Loading [MathJax]/jax/output/SVG/jax.js
Research article

Relative cluster tilting subcategories in an extriangulated category

  • Received: 02 August 2022 Revised: 04 January 2023 Accepted: 08 January 2023 Published: 31 January 2023
  • Let B be an extriangulated category which admits a cluster tilting subcategory T. We firstly introduce notions of T-cluster tilting subcategories and related subcategories. Then we prove there is a correspondence between T-cluster tilting subcategories of B and support τ-tilting pairs of modΩ(T_), which recovers several main results from the literature. Note that the generalization is nontrivial and we give a new proof technique.

    Citation: Zhen Zhang, Shance Wang. Relative cluster tilting subcategories in an extriangulated category[J]. Electronic Research Archive, 2023, 31(3): 1613-1624. doi: 10.3934/era.2023083

    Related Papers:

    [1] Yajun Ma, Haiyu Liu, Yuxian Geng . A new method to construct model structures from left Frobenius pairs in extriangulated categories. Electronic Research Archive, 2022, 30(8): 2774-2787. doi: 10.3934/era.2022142
    [2] Haiyu Liu, Rongmin Zhu, Yuxian Geng . Gorenstein global dimensions relative to balanced pairs. Electronic Research Archive, 2020, 28(4): 1563-1571. doi: 10.3934/era.2020082
    [3] Dongxing Fu, Xiaowei Xu, Zhibing Zhao . Generalized tilting modules and Frobenius extensions. Electronic Research Archive, 2022, 30(9): 3337-3350. doi: 10.3934/era.2022169
    [4] Agustín Moreno Cañadas, Robinson-Julian Serna, Isaías David Marín Gaviria . Zavadskij modules over cluster-tilted algebras of type $ \mathbb{A} $. Electronic Research Archive, 2022, 30(9): 3435-3451. doi: 10.3934/era.2022175
    [5] Shengxiang Wang, Xiaohui Zhang, Shuangjian Guo . The Hom-Long dimodule category and nonlinear equations. Electronic Research Archive, 2022, 30(1): 362-381. doi: 10.3934/era.2022019
    [6] Jiangsheng Hu, Dongdong Zhang, Tiwei Zhao, Panyue Zhou . Balance of complete cohomology in extriangulated categories. Electronic Research Archive, 2021, 29(5): 3341-3359. doi: 10.3934/era.2021042
    [7] János Kollár . Relative mmp without $ \mathbb{Q} $-factoriality. Electronic Research Archive, 2021, 29(5): 3193-3203. doi: 10.3934/era.2021033
    [8] Yilin Wu, Guodong Zhou . From short exact sequences of abelian categories to short exact sequences of homotopy categories and derived categories. Electronic Research Archive, 2022, 30(2): 535-564. doi: 10.3934/era.2022028
    [9] Peigen Cao, Fang Li, Siyang Liu, Jie Pan . A conjecture on cluster automorphisms of cluster algebras. Electronic Research Archive, 2019, 27(0): 1-6. doi: 10.3934/era.2019006
    [10] Ali Aytekin, Kadir Emir . Colimits of crossed modules in modified categories of interest. Electronic Research Archive, 2020, 28(3): 1227-1238. doi: 10.3934/era.2020067
  • Let B be an extriangulated category which admits a cluster tilting subcategory T. We firstly introduce notions of T-cluster tilting subcategories and related subcategories. Then we prove there is a correspondence between T-cluster tilting subcategories of B and support τ-tilting pairs of modΩ(T_), which recovers several main results from the literature. Note that the generalization is nontrivial and we give a new proof technique.



    In [1] (see [2] for type A), the authors introduced cluster categories which were associated to finite dimensional hereditary algebras. It is well known that cluster-tilting theory gives a way to construct abelian categories from some triangulated and exact categories.

    Recently, Nakaoka and Palu introduced extriangulated categories in [3], which are a simultaneous generalization of exact categories and triangulated categories, see also [4,5,6]. Subcategories of an extriangulated category which are closed under extension are also extriangulated categories. However, there exist some other examples of extriangulated categories which are neither exact nor triangulated, see [6,7,8].

    When T is a cluster tilting subcategory, the authors Yang, Zhou and Zhu [9, Definition 3.1] introduced the notions of T[1]-cluster tilting subcategories (also called ghost cluster tilting subcategories) and weak T[1]-cluster tilting subcategories in a triangulated category C, which are generalizations of cluster tilting subcategories. In these works, the authors investigated the relationship between C and modT via the restricted Yoneda functor G more closely. More precisely, they gave a bijection between the class of T[1]-cluster tilting subcategories of C and the class of support τ-tilting pairs of modT, see [9, Theorems 4.3 and 4.4].

    Inspired by Yang, Zhou and Zhu [9] and Liu and Zhou [10], we introduce the notion of relative cluster tilting subcategories in an extriangulated category B. More importantly, we want to investigate the relationship between relative cluster tilting subcategories and some important subcategories of modΩ(T)_ (see Theorem 3.9 and Corollary 3.10), which generalizes and improves the work by Yang, Zhou and Zhu [9] and Liu and Zhou [10].

    It is worth noting that the proof idea of our main results in this manuscript is similar to that in [9, Theorems 4.3 and 4.4], however, the generalization is nontrivial and we give a new proof technique.

    Throughout the paper, let B denote an additive category. The subcategories considered are full additive subcategories which are closed under isomorphisms. Let [X](A,B) denote the subgroup of HomB(A,B) consisting of morphisms which factor through objects in a subcategory X. The quotient category B/[X] of B by a subcategory X is the category with the same objects as B and the space of morphisms from A to B is the quotient of group of morphisms from A to B in B by the subgroup consisting of morphisms factor through objects in X. We use Ab to denote the category of abelian groups.

    In the following, we recall the definition and some properties of extriangulated categories from [4], [11] and [3].

    Suppose there exists a biadditive functor E:Bop×BAb. Let A,CB be two objects, an element δE(C,A) is called an E-extension. Zero element in E(C,A) is called the split E-extension.

    Let s be a correspondence, which associates any E-extension δE(C,A) to an equivalence class s(δ)=[AxByC]. Moreover, if s satisfies the conditions in [3, Definition 2.9], we call it a realization of E.

    Definition 2.1. [3, Definition 2.12] A triplet (B,E,s) is called an externally triangulated category, or for short, extriangulated category if

    (ET1) E:Bop×BAb is a biadditive functor.

    (ET2) s is an additive realization of E.

    (ET3) For a pair of E-extensions δE(C,A) and δE(C,A), realized as s(δ)=[AxByC] and s(δ)=[AxByC]. If there exists a commutative square,

    then there exists a morphism c:CC which makes the above diagram commutative.

    (ET3)op Dual of (ET3).

    (ET4) Let δ and δ be two E-extensions realized by AfBfD and BgCgF, respectively. Then there exist an object EB, and a commutative diagram

    and an E-extension δ realized by AhChE, which satisfy the following compatibilities:

    (i). DdEeF realizes E(F,f)(δ),

    (ii). E(d,A)(δ)=δ,

    (iii). E(E,f)(δ)=E(e,B)(δ).

    (ET4op) Dual of (ET4).

    Let B be an extriangulated category, we recall some notations from [3,6].

    ● We call a sequence XxYyZ a conflation if it realizes some E-extension δE(Z,X), where the morphism x is called an inflation, the morphism y is called an deflation and XxYyZδ is called an E-triangle.

    ●When XxYyZδ is an E-triangle, X is called the CoCone of the deflation y, and denote it by CoCone(y); C is called the Cone of the inflation x, and denote it by Cone(x).

    Remark 2.2. 1) Both inflations and deflations are closed under composition.

    2) We call a subcategory T extension-closed if for any E-triangle XxYyZδ with X, ZT, then YT.

    Denote I by the subcategory of all injective objects of B and P by the subcategory of all projective objects.

    In an extriangulated category having enough projectives and injectives, Liu and Nakaoka [4] defined the higher extension groups as

    Ei+1(X,Y)=E(Ωi(X),Y)=E(X,Σi(Y)) for i0.

    By [3, Corollary 3.5], there exists a useful lemma.

    Lemma 2.3. For a pair of E-triangles LlMmN and DdEeF. If there is a commutative diagram

    f factors through l if and only if h factors through e.

    In this section, B is always an extriangulated category and T is always a cluster tilting subcategory [6, Definition 2.10].

    Let A, BB be two objects, denote by [¯T](A,ΣB) the subset of B(A,ΣB) such that f[¯T](A,ΣB) if we have f: ATΣB where TT and the following commutative diagram

    where I is an injective object of B [10, Definition 3.2].

    Let M and N be two subcategories of B. The notation [¯T](M,Σ(N))=[T](M,Σ(N)) will mean that [¯T](M,ΣN)=[T](M,ΣN) for every object MM and NN.

    Now, we give the definition of T-cluster tilting subcategories.

    Definition 3.1 Let X be a subcategory of B.

    1) [11, Definition 2.14] X is called T-rigid if [¯T](X,ΣX)=[T](X,ΣX);

    2) X is called T-cluster tilting if X is strongly functorially finite in B and X={MC[¯T](X,ΣM)=[T](X,ΣM) and [¯T](M,ΣX)=[T](M,ΣX)}.

    Remark 3.2. 1) Rigid subcategories are always T-rigid by [6, Definition 2.10];

    2) T-cluster tilting subcategories are always T-rigid;

    3) T-cluster tilting subcategories always contain the class of projective objects P and injective objects I.

    Remark 3.3. Since T is a cluster tilting subcategory, XB, there exists a commutative diagram by [6, Remark 2.11] and Definition 2.1((ET4)op), where T1, T2T and h is a left T-approximation of X:

    Hence XB, there always exists an E-triangle

    Ω(T1)fXΩ(T2)X with TiT.

    By Remark 3.2(3), PT and B=CoCone(T,T) by [6, Remark 2.11(1),(2)]. Following from [4, Theorem 3.2], B_=B/T is an abelian category. fB(A,C), denote by f_ the image of f under the natural quotient functor BB_.

    Let Ω(T)=CoCone(P,T), then Ω(T)_ is the subcategory consisting of projective objects of B_ by [4, Theorem 4.10]. Moreover, modΩ(T)_ denotes the category of coherent functors over the category of Ω(T)_ by [4, Fact 4.13].

    Let G: BmodΩ(T)_, MHomB_(,M)Ω(T)_ be the restricted Yoneda functor. Then G is homological, i.e., any E-triangle XYZ in B yields an exact sequence G(X)G(Y)G(Z) in modΩ(T)_. Similar to [9, Theorem 2.8], we obtain a lemma:

    Lemma 3.4. Denote proj(modΩ(T)_) the subcategory of projective objects in modΩ(T)_. Then

    1) G induces an equivalence Ω(T)proj(modΩ(T)_).

    2) For NmodΩ(T)_, there exists a natural isomorphism

    HommodΩ(T)_(G(Ω(T)),N)N(Ω(T)).

    In the following, we investigate the relationship between B and modΩ(T)_ via G more closely.

    Lemma 3.5. Let X be any subcategory of B. Then

    1) any object XX, there is a projective presentation in mod Ω(T)_

    PG(X)1πG(X)PG(X)0G(X)0.

    2) X is a T-rigid subcategory if and only if the class {πG(X)XX} has property ((S) [9, Definition 2.7(1)]).

    Proof. 1). By Remark 3.3, there exists an E-triangle:

    Ω(T1)fXΩ(T0)X

    When we apply the functor G to it, there exists an exact sequence G(Ω(T1))G(Ω(T0))G(X)0. By Lemma 3.4(1), G(Ω(Ti)) is projective in mod Ω(T)_. So the above exact sequence is the desired projective presentation.

    2). For any X0X, using the similar proof to [9, Lemma 4.1], we get the following commutative diagram

    where α=HommodΩT_(πG(X),G(X0)). By Lemma 3.4(2), both the left and right vertical maps are isomorphisms. Hence the set {πG(X)XX} has property ((S) iff α is epic iff HomB_(fX,X0) is epic iff X is a T-rigid subcategory by [10, Lemma 3.6].

    Lemma 3.6. Let X be a T-rigid subcategory and T1 a subcategory of T. Then XT1 is a T-rigid subcategory iff E(T1,X)=0.

    Proof. For any MXT1, then M=XT1 for XX and T1T1. Let h: XT be a left T-approximation of X and y: T1Σ(X) for XX any morphism. Then there exists the following commutative diagram

    with P1P, f=(h001) and β=(i000i1).

    When XT1 is a T-rigid subcategory, we can get a morphism g: XT1Σ(X)Σ(T1) such that βg=(10)y(0 1)f. i.e., b: T1I such that y=i0b. So E(T1,X)=0 and then E(T1,X)=0.

    Let γ=(r11r12r21r22): TT1Σ(X)Σ(T1) be a morphism. As X is T-rigid, r11h: XΣ(X) factors through i0. Since E(T,X)=0, r12: T1Σ(X) factors through i0. As T is rigid, the morphism r21h: XTΣ(T1) factors through i1, and the morphism r22: T1Σ(T1) factors through i1. So the morphism γf can factor through β=(i000i1). Therefore XT1 is an T-rigid subcategory.

    For the definition of τ-rigid pair in an additive category, we refer the readers to see [9, Definition 2.7].

    Lemma 3.7. Let U be a class of T-rigid subcategories and V a class of τ-rigid pairs of modΩ(T)_. Then there exists a bijection φ: UV, given by : X(G(X),Ω(T)Ω(X)).

    Proof. Let X be T-rigid. By Lemma 3.5, G(X) is a τ-rigid subcategory of mod Ω(T)_.

    Let YΩ(T)Ω(X), then there exists X0X such that Y=Ω(X0). Consider the E-triangle Ω(X0)PX0 with PP. XX, applying HomB(,X) yields an exact sequence HomB(P,X)HomB(Ω(X0),X)E(X0,X)0. Hence in B_=B/T, HomB_(Ω(X0),X)E(X0,X).

    By Remark 3.3, for X0, there is an E-triangle Ω(T1)Ω(T2)X0 with T1, T2T. Applying HomB_(,X), we obtain an exact sequence HomB_(Ω(T2),X)HomB_(Ω(T1),X)E(X0,X)E(Ω(T2),X). By [10, Lemma 3.6], HomB_(Ω(T2),X)HomB_(Ω(T1),X) is epic. Moreover, Ω(T2)_ is projective in B_ by [4, Proposition 4.8]. So E(Ω(T2),X)=0. Thus E(X0,X)=0. Hence XX,

    G(X)(Y)=HomB_(Ω(X0),X)=0.

    So (G(X),Ω(T)Ω(X)) is a τ-rigid pairs of modΩ(T)_.

    We will show φ is a surjective map.

    Let (N,σ) be a τ-rigid pair of modΩ(T)_. NN, consider the projective presentation

    P1πNP0N0

    such that the class {πN|NN} has Property (S). By Lemma 3.4, there exists a unique morphism fN: Ω(T1)Ω(T0) in Ω(T)_ satisfying G(fN)=πN and G(Cone(fN))N. Following from Lemma 3.5, X1:= {cone(fN)NN} is a T-rigid subcategory.

    Let X=X1Y, where Y={TTΩ(T)σ}. For any T0Y, there is an E-triangle Ω(T0)PT0 with PP. For any Cone(fN)X1, applying HomB_(,Cone(fN)), yields an exact sequence HomB_(Ω(T0),Cone(fN))E(T0,Cone(fN))E(P,Cone(fN))=0. Since (N,σ) is a τ-rigid pair, HomB_(Ω(T0),Cone(fN))=G(Cone(fM))(Ω(T0))=0. So E(T0,Cone(fN))=0. Due to Lemma 3.6, X=X1Y is T-rigid. Since YT, we get G(Y)=HomB_(,T)Ω(T)=0 by [4, Lemma 4.7]. So G(X)=G(X1)=N.

    It is straightforward to check that Ω(T)Ω(X1)=0. Let XΩ(T)Ω(X), then XΩ(T) and XΩ(X)=Ω(X1)σ. So we can assume that X=Ω(X1)E, where Eσ. Then Ω(X1)EΩ(T). Since EΩ(T), we get Ω(X1)Ω(T)Ω(X1)=0. So Ω(T)Ω(X)σ. Clearly, σΩ(T). Moreover, σΩ(X). So σΩ(T)Ω(X). Hence Ω(T)Ω(X)=σ. Therefore φ is surjective.

    Lastly, φ is injective by the similar proof method to [9, Proposition 4.2].

    Therefore φ is bijective.

    Lemma 3.8. Let T be a rigid subcategory and AaBCδ an E-triangle satisfying [¯T](C,Σ(A))=[T](C,Σ(A)). If there exist an E-extension γE(T,A) and a morphism t: CT with TT such that tγ=δ, then the E-triangle AaBCδ splits.

    Proof. Applying HomB(T,) to the E-triangle AIiΣ(A)α with II, yields an exact sequence HomB(T,A)E(T,X)E(T,I)=0. So there is a morphism dHomB(T,Σ(A)) such that γ=dα. So δ=tγ=tdα=(dt)α. So we have a diagram which is commutative:

    Since [¯T](C,Σ(A))=[T](C,Σ(A)) and dt[T](C,Σ(A)), dt can factor through i. So 1A can factor through a and the result follows.

    Now, we will show our main theorem, which explains the relation between T-cluster tilting subcategories and support τ-tilting pairs of modΩ(T)_.

    The subcategory X is called a preimage of Y by G if G(X)=Y.

    Theorem 3.9. There is a correspondence between the class of T-cluster tilting subcategories of B and the class of support τ-tilting pairs of modΩ(T)_ such that the class of preimages of support τ-tilting subcategories is contravariantly finite in B.

    Proof. Let φ be the bijective map, such that X(G(X),Ω(TΩ(X))), where G is the restricted Yoneda functor defined in the argument above Lemma 3.4.

    1). The map φ is well-defined.

    If Xis T-cluster tilting, then X is T-rigid. So φ(X) is a τ-rigid pair of modΩ(T)_ by Lemma 3.7. Therefore Ω(T)Ω(X)KerG(X). Assume Ω(T0)Ω(T) is an object of KerG(X). Then HomB_(Ω(T0),X)=0. Applying HomB_(,X) with XX to Ω(T0)PT0 with PP, yields an exact sequence

    HomB_(P,X)HomB_(Ω(T),X)E(T0,X)0.

    Hence we get E(T0,X)HomB_(Ω(T0),X)=0.

    Applying HomB(T0,) to XIΣ(X), we obtain

    (3.1)      [¯T](T0,Σ(X))=[T](T0,Σ(X)).

    For any ba: XaRbΣ(T0) with RT, as T is rigid, we get a commutative diagram:

    Hence we get (3.2)[¯T](X,Σ(T0))=[T](X,Σ(T0)).

    By the equalities (3.1) and (3.2) and X being a T-rigid subcategory, we obtain

    [¯T](X,Σ(XT0))=[T](X,Σ(XT0)) and [¯T](XT0,Σ(X))=[T](XT0,Σ(X)).

    As X is T-cluster tilting, we get XT0X. So T0X. And thus Ω(T0)Ω(T)Ω(X). Hence KerG(X)=Ω(T)Ω(X).

    Since X is functorially finte, similar to [6, Lemma 4.1(2)], Ω(T)Ω(T), we can find an E-triangle Ω(T)fX1X2, where X1, X2X and f is a left X-approximation. Applying G, yields an exact sequence

    G(Ω(R))G(f)G(X1)G(X2)0.

    Thus we get a diagram which is commutative, where HomB_(f,X) is surjective.

    By Lemma 3.4, the morphism G(f) is surjective. So G(f) is a left G(X)-approximation and (G(X),Ω(T)Ω(X)) is a support τ-tilting pair of modΩ(T)_ by [3, Definition 2.12].

    2). φ is epic.

    Assume (N,σ) is a support τ-tilting pair of modΩ(T)_. By Lemma 3.7, there is a T-rigid subcategory X satisfies G(X)=N. So Ω(T)Ω((T)), there is an exact sequence G(Ω(T))αG(X3)G(X4)0, such that X3, X4X and α is a left G(X)-approximation. By Yoneda's lemma, we have a unique morphism in modΩ((T))_:

    β: Ω(T)X3 such that α=G(β) and G(cone(β))G(X4).

    Moreover, XX, consider the following commutative diagram

    By Lemma 3.4, G() is surjective. So the map HomB_(β,X) is surjective.

    Denote Cone(β) by YR and Xadd{YRΩ(T)Ω(T)} by ˜X.

    We claim ˜X is T-rigid.

    (I). Assume a: YRa1T0a2Σ(X) with T0T and XX. Consider the following diagram:

    Since X is T-rigid, f: X3I such that aγ=if. So there is a morphism g:Ω(T)X making the upper diagram commutative. Since HomB_(β,X) is surjective, g factors through β. Hence a factors through i, i.e., [¯T](YR,Σ(X))=[T](YR,Σ(X)).

    (II). For any morphism b: Xb1T0b2Σ(YR) with T0T and XX. Consider the following diagram:

    By [3, Lemma 5.9], RΣ(X3)Σ(YT) is an E-triangle. Because T is rigid, b2 factors through γ1. By the fact that X is T-rigid, b=b2b1 can factor through iX. Since γ1iX=iY, we get that b factors through iY. So [¯T](X,Σ(YT))=[T](X,Σ(YT)).

    By (I) and (II), we also obtain [¯T](YT,Σ(YT))=[T](YT,Σ(YT)).

    Therefore ˜X=Xadd{YTΩ(T)Ω(T)} is T-rigid.

    Let MB satisfying [¯T](M,Σ(˜X))=[T](M,Σ(˜X)) and [¯T](˜X,ΣM)=[T](˜X,ΣM). Consider the E-triangle:

    Ω(T5)fΩ(T6)gM

    where T5, T6T. By the above discussion, there exist two E-triangles:

    Ω(T6)uX6vY6 and Ω(T5)uX5vY5.

    where X5, X6X, u and u are left X-approximations of Ω(T6), Ω(T5), respectively. So there exists a diagram of E-triangles which is commutative:

    We claim that the morphism x=uf is a left X-approximation of Ω(T5). In fact, let XX and d: Ω(T5)X, we can get a commutative diagram of E-triangles:

    where PP. By the assumption, [¯T](M,Σ(X))=[T](M,Σ(X)). So d2h factors through iX. By Lemma 2.3, d factors through f. Thus f1: Ω(T6)X such that d=f1f. Moreover, u is a left X-approximation of Ω(T6). So u1: X6X such that f1=u1u. Thus d=f1f=u1uf=u1x. So x=uf is a left X-approximation of Ω(T5).

    Hence there is a commutative diagram:

    By [3, Corollary 3.16], we get an E-triangle X6(yλ)NX5Y5xδ5

    Since u is a left X-approximation of Ω(T5), there is also a commutative diagram with PP:

    such that δ5=tμ. So xδ5=xtμ=txμ. By Lemma 3.8, the E-triangle xδ5 splits. So NX5X6Y5˜X. hence N˜X.

    Similarly, consider the following commutative diagram with PP:

    and the E-triangle MNYgδ6. Then t: YT6 such that δ6=tδ. Then gδ6=gtδ=t(gδ). Since [¯T](˜X,ΣM)=[T](˜X,ΣM), the E-triangle gδ6 splits by Lemma 3.5 and M is a direct summands of N. Hence M˜X.

    By the above, we get ˜X is a T-cluster tilting subcategory.

    By the definition of YR, G(YR)G(X). So G(˜X)G(X)N. Moreover, σ=Ω(T)Ω(X)Ω(T)Ω(˜X) and Ω(T)Ω(˜X)kerG(X)=σ. So Ω(T)Ω(˜X)=σ. Hence φ is surjective.

    3). φ is injective following from the proof of Lemma 3.7.

    By [4, Proposition 4.8 and Fact 4.13], B_modΩ(T)_. So it is easy to get the following corollary by Theorem 3.9:

    Corollary 3.10. Let X be a subcategory of B.

    1) X is T-rigid iff X_ is τ-rigid subcategory of B_.

    2) X is T-cluster tilting iff X_ is support τ-tilting subcategory of B_.

    If let H=CoCone(T,T), then H can completely replace B and draw the corresponding conclusion by the proof Lemma 3.7 and Theorem 3.9, which is exactly [12, Theorem 3.8]. If let B is a triangulated category, then Theorem 3.9 is exactly [9, Theorem 4.3].

    This research was supported by the National Natural Science Foundation of China (No. 12101344) and Shan Dong Provincial Natural Science Foundation of China (No.ZR2015PA001).

    The authors declare they have no conflict of interest.



    [1] B. Buan, R. Marsh, M. Reineke, I. Reiten, G. Todorov, Tilting theory and cluster combinatorics, Adv. Math., 204 (2006), 572–618. https://doi.org/10.1016/j.aim.2005.06.003 doi: 10.1016/j.aim.2005.06.003
    [2] P. Caldero, F. Chapoton, R. Schiffler, Quivers with relations arising from clusters (An case), Trans. Am. Math. Soc., 358 (2006), 1347–1364. https://doi.org/10.1090/s0002-9947-05-03753-0 doi: 10.1090/s0002-9947-05-03753-0
    [3] H. Nakaoka, Y. Palu, Extriangulated categories, Hovey twin cotorsion pairs and model structures, Cah. Topol. Geom. Differ. Categ., 60 (2019), 117–193.
    [4] Y. Liu, H. Nakaoka, Hearts of twin cotorsion pairs on extriangulated categories, J. Algebra, 528 (2019), 96–149. https://doi.org/10.1016/j.jalgebra.2019.03.005 doi: 10.1016/j.jalgebra.2019.03.005
    [5] T. Zhao, Z. Huang, Phantom ideals and cotorsion pairs in extriangulated categories, Taiwan. J. Math., 23 (2019), 29–61. https://doi.org/10.11650/TJM/180504 doi: 10.11650/TJM/180504
    [6] P. Zhou, B. Zhu, Cluster tilting subcategories in extriangulated categories, Theory Appl. Categ., 34 (2019), 221–242.
    [7] J. He, P. Zhou, On the relation between n-cotorsion pairs and (n+1)-cluster tilting subcategories, J. Algebra Appl., 21 (2022), 2250011. https://doi.org/10.1142/S0219498822500116 doi: 10.1142/S0219498822500116
    [8] P. Zhou, B. Zhu, Triangulated quotient categories revisited, J. Algebra, 502 (2018), 196–232. https://doi.org/10.1016/j.jalgebra.2018.01.031 doi: 10.1016/j.jalgebra.2018.01.031
    [9] W. Yang, P. Zhou, B. Zhu, Triangulated categories with cluster-tilting subcategories, Pac. J. Math., 301 (2019), 703–740. https://doi.org/10.2140/PJM.2019.301.703 doi: 10.2140/PJM.2019.301.703
    [10] Y. Liu, P. Zhou, Relative rigid objects in extriangulated categories, J. Pure Appl. Algebra, 226 (2022), 106923. https://doi.org/10.1016/J.JPAA.2021.106923 doi: 10.1016/J.JPAA.2021.106923
    [11] Y. Liu, P. Zhou, On the relation between relative rigid and support tilting, preprint, arXiv: 2003.12788V1. https://doi.org/10.48550/arXiv.2003.12788
    [12] Y. Liu, P. Zhou, Relative rigid subcategories and τ-tilting theory, Algebras Representation Theory, 25 (2022), 1699–1722. https://doi.org/10.1007/s10468-021-10082-6 doi: 10.1007/s10468-021-10082-6
  • Reader Comments
  • © 2023 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)
通讯作者: 陈斌, bchen63@163.com
  • 1. 

    沈阳化工大学材料科学与工程学院 沈阳 110142

  1. 本站搜索
  2. 百度学术搜索
  3. 万方数据库搜索
  4. CNKI搜索

Metrics

Article views(1857) PDF downloads(134) Cited by(0)

Other Articles By Authors

/

DownLoad:  Full-Size Img  PowerPoint
Return
Return

Catalog