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...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...

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...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...

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...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...

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...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...

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...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...

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.... 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...

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...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-...

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...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...

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... 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...

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...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...

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...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...

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...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 ...

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...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...

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... 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...

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...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...

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...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...

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.... 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...

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...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]). ...

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...· · · 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...

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...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...

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...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...

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...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 ...

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...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...

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... 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...

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...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...

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..., 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 ...

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.... 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...

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...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...

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...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...

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...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...

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...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...

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...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...

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...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...

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...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...

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...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...

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...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...

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... 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...