Weakly Gorenstein comodules over triangular matrix coalgebras

: 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.


Introduction
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 M C 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 = A M 0 B . 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 = C U 0 D 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: 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.
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.

Preliminaries
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 ∆ and counit ε. We recall that a let C-comodule is a k-vector space M together with a k-linear map ρ M : The k-vector space of all C-comodule homomorphisms from M to N is denoted by Hom C (M, N). Similarly we can define a right C-comodule. Let M C and C M denotes the category of right and left C-comodules respectively. For any M ∈ M C and N ∈ C M. Following [5,9], we recall that the cotensor product M C N is the k-vector space where ρ M and ρ N 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 ρ − M : M → E ⊗ M, and also a right C-comodule with structure map ρ + M : M → M ⊗ C such that (I ⊗ ρ + M )ρ − M = (ρ − M ⊗ I)ρ + M , we then say that M is an (E, C)-bicomodule. We let E M C denote the category of (E, C)-bicomodules. Let N be a (C, D)-bicomodule. Then M C N acquires a structure of (E, D)-bicomodule with structure maps For every right exact exact linear functor F : M C → M D preserving direct sums, there exists a (C, D)-bicomodule X, in fact X = F(C), such that F is naturally isomorphic to the funtor − C X (See [19, Proposition 2.1]). Since every comodule is the union of its finite-dimensional subcomodules, there is a functorial isomorphism If U is a quasi-finite right D-comodule, we denote the left adjoint functor of − C U by h D (U, −). Then for any right C-comodule W and any D-comodule N, we have that The functor h D (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:  For two k-coalgebras C and D, let U be a C-D-bicomodule with the left C-coaction on U, u → u [−1] ⊗ u [0] , and the right D-coaction on U, u → u [0] ⊗ u [1] (using Sweedler's convention with the summations symbol omitted). We recall from [4,13,14,21] that Γ = C U 0 D can be made into a coalgebra by defining the comutiplication ∆ : Γ → Γ ⊗ Γ and the counit ε : Γ → K as follows The coalgebra Γ is called a triangular matrix coalgebra. We know from [14] that the right comodule category M Γ 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 ϕ ∈ Hom D (Y, X C U) is the right D-comodule morphism. For any two objects (X, Y, ϕ) and (X , Next we define some exact functors between the right comodule category M Γ and the comodule representation category R(Γ).

then there exists an isomorphism
for any right D-comodule Y and left D-comodule Y and any 1 ≤ i ≤ n.

Weakly Gorenstein injective comodules over triangular matrix coalgebras
Recall from [1,10,16] that for an exact sequence of injective right C-comodules if Hom C (I, E C ) is also exact for any injective right C-comodule I, then E C is said to be complete. For a right C-comodule M, if M ker(E 0 → E 1 ), then M is called Gorenstein injective. If there exists an exact sequence of right C-comodules with E i (i ≥ 0) injectives and which remains exact whenever Hom C (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 M C is an exact sequence of injective right C-comodules, then M C U is exact.   (1) (X, Y, ϕ) is a weakly Gorenstein injective Γ-comodule.
(2) ⇒ (1) If X ∈ WGI(C), then there exists the following exact sequence with each E i (i ≥ 0) injective. By the assumption that U is weakly compatible, it follows that the sequence X C C U is exact and E i C U(i ≥ 0) are injective. Here kerϕ ∈ WGI(D). Thus there exists an exact sequence as follows with each I i (i ≥ 0) 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 sequencē Next we only need to prove that Hom Γ (E,L Γ ) is exact for any injective right Γ-comodule E.
The following result can be viewed as an application of the above theorem on Gorenstein injective comodules. (X, Y, ϕ) ∈ GI(Γ) ⇔ X ∈ GI(C), kerϕ ∈ GI(D) , and ϕ : Y → X C U is surjective.

Weakly Gorenstein coflat comodules over triangular matrix coalgebras
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.
such that M = ker(E 0 → E 1 ), and E C C Q is exact for any projective left C-comodule Q.
with each E i (i ≥ 0) injective, and M C C Q 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.  (1) M is Gorenstein coflat; (2) There is an exact sequence of injective right C-comodules such that M = ker(E 0 → E 1 ), and E C C Q 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) Tor i C (M, P) = 0 for any projective left C-comodule P and any i ≥ 1; (3) Tor i C (M, Q) = 0 for any finite-dimensional projective left C-comodule P and any i ≥ 1. 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 Q ⊕ λ∈Λ Q λ , where each Q λ is a finite-dimensional projective. Choose an exact sequence Thus we obtain the following exact sequence of right Γ-comodules Let Q be any finite-dimensional projective left Γ-comodule. Then Q ⊕ i∈Λ Q i , Q i is indecomposable and projective. Here Q * i is indecomposable and injective. Thus there is an indecomposable and injective right C-comodule E i (i ∈ Λ) such that Therefore, L Γ C Q is exact since ⊕ i∈Λ Hom C (E i , X C ) is exact. That is, (X, Y, ϕ) ∈ WGC(Γ).
Similarly, applying U D to L Γ , we get an exact sequence as follows This yields the following commutative diagram with exact rows and columns 0 0 kerϕ π K 0 Therefore, ϕ is surjective, and kerϕ ∈ WGC(D).
It is clearly that Gorenstein coflat comodules is weakly Gorenstein coflat comodules. Thus the above result holds for Gorenstein coflat comodules.