2011
(Show Context)
Citation Context ...hed class of such algebras, the socalled n-representation-finite algebras, were introduced by Iyama-Oppermann in [31] and have been studied in greater detail by Herschend-Iyama in the case n = 2, see =-=[25]-=-. In a parallel direction, 2-cluster-tilting subcategories of the module category of a preprojective algebra of Dynkin type, which has infinite global dimension, are central in Geiß-Leclerc-Schröer’s... |

11 | Tubular cluster algebras I: categorification, - Barot, Geiss - 2012 |

11 |
Auslander algebras and initial seeds for cluster algebras
- Geiß, Leclerc, et al.
(Show Context)
Citation Context ...s abelian category hence the stable categorymodΛ is triangulated. It is known that the standard 2-cluster-tilting Λ-module T corresponding to the linear orientation of An satisfies ℧ 2(T ) ∼= T , see =-=[21]-=-. It follows that addT ⊆ modΛ is a Frobenius 2-exact category and thus addT ⊆ modΛ is an algebraic 4-angulated category. • Let Λ be an n-representation-finite algebra. Then, [32, Cor. 3.7] implies tha... |

10 | The self-injective cluster-tilted algebras,
- Ringel
- 2008
(Show Context)
Citation Context ...tegory [37]. Moreover, a basic 2-cluster-tilting object T ∈ C satisfies Σ2T ∼= T if and only if C(T, T ) is a selfinjective algebra, see [32, Cor. 3.8]. All such algebras were classified by Ringel in =-=[43]-=-. In particular, such algebras exist only if Q is a Dynkin quiver of type D (including D3 = A3). Hence, if T is a 2-cluster-tilting object in C such that Σ2T ∼= T , then addT ⊆ C is an algebraic 4-ang... |

7 | Ampleness of two-sided tilting complexes, - Minamoto - 2012 |

4 |
The Grothendieck group of an n-angulated category
- Bergh, Thaule
(Show Context)
Citation Context ...es which are closed under the n-ABELIAN AND n-EXACT CATEGORIES 3 n-th power of the shift functor [20, Thm. 1]. The properties of (n + 2)-angulated categories have been investigated by Bergh-Thaule in =-=[12, 13, 14]-=-. The aim of this article is to introduce n-abelian categories which are categories inhabited by certain exact sequences with n + 2 terms, called n-exact sequences. The case n = 1 corresponds to the c... |

4 | n-representation infinite algebras
- Herschend, Iyama, et al.
(Show Context)
Citation Context ...iated to KQ where Q is a Dynkin quiver ~Am, see [25, Sec. 9.2]. 6.2. n-representation infinite algebras. The class of n-representation-infinite algebras was introduced by Herschend-Iyama-Oppermann in =-=[27]-=- as a higher analog of representation-infinite hereditary algebras from the viewpoint of higher AuslanderReiten theory. These class of algebras complements that of n-representation-finite algebras. Le... |

3 |
The axioms for n-angulated categories
- Bergh, Thaule
- 2013
(Show Context)
Citation Context ...es which are closed under the n-ABELIAN AND n-EXACT CATEGORIES 3 n-th power of the shift functor [20, Thm. 1]. The properties of (n + 2)-angulated categories have been investigated by Bergh-Thaule in =-=[12, 13, 14]-=-. The aim of this article is to introduce n-abelian categories which are categories inhabited by certain exact sequences with n + 2 terms, called n-exact sequences. The case n = 1 corresponds to the c... |

3 |
Higher n-angulations from local algebras. arXiv:1311.2089
- Bergh, Thaule
- 2013
(Show Context)
Citation Context ...es which are closed under the n-ABELIAN AND n-EXACT CATEGORIES 3 n-th power of the shift functor [20, Thm. 1]. The properties of (n + 2)-angulated categories have been investigated by Bergh-Thaule in =-=[12, 13, 14]-=-. The aim of this article is to introduce n-abelian categories which are categories inhabited by certain exact sequences with n + 2 terms, called n-exact sequences. The case n = 1 corresponds to the c... |

2 | Representation theory of geiglelenzing complete intersections
- Herschend, Iyama, et al.
(Show Context)
Citation Context ...nstruct (n + 2)-angulated categories is explained in Theorem 5.16. Now we explain the notion of 2-exact category with concrete examples. The first example is due to Herschend-Iyama-Minamoto-Oppermann =-=[26]-=- (see also Theorem 6.8 and the example after it). Let cohP2K be the category of coherent sheaves over the projective plane over K, and denote the category of vector bundles over P 2 K by vectP 2 K . N... |

1 |
Framed triangulated categories.Math
- Bondal, Kapranov
- 1990
(Show Context)
Citation Context ... Thm. I.2.6]. Triangulated categories arising in this way have been called algebraic by Keller in [36]. Algebraic triangulated categories have a natural dg-enhancement in the sense of Bondal-Kapranov =-=[15]-=-, thus are often considered as a more reasonable class than that of general triangulated categories. Recently, a new class of additive categories appeared in representation theory. The 2-cluster-tilti... |

1 | τ2-stable tilting complexes over weighted projective lines. arXiv:1402.6036 - Jasso - 2014 |

1 | Sous les catégories dérivées. Comptes Rendus des Séances de l’Académie des Sciences - Keller, Vossieck - 1987 |