Research article

Weakly Gorenstein comodules over triangular matrix coalgebras

  • Received: 10 March 2022 Revised: 14 June 2022 Accepted: 15 June 2022 Published: 20 June 2022
  • MSC : 18G10, 18G25

  • In this paper, we characterise weakly Gorenstein injective and weakly Gorenstein coflat comodules over triangular matrix coalgebras by introducing the class of weakly compatible bicomodules. In particular, Gorenstein injective and Gorenstein coflat comodules are investigated.

    Citation: Dingguo Wang, Chenyang Liu, Xuerong Fu. Weakly Gorenstein comodules over triangular matrix coalgebras[J]. AIMS Mathematics, 2022, 7(8): 15471-15483. doi: 10.3934/math.2022847

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  • In this paper, we characterise weakly Gorenstein injective and weakly Gorenstein coflat comodules over triangular matrix coalgebras by introducing the class of weakly compatible bicomodules. In particular, Gorenstein injective and Gorenstein coflat comodules are investigated.



    The homological theory of comodules over coalgebras and Hopf algebras was introduced by Doi [5]. Auslander and Bridger defined Gorenstein projective modules by G-dimensions for finitely generated modules in [2]. Enoch and Jenda [6] developed the relative homological algebra, especially the Gorenstein homological algebra. Since then, the Gorenstein homological algebra has been developed rapidly and has obtained fruitful results in this field [12,18]. Asensio et al in [1] introduced Gorenstein injective comodules which is a generalization of injective comodules over any coalgebra. A coalgebra C is said to be right semiperfect [15] if the category MC has enough projectives. Recently, Meng introduced weakly Gorenstein injective and weakly Gorenstein coflat comodules over any coalgebra in [16], which proved that, for a left semiperfect coalgebra, weakly Gorenstein injective comodules is equivalent with weakly Gorenstein coflat comodules.

    Triangular matrix rings play a significant role in the study of classical ring theory and representation theory of algebras. Given two rings A, B, and A-B-bimodule M, one can form the upper triangular matrix ring Λ=(AM0B). A number of researchers have investigated the triangular matrix rings (algebras). The readers can review [11,20,22,23,24] and references therein for more details. Zhang studied the structure of Gorenstein-projective modules over triangular matrix algebras in [24]. Under some mild conditions, Zhang and Xiong [22] described all the modules in Λ, and obtained criteria for the Gorensteinness of Λ. As applications, they determined all the Gorenstein-projective Λ-modules if Λ is Gorenstein. Dually, given coalgebras C and D, a C-D-bicomodule U, Γ=(CU0D) can be made into a coalgebra, which is called triangular matrix coalgebras. The reader may refer to [8,10,13,14] and references therein. The comodule representation category over the Morita-Takeuchi context coalgebra Γ was studied in [10]. Moreover, the authors explicitely determined all Gorenstein injective comodules over the Morita-Takeuchi context coalgebra Γ.

    Motivated by the research mentioned above, we devote this paper to studying weakly Gorenstein injective and weakly Gorenstein coflat comodules over triangular matrix coalgebras by means of the relative homological theory in comodule categories.

    The main theorems of this paper are the following:

    Theorem 1.1 (Theorem 3.3). Let Γ=(CU0D) be the triangular matrix coalgebra, and U be a weakly compatible C-D-bicomodule. The following conditions are equivalent:

    (1) (X,Y,φ) is a weakly Gorenstein injective Γ-comodule.

    (2) X is a weakly Gorenstein injective C-comodule, kerφ is a weakly Gorenstein injective D-comodule, and φ:YXCU is surjetive.

    Theorem 1.2 (Theorem 4.5). Let Γ=(CU0D) be the triangular matrix coalgebra which is semiperfect, and U be a weakly compatible C-D-bicomodule. The following conditions are equivalent:

    (1) (X,Y,φ) is a weakly Gorenstein coflat Γ-comodule.

    (2) X is a weakly Gorenstein coflat C-comodule, kerφ is a weakly Gorenstein coflat D-comodule, and φ:YXCU is surjetive.

    In this section, we include some details to establish notation and for sake of completeness.

    Throughout this paper we fix an arbitrary field k. The reader is referred to [17] for the coalgebra and comodule terminology. Let C be a k-coalgebra with comultiplication Δ and counit ε. We recall that a let C-comodule is a k-vector space M together with a k-linear map ρM:MCM such that (Δid)ρM=(idρM)ρM and (εid)ρM=idM. A k-linear map f:MN between two left C-comodules M and N is a C-comodule homomorphism if ρNf=(idf)ρM. The k-vector space of all C-comodule homomorphisms from M to N is denoted by HomC(M,N). Similarly we can define a right C-comodule. Let MC and CM denotes the category of right and left C-comodules respectively. For any MMC and N CM. Following [5,9], we recall that the cotensor product MCN is the k-vector space

    where ρM and ρN are the structure maps of M and N, respectively.

    Let C,D and E be three coalgebras. If M is a left E-comodule with structure map ρM:MEM, and also a right C-comodule with structure map ρ+M:MMC such that (Iρ+M)ρM=(ρMI)ρ+M, we then say that M is an (E,C)-bicomodule. We let EMC denote the category of (E,C)-bicomodules. Let N be a (C,D)-bicomodule. Then MCN acquires a structure of (E,D)-bicomodule with structure maps

    ρMCI:MCN(EkM)CNEk(MCN), and 
    ICρ+N:MCNMC(NkD)(MCN)kD.

    It is known that MkN=MkN, MCCM,CCNN, (MCN)DLMC(NDL) for any L DM, i.e. the cotensor product is associative and the functors

    MC: CMMk and CN:MCMk

    are left exact, commute with arbitrary direct sums.

    For every right exact exact linear functor F:MCMD preserving direct sums, there exists a (C,D)-bicomodule X, in fact X=F(C), such that F is naturally isomorphic to the funtor CX (See [19,Proposition 2.1]). Since every comodule is the union of its finite-dimensional subcomodules, there is a functorial isomorphism

    MCNHomC(N,M)

    for any MMC and finite-dimensional N CM, where N=Homk(N,k) is equipped with the k-dual right C-comodule structure map

    NHomK(N,C)NC, α(Iα)ρN, αN (see[5,P.32]).

    This implies that the functor MC from CM to Mk is exact if and only if so is the functor HomC(,M) from MC to Mk, i.e., the functors MC (resp. CN) is exact if and only if M (resp. N) is an injective right (resp. left) C-comodule.

    Let U be a C-D-bicomodule, then we may consider the functor CU:MCMD. Unfortunately, in general, CU does not have a left adjoint functor. However, Takeuchi proved the following results:

    Theorem 2.1. [19,Proposition 1.10] Let C and D be two coalgebras, and U be a C-D-bicomodule. Then the functor CU:MCMD has a left adjoint functor if and only if U is a quasi-finite right D-comodule, i.e., HomD(F,U) is finite-dimensional for all finite-dimensional FMD.

    If U is a quasi-finite right D-comodule, we denote the left adjoint functor of CU by hD(U,). Then for any right C-comodule W and any D-comodule N, we have that

    HomD(N,WCU)HomC(hD(U,N),W),

    that is, (hD(U,),CU) is a adjoint pair with unit ϵ:IdMDhD(U,)CU and counit δ:hD(U,CU)IdMC. The functor hD(U,) has a behavior similar to the usual Hom functor of algebras.

    Proposition 2.2. Let C,D and E be three coalgebras, M and N be a (D,C)-bicomodule and an (E,C)-bicomodule, respectively, such that M is quasi-finite as right C-comodule. Then the following assertions hold:

    (a) We have hC(M,N)=limHomC(Nλ,M)=lim(MCNλ), where {Nλ} is the family of finite dimensional subcomodules of C-comodule N (See the proof of [19,Proposition 1.3, P.633] and [5,P.32]).

    (b) The vector space hC(M,N) is an (E,D)-bicomodule (See [19,1.7-1.9, P.634]).

    (c) The functor hC(M,) is right exact and preserves all direct limits and direct sums (See [19,1.6, P.634]).

    (d) The functor hC(M,) is exact if and only if M is injective as right C-comodule (See [19,1.12, P.635]).

    Remark 2.3. The set CoendC(M)=hC(M,M) has an structure of k-coalgebra and then M becomes a (CoendC(M),C)-bicomodule, see [19,Proposition 2.1] for details. Symmetrically, M DMC is quasi-finite as left D-comodule if and only if the functor MC: CM DM has a left adjoint functor. In this case we denote by hD(,M) that functor.

    For two k-coalgebras C and D, let U be a C-D-bicomodule with the left C-coaction on U, uu[1]u[0], and the right D-coaction on U, uu[0]u[1] (using Sweedler's convention with the summations symbol omitted). We recall from [4,13,14,21] that Γ=(CU0D) can be made into a coalgebra by defining the comutiplication Δ:ΓΓΓ and the counit ε:ΓK as follows

    Δ((cu0d))=(c1000)(c2000)+(u[1]000)(0u[0]00)+(0u[0]00)(000u[1])+(000d1)(000d2)ε((cu0d))=εC(c)+εD(d).

    The coalgebra Γ is called a triangular matrix coalgebra.

    We know from [14] that the right comodule category MΓ and the comodule representation category R(Γ) are equivalent. The objects of R(Γ) are the triples (X,Y,φ), where X is a right C-comodule, Y is a right D-comodule, and φHomD(Y,XCU) is the right D-comodule morphism. For any two objects (X,Y,φ) and (X,Y,φ) in R(Γ), the morphism from (X,Y,φ) to (X,Y,φ) in R(Γ) is a pair of homomorphisms

    α = (α1,α2):(X,Y,φ)(X,Y,φ),

    where α1HomC(X,X) and α2HomD(Y,Y) such that the following diagram is commutative

    Next we define some exact functors between the right comodule category MΓ and the comodule representation category R(Γ).

    (1) For any objects X and X in MC, and any right C-comodule morphism α:XX, the functor TC:MCR(Γ) is given by TC(X)=(X,XCU,IdXCU) and TC(α)=(α,αIdU).

    (2) The functor UC:R(Γ)MC is defined by UC(X,Y,φ)=X, UC(α,β)=α for any objects (X,Y,φ) and (X,Y,φ) in R(Γ) and any right Γ-comodule morphism (α,β):(X,Y,φ)(X,Y,φ).

    (3) The functor UD:R(Γ)MD is defined by UD(X,Y,φ)=Y, UD(α,β)=β for any objects (X,Y,φ) and (X,Y,φ) in R(Γ) and any right Γ-comodule morphism (α,β):(X,Y,φ)(X,Y,φ).

    (4) The functor HD:MDR(Γ) is given by HD(Y)=(HomD(U,Y),Y,ϵY),HD(β)=(HomD(U,β),β) for any right D-comodule morphism β:YY.

    Remark 2.4. (i) If I is an indecomposable injective right C-comodule, then TC(I) is an indecomposable injective right Γ-comodule.

    (ii) If P is an indecomposable projective right D-comodule, then HD(P) is an indecomposable projective right Γ-comodule.

    (iii) (TC,UC) and (UD,HD) are adjoint pairs.

    Lemma 2.5. Let Γ=(CU0D) be a triangular matrix coalgebra, which is semiperfect. Then

    (1) UCLnHD()ExtnD(U,), UDLnHD()=0,for the left derived functor LnHD(n1).

    (2) If ExtiD(U,Y)=0 (1in), then there exists an isomorphism

    ToriD(Y,Y)ToriΓ(HD(Y),(X,Y,φ))

    for any right D-comodule Y and left D-comodule Y and any 1in.

    Proof. (1) By [10,Theorem 2], if the triangular matrix coalgebra Γ=(CU0D) is semiperfect, then coalgebras C and D are semiperfect. So, for any YMD, there exists an exact sequence in MD

    with P a projective right D-comodule. Then HD(P) is a right Γ-projective comodule. Applying the left derived functor to the above exact sequence, we have L1HD(P)=0. Thus we get the following exact sequence

    So

    (L1HD)Yker(HD(π))=(ker(HomD(U,π)),0,0)=(Ext1D(U,Y),0,0),

    and (LnHD)Y=(ExtnD(U,Y),0,0), for any positive integer n. Hence,

    UC(LnHD)Y=ExtnD(U,Y),UD(LnHD)Y=0

    for any positive integer n.

    (2) Assume that

    is a projective resolution of Y. Since ExtiD(U,Y)=0(1in), it follows from (1) that UCLnHD(Y)=0, UDLnHD(Y)=0, and LnHD(Y)=0. This implies that the following sequence

    is a projective resolution of HD(Y). For any left Γ-comodule (X,Y,φ), its dual space is (X,Y,φ)MΓ. Then we get HD(Pi)Γ(X,Y,φ)HomΓ((X,Y,φ),HD(Pi))(1in) and PiDYHomD((Y),Pi)(1in).

    For brevity, we denote HomΓ((X,Y,φ),HD(Pi)) by ((X,Y,φ),HD(Pi)). Since (UD,HD) is an adjoint pair, we have the commutative diagram with exact rows

    By the above isomorphism, we furthermore get the following commutative diagram with exact rows

    Consequently,

    ToriD(Y,Y)ToriΓ(HD(Y),(X,Y,φ))

    for any 1in.

    Recall from [1,10,16] that for an exact sequence of injective right C-comodules

    if HomC(I,EC) is also exact for any injective right C-comodule I, then EC is said to be complete. For a right C-comodule M, if Mker(E0E1), then M is called Gorenstein injective. If there exists an exact sequence of right C-comodules

    with Ei(i0) injectives and which remains exact whenever HomC(E,) is applied for any injective right C-comodule E, then we call M is weakly Gorenstein injective. From now on, we denote by GI(Γ) and WGI(Γ) the category of Gorenstein injective comodules and weakly Gorenstein injective comodules over Γ, respectively.

    As a generalization of compatible bicomodules, we now show the "weak analogue" of compatible bicomodules as follows.

    A C-D-bicomodule U is weakly compatible if the following two conditions hold:

    (1) If MC is an exact sequence of injective right C-comodules, then MCU is exact.

    (2) If MD is a complete exact sequence of injective right D-comodules, then HomD(U,MD) is exact.

    Lemma 3.1. [7] For the triangular matrix coalgebra Γ=(CU0D), if (X,Y,φ)GI(Γ), then

    ExtiC(I,(X,Y,φ))=0

    for any injective right Γ-comodule I and any i1.

    Lemma 3.2. [16] For the right C-comodule M, the followings are equivalent:

    (1) M is weakly Gorenstein injective;

    (2) ExtiC(E,M)=0 for any injective comodule E and any i1;

    (3) ExtiC(L,M)=0 for any finite-dimensional right C-comodule L and any i1.

    Theorem 3.3. Let Γ=(CU0D) be the triangular matrix coalgebra, and U be a weakly compatible C-D-bicomodule. The following conditions are equivalent:

    (1) (X,Y,φ) is a weakly Gorenstein injective Γ-comodule.

    (2) X is a weakly Gorenstein injective C-comodule, kerφ is a weakly Gorenstein injective D-comodule, and φ:YXCU is surjetive.

    Proof. (2)(1) If XWGI(C), then there exists the following exact sequence

    with each Ei(i0) injective. By the assumption that U is weakly compatible, it follows that the sequence XCCU is exact and EiCU(i0) are injective. Here kerφWGI(D). Thus there exists an exact sequence as follows

    with each Ii(i0) injective. By using "the Generalized Horseshoe Lemma", we get the following exact sequence

    So we have the following commutative diagrams with exact rows

    Hence there exists an exact sequence

    Next we only need to prove that HomΓ(E,ˉLΓ) is exact for any injective right Γ-comodule E.

    Because EiΛTC(Ei) with finite-dimensional indecomposable injective right C-comodule Ei(iΛ), where Λ is a finite index set. Then

    HomΓ(E,ˉLΓ)=HomΓ(iΛTC(Ei),ˉLΓ)iΛHomΓ(TC(Ei),ˉLΓ)iΛHomC(Ei,UC(ˉLΓ)).

    So HomΓ(E,ˉLΓ) is exact, that is, (X,Y,φ)WGI(Γ).

    (1)(2) If (X,Y,φ)WGI(Γ), then there exists an exact sequence of right Γ-comodule

    Applying UC to ˉLΓ, we get an exact sequence

    Hence HomC(E,XC)HomΓ(TC(E),ˉLΓ) for any injective right C-comodule E. Therefore, XWGI(C).

    By applying the exact functor UD to ˉLΓ again, we also get an exact sequence

    Hence, we get the following commutative diagram with exact rows and columns

    Thus we conclude that φ is surjective, and kerφWGI(D).

    The following result can be viewed as an application of the above theorem on Gorenstein injective comodules.

    Corollary 3.4. Let Γ=(CU0D)be the triangular matrix coalgebra with U a weakly compatible C-D-bicomodule, we have the following equivalence:

    (X,Y,φ)GI(Γ)XGI(C),kerφGI(D), and φ:YXCU is surjective.

    In this section, we first have the following key observation, which is very important for the proof of our main result. The reader may refer to [16] for more details.

    Definition 4.1. A right C-comodule M is called Gorenstein coflat if there is an exact sequence of injective right C-comodules

    such that M=ker(E0E1), and ECCQ is exact for any projective left C-comodule Q.

    Definition 4.2. A right C-comodule M is called weakly Gorenstein coflat if there is an exact sequence of right C-comodules

    with each Ei(i0) injective, and MCCQ is exact for any projective left C-comodule Q.

    We write WGC(Γ) and GC(Γ) for the category of weakly Gorenstein coflat and Gorenstein coflat comodules over Γ, respectively. Under the assumption of right semiperfect, the following result establishes the relation between weakly Gorenstein injective right C-comodules and weakly Gorenstein coflat right C-comodules.

    Remark 4.3. (1) The class of weakly Gorenstein injective right C-comodules is closed under extensions, cokernels of monomorphisms and direct summands. If C is a right semiperfect coalgebra, then the class of weakly Gorenstein injective right C-comodules is closed under direct products.

    (2) The class of weakly Gorenstein coflat right C-comodules is closed under extensions, cokernels of monomorphisms, direct sums, direct summands and direct limit.

    (3) Let C be a right semiperfect coalgebra and M a right C-comodule, then M is weakly Gorenstein coflat if and only if M is weakly Gorenstein injective.

    Lemma 4.4. [16] For a right C-comodule M, the following statements are equivalent:

    (1) M is Gorenstein coflat;

    (2) There is an exact sequence of injective right C-comodules

    such that M=ker(E0E1), and ECCQ is also exact for any finite-dimensional projective left C-comodule Q.

    Lemma 4.5. Let C be a semiperfect coalgebra, then the following statements are equivalent for any right C-comodule M:

    (1) M is weakly Gorenstein coflat;

    (2) ToriC(M,P)=0 for any projective left C-comodule P and any i1;

    (3) ToriC(M,Q)=0 for any finite-dimensional projective left C-comodule P and any i1.

    Proof. It is obvious for (1)(2) by the definition. We only need to show (3)(1). For any projective left C-comodule Q, then from the proof of [3,Corollary 2.4.22, P.100] we know that QλΛQλ, where each Qλ is a finite-dimensional projective. Choose an exact sequence

    with each Ei injective. Since

    MCCQMCC(λΛQλ)λΛ(MCCQλ)

    and ToriC(M,Qλ)=0 for all λΛ and i1, it follows that MCCQ is exact. That is, M is weakly Gorenstein coflat.

    (1)(3) is evident.

    Theorem 4.6. Let Γ=(CU0D) be the triangular matrix coalgebra which is semiperfect, and U be a weakly compatible C-D-bicomodule. The following conditions are equivalent:

    (1) (X,Y,φ) is a weakly Gorenstein coflat Γ-comodule.

    (2) X is a weakly Gorenstein coflat C-comodule, kerφ is a weakly Gorenstein coflat D-comodule, and φ:YXCU is surjetive.

    Proof. (2)(1) Let XWGC(C). Since U is weakly compatible, it follows that there exists an exact sequence of right C-comodules with each Ii injective

    and XCCU is exact. Suppose that kerφWGC(D), there exists the following exact sequence

    The Generalized Horseshoe Lemma yields the following exact sequence

    This gives rise to the following commutative diagram with exact rows

    Thus we obtain the following exact sequence of right Γ-comodules

    Let Q be any finite-dimensional projective left Γ-comodule. Then QiΛQi, Qi is indecomposable and projective. Here Qi is indecomposable and injective. Thus there is an indecomposable and injective right C-comodule Ei(iΛ) such that TC(Ei)Qi(iΛ). Thus

    LΓCQLΓC(iΛQi)iΛ(LΓCQi)iΛHomΓ(Qi,LΓ)iΛHomΓ(TC(Ei),LΓ)iΛHomC(Ei,UC(LΓ))iΛHomC(Ei,XC).

    Therefore, LΓCQ is exact since iΛHomC(Ei,XC) is exact. That is, (X,Y,φ)WGC(Γ).

    (1)(2) If (X,Y,φ)WGC(Γ), then there is the following exact sequence

    with (Ii,Ki(IiCU),(0,id)) injectives. By applying the functor UC to LΓ, we get the exact sequence

    For any finitely dimensional projective left C-comodule P, P is injective right C-comodule. Thus TC(P) is injective. This yields that

    XCCPHomC(P,XC)=HomC(P,UC(LΓ))HomΓ(TC(P),LΓ).

    Thus HomΓ(TC(P),LΓ) is exact since (X,Y,φ)WGC(Γ). So XCCP is also exact. That is, XWGC(C).

    Similarly, applying UD to LΓ, we get an exact sequence as follows

    This yields the following commutative diagram with exact rows and columns

    Therefore, φ is surjective, and kerφWGC(D).

    It is clearly that Gorenstein coflat comodules is weakly Gorenstein coflat comodules. Thus the above result holds for Gorenstein coflat comodules.

    Corollary 4.7. Let Γ=(CU0D) be the triangular matrix coalgebra which is semiperfect, and U be a weakly compatible C-D-bicomodule. Then

    (X,Y,φ)GC(Γ)XGC(C),kerφGC(D), and φ:YXCU is surjective.

    The authors would like to express their gratitude to the anonymous referee for their very helpful suggestions and comments which led to the improvement of our original manuscript. This work are supported by National Natural Science Foundation of China (Grant No. 11871301) and Natural Science Foundation of Shandong Province (Grant Nos. ZR2019MA060, ZR2020QA002).

    The authors declare no conflicts of interest.



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