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Gorenstein homological aspects of monomorphism categories via Morita rings

For any ring R the category of monomorphisms is a full subcategory of the morphsim category over R, where the latter is equivalent to the module category of the triangular matrix ring with entries the ring R. In this work, we consider the monomorphism category as a full subcategory of the module category over the Morita ring with all entries the ring R and zero bimodule homomorphisms. This approach provides an interesting link between Morita rings and monomorphism categories. The aim of this paper is two-fold. First, we construct Gorenstein-projective modules over Morita rings with zero bimodule homomorphisms and we provide sufficient conditions for such rings to be Gorenstein Artin algebras. This is the first part of our work which is strongly connected with monomorphism categories. In the second part, we investigate monomorphisms where the domain has finite projective dimension. In particular, we show that the latter category is a Gorenstein subcategory of the monomorphism category over a Gorenstein algebra. Finally, we consider the category of coherent functors over the stable category of this Gorenstein subcategory and show that it carries a structure of a Gorenstein abelian category.


Introduction and Main Results
This article deals with Gorenstein homological aspects of Morita rings and monomorphism categories. We highlight the connection between monomorphism categories and Morita rings and we explain how this setting is the proper framework for the aims and objectives of this paper. In what follows, we first give some background, motivation and indicate how these concepts are connected from our perspective.
For an abelian category A we denote by Mor A the category of morphisms over A . The monomorhism category Mon A is by definition the full subcategory of Mor A consisting of all monomorphisms in A . If R is a ring and A is the category Mod-R of R-modules, then the category of monomorphisms Mon (Mod-R) can be considered as a full subcategory of the module category Mod-T 2 (R), where T 2 (R) = R R 0 R , since it is known that the morphism category Mor (Mod-R) is equivalent to Mod-T 2 (R). Note that Mon A is an extension closed subcategory of the abelian category Mor A and therefore is an exact category in the sense of Quillen. Monomorphism categories appear quite naturally in various settings and are omnipresent in representation theory. In fact, there are connections with classification problems (Ringel,, Xiong, Zhang, Zhang [45]), with weighted projective lines (Kussin, Lenzing, Meltzer [31]), and with aspects of Gorenstein homological algebra (Beligiannis [10,11], Zhang [49], Chen [17]). In a series of papers [46][47][48], the authors determined the Gorenstein-projective modules over the triangular matrix algebra A N 0 B under some conditions on the bimodule A N B . In the special case where A = N = B and A is a Gorenstein Artin algebra, i.e. T 2 (A) = A A 0 A , the authors in [47] showed that a module over T 2 (A), i.e. a triple (X, Y, f ), is Gorenstein-projective if and only if (X, Y, f ) belongs to the monomorphism category mon(A) and the modules X, Y and Coker f are Gorenstein-projective. A natural extension of triangular matrix rings is the class of Morita rings with zero bimodule homomorphisms. First, recall that Morita rings are 2 × 2 matrix rings associated to Morita contexts (Bass [6], Cohn [21]). A Morita context over two unital associative rings R and S, is a tuple M = (R, N, M, S, φ, ψ), where S M R is an S-R-bimodule, R N S is an R-S-bimodule, φ : M ⊗ R N −→ S is an S-S-bimodule homomorphism, and ψ : N ⊗ S M −→ R is an R-R-bimodule homomorphism, satisfying certain associativity conditions. If M is a Morita context, there is the associative ring Λ (φ,ψ) = A ANB B MA B called the Morita ring of M, where Λ (φ,ψ) = A ⊕ N ⊕ M ⊕ B as an abelian group and the matrix multiplication is given by the ordinary operation on matrices using the maps φ and ψ, see subsection 2.1. In this paper the Morita rings that we deal with admit the structure of an Artin algebra. We state this as: Morita rings which are Artin algebras, in order to distinguish it from the notion of Morita algebras due to Kerner-Yamagata [29]. We refer to [33,43] for the terminology of Morita rings, and to [25,Introduction], among other papers, for a thorough discussion of Morita rings as well as examples and situations where Morita rings appear.
A particular case of interest for us is when A = B = N = M = R, where R is an associative unital ring and φ = ψ = 0. This class of matrix rings provides on one hand a natural extension of the triangular matrix ring T 2 (R) and on the other provides the link between monomorphsim categories and Morita rings. We now explain this. For an abelian category A we define the double morphism category DMor(A ), see subsection 2.3, whose objects are pairs of morphisms in A : such that f • g = 0 and g • f = 0 and morphisms are commutative squares defined in a natural way. In particular, if R is a ring then the double morphism category DMor(Mod-R) of Mod-R is equivalent to the category of modules over the Morita ring ∆ (0,0) = R R R R with zero bimodule homomorphisms. Consider now the upper triangular matrix ring T 2 (R). The monomorphism category Mon(R) is a full subcategory of Mod-T 2 (R) defined by Mon(R) = {(X, Y, f ) | f : X −→ Y is a monomorphism}. As mentioned already, a natural extension of the ring T 2 (R) is the Morita ring ∆ (0,0) , since there is a full embedding Mod-T 2 (R) −→ Mod-∆ (0,0) (see subsection 3.3). Then we define the monomorphism category Mono(R) as the full subcategory of Mod-∆ (0,0) defined by all tuples (X, Y, f, 0) such that f : X −→ Y is a monomorphism. It follows easily that the exact categories Mon(R) and Mono(R) are equivalent. However, there are many properties which are not satisfied when we view the monomorphism category inside the double morphism category. From the discussion so far, one of the main problems considered in this paper is as follows : Problem : Construct Gorenstein-projective modules over Morita rings with zero bimodule homomorphisms.
This problem provides the link between the monomorhism categories and Morita rings as explained above. In particular, we obtain in Section 3 families of Gorenstein-projective modules over ∆ (0,0) which belong to the monomorphism category Mono(R). The situation mentioned above and the important role that monomorphism categories as well as Morita rings play in various different contexts, provide a strong motivation for studying monomorphism categories in the context of Morita rings using homological and representation-theoretic tools. Our aim in this paper is two-fold and can be summarized as follows : (i) Construct Gorenstein-projective modules over Morita rings with zero bimodule homomorphisms and provide sufficient conditions for such rings to be Gorenstein Artin algebras.
(ii) Construct Gorenstein abelian categories from exact subcategories of the monomorphism category.
The organization and the main results of the paper are as follows. In Section 2 we collect preliminary notions and results on Morita rings and monomorphism categories that will be useful throughout the paper and we fix notation. Moreover we introduce the double morphism category of an abelian category and we define the monomorphism category in this general setting. The rest of the paper is divided into two parts that we describe briefly below and we present the main results.
The first part of our paper deals with item (i) and consists of Sections 3 and 4. For an algebra Λ, we denote by Gproj Λ the full subcategory of mod-Λ consisting of the finitely generated Gorenstein-projective Λ-modules. Let Λ (0,0) be a Morita ring which is an Artin algebra and has zero bimodule homomorphisms. In order to construct Gorenstein-projective modules over Λ (0,0) , we need to assume some natural conditions on the bimodules A N B and B M A , similar to the conditions considered by Zhang [48] in the triangular matrix case. We refer to these assumptions as the compatibility conditions on B M A and A N B , see Section 3. These conditions have a nice interpretation via finiteness of the projective dimension of the bimodules N and M , see Corollary 3.11. Our first main result is Theorem A (i), which provides a method to construct Gorenstein-projective modules over Morita rings with zero bimodule homomorphisms. We refer to Theorem 3.10 for the proof as well as its dual version. On the other hand, we give sufficient conditions As an application of Theorem A (i), we construct examples of Gorenstein-projective modules over the double morphism category DMor(mod-Λ). We remark that we obtain Gorenstein-projective modules that lie in mono(Λ), see Corollary 3.6. Also, from Theorem A (ii) we get examples of Morita rings which are Gorenstein algebras (Corollary 4.15). This constitutes the first part of this work which provides the link with monomorphism categories via Morita rings that we study further in the second part.
In Section 5 we study the subcategory C of mono(Λ), where Λ is an Artin algebra, consisting of all monomorphisms f : X −→ Y such that the projective dimension of X is finite. Our third main result is Theorem B (i) where assuming that Λ is Gorenstein we show that C is a Gorenstein subcategory of mono(Λ). We refer to Theorem 5.5 for its proof and to Definition 5.2 for the precise notion of Gorenstein subcategories in the setting of exact categories. Moreover, inspired by recent work of Matsui and Takahashi [32] we continue our study on the exact subcategory C of mono(Λ) by considering the category of coherent functors mod-C over the stable category C of C . Also, we define the subcategory Ω n (C ) of C consisting of all nth syzygies of objects in C (subsection 5.2). In this context, the fourth main result of this paper is Theorem B (ii), see Corollary 5.12, which shows that the category of coherent functors over C is a Gorenstein abelian category in the sense of [12]. Finally, using a result of Beligiannis [8] we realize the singularity category [32] of mod-C as the stable category of Cohen-Macaulay objects over mod-C .
Theorem B. (Gorenstein categories) Let Λ be an n-Gorenstein Artin algebra. ( For the category of coherent functors over C and Ω n (C ) the following hold : (a) mod-C is a 3n-Gorenstein abelian category. (b) mod-Ω n (C ) is a Frobenius abelian category. Moreover, there are the following triangle equivalences : above is a consequence of Theorem 5.8 which provides sufficient conditions on a subcategory B of an exact category A with enough projectives such that mod-B is a Gorenstein abelian category. It should be noted that this result generalizes, and is inspired by, a result of Matsui and Takahashi [32]. Conventions and Notation. We compose morphisms in a given category in a diagrammatic order. Our subcategories are assumed to be closed under isomorphisms and direct summands. For a ring R we usually work with left R-modules and the corresponding category is denoted by Mod-R. By a module over an Artin algebra Λ, we mean a finitely generated left Λ-module and we denote by mod-Λ the category of finitely generated left Λ-modules. For all unexplained notions and results concerning the representation theory of Artin algebras we refer to [5].

Morita Rings and Monomorphism Categories
In this section we fix notation and we collect several preliminary results on Morita rings and monomorphism categories that will be used throughout the paper.
where the addition of elements of Λ (φ,ψ) is componentwise and multiplication is given by We assume that φ(m ⊗ n)m ′ = mψ(n ⊗ m ′ ) and nφ(m ⊗ n ′ ) = ψ(n ⊗ m)n ′ for all m, m ′ ∈ M and n, n ′ ∈ N . This condition ensures that Λ (φ,ψ) is an associative ring. The description of the modules over a Morita ring Λ (φ,ψ) is well known, see for instance [24], but for completeness and due to our needs we also include it here. We introduce the following category.
Let M(Λ) be the category whose objects are tuples (X, such that the following diagrams are commutative : We denote by Ψ X and Φ Y the following compositions : is a pair of homomorphisms (a, b), where a : X −→ X ′ is an A-morphism and b : Y −→ Y ′ is a B-morphism, such that the following diagrams are commutative : The relationship between Mod-Λ (φ,ψ) and M(Λ) is given via the functor F : M(Λ) −→ Mod-Λ (φ,ψ) which is defined on objects (X, Y, f, g) of M(Λ) as follows : F(X, Y, f, g) = X ⊕ Y as abelian groups, with a Λ (φ,ψ) -module structure given by Then the functor F turns out to be an equivalence of categories, see [24,Theorem 1.5]. We refer to [25] for an extensive discussion of Morita rings mainly in the context of Artin algebras. We also refer to [35,Chapter 3] for a thorough discussion on the abelian structure of Morita rings in a more general setting. For our purpose and for completeness, we summarize in the next remark several properties of Morita rings that we need throughout the paper. (ii) From now on we identify the modules over Λ (φ,ψ) with the objects of M(Λ). ψ) and consider the maps c : Ker a −→ X and d : Ker b −→ Y . Then the kernel of (a, b) is the object (Ker a, Ker b, h, j) where the maps h and j are induced from the following commutative diagrams : Similarly, we derive a description for the cokernel of the morphism (a, b).
As in [25] we define the following functors :  (i) The indecomposable projective Λ (φ,ψ) -modules are objects of the form : where P is an indecomposable projective A-module and Q is an indecomposable projective Bmodule. where I is an indecomposable injective A-module and J is an indecomposable injective B-module.
We continue now with examples of Morita rings which will be used in the sequel. (ii) Any pair (A, P A ), where A is a ring and P A is a right A-module, induces a Morita ring as follows : It is well known that if the A-module P is a progenerator, then the rings A and End A (P ) are Morita equivalent. (iii) Let Λ be an Artin algebra and X, Y two Λ-modules. Then the endomorphism algebra End Λ (X⊕Y ) is a Morita ring : The bimodule homomorphisms φ : be a Morita ring where the bimodule morphisms φ and ψ are zero. Then we have an isomorphism of rings : For the notion of trivial extension of rings and for the above isomorphism we refer to [22], see also [25,Proposition 2.5].
(v) Suppose that we have the following Morita ring : where every entry is a ring A. Then, it follows from the associativity of the multiplication that φ = ψ, see [25, Corollary 2.13] for more details. A special case is when φ = 0, that is In the next subsection we analyze the module category of ∆ (0,0) via recollements of abelian categories.
We close this subsection with the next result, which shows that always a Morita ring gives rise to a recollement situation. This provides a way to relate the module category of a Morita ring with the module categories of its underlying rings. For the proof see [25,Proposition 2.4] and for more details on recollements of abelian categories we refer to [23,36]. Proposition 2.4. Let Λ (φ,ψ) be a Morita ring. Then the following diagrams : are recollements of abelian categories, that is : is an adjoint triple.  Our aim next is to analyze the recollement of the module category of the Morita ring ∆ (0,0) = R R R R , where R is a unital associative ring. For this reason, we introduce in the next subsection the double morphism category of an abelian category. This construction can be considered as an abstract model for the category of modules over ∆ (0,0) .

2.2.
The Double Morphism Category. Let A be an abelian category. The double morphism category of A , denote by DMor(A ), has as objects diagrams of the form : where f and g are morphisms in A such that f • g = 0 and g • f = 0. We simply denote the objects as such that the following diagram commutes : is commutative, where α = (a 1 , a 2 ), β = (b 1 , b 2 ) and γ = (c 1 , c 2 ). Since the endofunctor H is (right) exact, it follows from [22] that the trivial extension (A × B) ⋉ H is an abelian category. Proposition 2.5. Let A be an abelian category.
(i) There is an equivalence of categories : In particular, the double morphism category DMor(A ) is abelian. (ii) There is a recollement of abelian categories : Proof. (i) Let (X, Y, f, g) be an object of DMor(A ). We define the functor and given a morphism (a, b) : (X, Y, f, g) −→ (X ′ , Y ′ , f ′ , g ′ ) in DMor(A ) then F(a, b) = H(a, b). The functor F is well defined since the following composition is an object of (A × A ) ⋉ H, then since H(a 1 , a 2 ) • (a 1 , a 2 ) = 0 we infer that (X, Y, a 2 , a 1 ) ∈ DMor(A ) such that F(X, Y, a 2 , a 1 ) = (a 1 , a 2 ). This shows that the functor F is essentially surjective. Hence, the categories DMor In the following remark we collect some interesting properties of the recollement diagram of a double morphism category.
It is easy to check that U ′ A is the middle functor of the adjoint triple (H A , U ′ A , T A ) and therefore we obtain a recollement of abelian categories (A , DMor(A ), A ). Then the following commutative diagram : We close this subsection with the next example, which was the starting point for introducing the double morphism category of an abelian category.
For later use, we fix the above notation for the functors of the recollement of Mod-∆ (0,0) . In particular, and relative to subsection 2.1 and Proposition 2.5, we have : (i) The functor T 1 : Mod-R −→ Mod-∆ (0,0) is given by T 1 (X) = (X, X, Id X , 0) on the objects X ∈ Mod-R and for an R-morphism a : X −→ X ′ then T 1 (a) = (a, a). Moreover, the functor T 1 is exact. Similarly, the functor T 2 : Mod-R −→ Mod-∆ (0,0) is given by T 2 (X) = (X, X, 0, Id X ) on the objects X ∈ Mod-R and for an R-morphism a : X −→ X ′ then T 2 (a) = (a, a). Note that in this case T 2 is precisely the functor H 1 appearing in the recollement (2.4).
where c is the restriction map of b to Ker g. This is the functor P A in Proposition 2.5.

Monomorphism Categories. Let
A be an abelian category. We denote by MorA the category of morphisms of A . Recall that the objects of MorA are triples (X, Y, f ), where f : X −→ Y is a morphism in A , and given two objects (X, Y, f ) and (X ′ , Y ′ , f ′ ) then a morphism is a pair (a, b) of maps in A such that the following diagram commutes : MorA is a special case of a trivial extension of abelian categories, see [22], it follows that MorA is an abelian category. Our particular interest is the monomorphism category MonA of A , which is the full subcategory of MorA consisting of monomorphisms in A . The monomorphism category MonA is an extension closed additive subcategory of MorA , and this implies that MonA is an exact category in the sense of Quillen [39]. Due to our needs in the sequel (see Section 5) we recall the notion of exact categories. Our approach follows the appendix of Keller [28, Appendix A], see also [16].
Let A be an additive category. A pair of composable morphisms X f / / Y g / / Z is called exact, if f is the kernel of g and g is the cokernel of f . Two exact pairs (f, g) and (f ′ , g ′ ) are isomorphic if there are isomorphisms a : E be a class of exact pairs which is closed under isomorphisms. A pair (f, g) in E is called a conflation, while the map f is called an inflation and the map g is called a deflation. Then the class E is an exact structure of A and (A , E ) is called an exact category, if the following axioms hold : (Ex 0) : The identity morphism of the zero object is a deflation. (Ex 1) : The composition of two deflations is a deflation. (Ex 2) : If g : Y −→ Z is a deflation and c : Z ′ −→ Z is a morphism, then there exists a pullback diagram : is an inflation and a : X −→ X ′ is a morphism, then there exists a pushout diagram : Let (A , E ) be an exact category. Recall that an object P in A is projective if the functor Hom A (P, −) sends conflations to short exact sequences, and A has enough projectives if for any object A in A there exists a deflation g : P −→ A with P ∈ Proj A . Dually we have the notions of injective objects and enough injectives.
Of particular interest in representation theory is the morphism category over the category of finitely generated modules over an Artin algebra Λ. In particular, there is an equivalence of categories between the morphism category Mor(mod-Λ) and the module category mod-T 2 (Λ), see [5]. In this case, the monomorphism category Mon(mod-Λ), simply denoted by mon(Λ), is the full subcategory of mod-T 2 (Λ) consisting of all monomorphisms of Λ-modules. Note that this holds for any ring R, i.e. Mod-T 2 (R) ≃ Mor(Mod-R).
We now return to the double morphism category. For an abelian category A , define the monomorphism categories of DMor(A ) as follows : In the next result we show that the above monomorphism categories are exact and that they are equivalent. Proof. There is a functor F : Note that the functor F is an exact full embedding and using From now on the monomorphism category of an abelian category A , denoted by Mono(A ), is the category Mono 1 (A ), i.e. the image of Mon(A ) via the functor F. Note that when A is exact, the monomorphism category of A is the inflation category of A , that is, the morphisms of A which are inflations. We continue to call this category the monomorphism category of A and we denote it by Mono(A ) as well. The next result provides a description of the projective and injective objects in Mono(A ) and will be used in Section 5. Note that this is proved in [17, Lemma 2.1] for the monomorphism subcategory Mon(A ) of Mor(A ), but for completeness we provide a proof in our setting. Lemma 2.9. Let A be an exact (abelian) category with enough projective and injective objects. Then the monomorphism category Mono(A ) has enough projective and injective objects, in particular : Proof. We show (i), statement (ii) follows similarly. We first claim that if P is a projective object of A , then the objects Z 2 (P ) = (0, P, 0, 0) and T 1 (P ) = (P, P, Id P , 0) are projectives in Mono(A ). Indeed, let (X, Y, f, 0) be an object in Mono(A ) and let (f, g) be a conflation of A . It is easy to check that we have the isomorphisms where all rows and columns are conflations in A . From the above isomorphisms and since the functor Hom A (P, −) sends conflations to short exact sequences, we get that the objects Z 2 (P ) and T 1 (P ) lie in Proj(Mono(A )). Now we show that the exact category Mono(A ) has enough projective objects. Let (X, Y, f, 0) be an object in Mono(A ). Since A has enough projectives, there are deflations a : P −→ X and b : Q −→ Z with P, Q ∈ Proj A . Then, from [16,Corollary 8.13], we have the commutative diagram : where all rows and columns are conflations of A .
Then we have the following conflation in Mono(A ) : Hence, the exact category Mono(A ) has enough projective objects.
Let Λ be an Artin algebra and consider the Morita ring ∆ (0,0) which is an Artin algebra. In this case, the monomorphism category of Λ is the following full subcategory of mod-∆ (0,0) , i.e. of DMor(mod-Λ) : We close this section with the next result where we collect some useful properties of mono(Λ) that we need in the sequel. Lemma 2.10. Let Λ be an Artin algebra. Then the following statements hold.
(i) The monomorphism category mono(Λ) is an exact category which is closed under kernels.
(ii) We have the adjoint triples (Cok, Z 2 , U 2 ) and (U 2 , T 1 , U 1 ) : The above functors are exact and preserve projective objects, and Z 2 and T 1 are fully faithful.
Proof. (i) From Lemma 2.8 the monomorphism category mono(Λ) is exact, since it is an extension closed subcategory of mod- . Consider the following exact commutative diagram : We infer that mono(Λ) is closed under kernels. (ii) It is easy to check that the above functors form adjoint pairs, see Proposition 2.4 and Example 2.7. Since Cok, Z 2 , U 2 and T 1 are left adjoint functors of exact functors it follows that they preserve projective objects. The functor U 1 preserves projectives by the description of proj(mono(Λ)) given in Lemma 2.9, see also Example 2.7. Moreover, it follows easily from the definition that the functors Z 2 , U 2 , T 1 and U 1 are exact, and moreover that Z 2 and T 1 are fully faithful, see again Example 2.7. It remains to show that the cokernel functor Cok : be a conflation in mono(Λ). Then we have the following exact commutative diagram : where the maps f 1 , f 2 and f 3 are monomorphisms. From the Snake Lemma in the above diagram, it follows that the sequence 0 −→ Coker f 1 −→ Coker f 2 −→ Coker f 3 −→ 0 is exact and therefore the functor

Gorenstein-Projective Modules over Morita Rings
Our aim in this section is to study Gorenstein-projective modules over Morita rings. In particular we provide a method for constructing Gorenstein-projective modules over Morita rings, which are Artin algebras and satisfy certain conditions, from Gorenstein-projective modules of the underlying algebras. This section is divided into three subsections and the main result is stated in the second subsection. We start by recalling the notion of Gorenstein-projective modules and we also fix notation.
Let Λ be an Artin algebra. An acyclic complex of projective Λ-modules for some totally acyclic complex P • of projective Λ-modules. We denote by Gproj Λ the full subcategory of mod-Λ consisting of the finitely generated Gorenstein-projective Λ-modules. Moreover, we denote {X ∈ mod-Λ | Ext 1 Λ (Gproj Λ, X) = 0} by (Gproj Λ) ⊥ . Recall also from [10,11], that an Artin algebra Λ is said to be of finite Cohen-Macaulay type, if the category Gproj Λ is of finite representation type, i.e. the set of isomorphism classes of indecomposable finitely generated Gorenstein-projective modules is finite. Finally, for a Λ-module X we denote by add X the full subcategory of mod-Λ consisting of all direct summands of finite direct sums of X.

Lifting Gorenstein-Projective Modules. From Proposition 2.4 it follows that the functors T
preserve projective modules. In this subsection we investigate when the functors T A and T B preserve Gorenstein-projective modules. The first step towards this problem, is to examine when the above functors preserve totally acyclic complexes. Under some conditions, this is achieved in the next result.
be a Morita ring which is an Artin algebra. Proof. We prove only (i), the statement (ii) is dual. Assume that is a totally acyclic complex of projectives in mod-A. Then, by the assumption on the functor M ⊗ A − and Remark 2.1 (iv), we obtain that the following complex : . In fact, from Proposition 2.2 (i) it is enough to consider only the complexes Hom Λ (φ,ψ) (T A (P • ), T A (P )) and Hom Λ (φ,ψ) (T A (P • ), T B (Q)), where P lies in proj A and Q lies in proj B. In the first case, the complex Hom Λ (φ,ψ) (T A (P • ), T A (P )) is acyclic since the complex Hom A (P • , P ) is acyclic and from Proposition 2.4 the functor T A is fully faithful. Let Q be a projective B-module. Then, by using the adjoint pair (T A , U A ), we have the following commutative diagram : Since N ⊗ B Q is a direct sum of summands of N and add A N ⊆ (Gproj A) ⊥ , it follows that the complex Hom A (P • , N ⊗ B Q) is acyclic and therefore the complex Hom Λ (φ,ψ) (T A (P • ), T B (Q)) is also acyclic. We infer that the complex T A (P • ) is totally acyclic. Conversely, assume that P • is a complex of A-modules such that T A (P • ) is totally acyclic. If we apply the functor U A to the complex T A (P • ), we get that the complex P • : · · · −→ P −1 −→ P 0 −→ P 1 −→ · · · is acyclic. Note that since the functor T A is right exact and fully faithful it follows that each P i lies in proj A, see Proposition 2.4. Then, for every projective A-module P , we derive as above that the complex Hom A (P • , P ) is acyclic. We remark that in this direction we did not make use of our assumptions.
We refer to the above conditions as the compatibility conditions on the bimodules A N B and B M A . Note that this example generalizes the situation mentioned in (i).
As a consequence of Proposition 3.1 we have the following result, which provides sufficient conditions such that the functors T A and T B lift Gorenstein-projective modules. In particular, we derive that Cohen-Macaulay finiteness of the Morita ring is inherited to the underlying algebras as well.
is of finite Cohen-Macaulay type, then A is also of finite Cohen-Macaulay type.
is of finite Cohen-Macaulay type, then B is also of finite Cohen-Macaulay type.
Now we turn our attention to the algebra ∆ (φ,φ) = Λ Λ Λ Λ . We recall the following.  In the next result we show the one direction of Proposition 3.4 (ii) without assuming Λ to be Gorenstein.
Proof. Let (X, Y, f, g) be a Gorenstein-projective ∆ (φ,φ) -module. Then from Proposition 2.2, there exists a totally acyclic complex of the following form : where P i and Q i are projective Λ-modules. Then, if we apply the exact functor U 1 : mod-∆ (φ,φ) −→ mod-Λ, we get the exact sequence of projective Λ-modules : 9 9 s s s s s s s s s s We claim that the above complex is totally acyclic. Let P be a projective Λ-module. Then from Example 2.7, we have the following isomorphisms : Since the complex Hom ∆ (φ,φ) (T • , T 2 (P )) is acyclic, it follows from the above isomorphisms that the complex Hom Λ (P • , P ) is also acyclic. We infer that the complex P • is totally acyclic and therefore the Λ-module X is Gorenstein-projective. Similarly we show that Y is a Gorenstein-projective Λ-module.
As a consequence of Corollary 3.3 and Lemma 3.5 we obtain the following. Note that if Λ is Gorenstein, Proposition 3.4 (ii) gives a direct proof of the next result.
Corollary 3.6. Let Λ be an Artin algebra and let ∆ (φ,φ) = Λ Λ Λ Λ . Then for a Λ-module X the following statements are equivalent : In the special case where φ = 0, the Gorenstein-projective modules T 1 (X) lie in the monomorphism category mono(Λ) as defined in subsection 2.3. We close this subsection with the next example.
Example 3.7. Let K be a field and R = K[[X 1 , X 2 ]]/(X 1 X 2 ). Consider the Morita ring ∆ (0,0) = R R R R . By [19, Example 4.1.5] the R-modules X 1 and X 2 are Gorenstein-projective, where X i is the residue class in R of X i for i = 1, 2. Thus, from Corollary 3.6 and for i = 1, 2 it follows that the objects T 1 (X i ) = (X i , X i , Id Xi , 0) and T 2 (X i ) = (X i , X i , 0, Id Xi ) are Gorenstein-projective ∆ (0,0) -modules.

3.2.
Constructing Gorenstein-Projective Modules. In this subsection, we construct Gorensteinprojective modules over Morita rings Λ (0,0) which are Artin algebras and satisfy the compatibility conditions on the bimodules A N B and B M A , as discussed in subsection 3.1.
Before we proceed to the main result of this subsection (see Theorem 3.10), we need some preparations.
be a Morita ring. Then for every A-module X and B-module Y we have the following exact sequences in Mod-Λ (0,0) : (Hom A (N, X)) In the same way we derive that the rest sequences are exact, the details are left to the reader.
Lemma 3.9. Let Λ (0,0) be a Morita ring. Then for every X, X ′ ∈ Mod-A and Y, Y ′ ∈ Mod-B, we have the following isomorphisms : and Proof. We show the first isomorphism. From Proposition 2.4, we have the adjoint pair (Q A , Z A ) and from the recollement (Mod-A, Mod-Λ (0,0) , Mod-B) it follows that Q A T B = 0. Then, we have the isomorphism and it remains to compute the object Q A T A (X). From the counit of the adjoint pair (T B , U B ) we have the following exact sequence in Mod-Λ (0,0) : . This implies that Q A T A (X) ∼ = X and therefore we have the isomorphism The second isomorphism follows similarly by using the adjoint pair (Q B , Z B ).
We are ready to prove the main result of this section which constructs Gorenstein-projective modules over Morita rings Λ (0,0) . This result constitutes the first part of Theorem A presented in the Introduction.
(β) Assume that there exists a Gorenstein-projective A-module Z with a monomorphism t : M ⊗ A Z −→ Y , for some B-module Y , such that Coker t lies in Gproj B and there is a monomorphism s : N ⊗ B Coker t −→ X with Coker s = Z and for some A-module X. Then the tuple is a Gorenstein-projective Λ (0,0) -module, where π X : X −→ Coker s and π Y : Y −→ Coker t.
Proof. We prove (α), the statement (β) is dual. The proof for (α) is divided into four steps. In the first two steps we construct (co)resolutions of X and Y by objects coming from the totally acyclic complexes of Coker s and Z. Then in the third step we lift this (co)resolutions to mod-Λ (0,0) and in the final step we show that this construction is indeed a totally acyclic complex of the object (X, Y, (Id M ⊗π X )• t, (Id N ⊗π Y )• s).
Step 1 : Since the A-module Coker s is Gorenstein-projective, there exists a totally acyclic complex of projective A-modules : such that Ker d 0 P = Coker s and let d −1 P = λ −1 P • κ −1 P be the canonical factorization through Coker s. Also, since the B-module Z is Gorenstein-projective there exists a totally acyclic complex of projective B-modules : Q be the canonical factorization through Z. Then from the assumption (iii), it follows that the complex of A-modules N ⊗ B Q • is acyclic. Applying to the exact sequence : , we get the exact sequence : (ii). This implies that there is a map γ 0 : X −→ N ⊗ B Q 0 such that s • γ 0 = Id N ⊗κ −1 Q and therefore we obtain the map Then from the Horseshoe Lemma, see also [48, Lemma 1.6], we obtain the exact commutative diagram : Note that the existence of the maps γ i follow by using the assumption (ii). In the same way, we can construct a resolution of X by objects of the form P i ⊕ (N ⊗ B Q i ) but now we use that the modules P −i , i ≥ 1, are projective. In particular, we get the map , and for every i ≥ 2 we have the maps : similarly as described above. Thus, summarizing the construction so far, we have constructed the following exact sequence : Step 2 : We construct an exact sequence similar to ( * ) for the B-module Y . Since add B M ⊆ (Gproj B) ⊥ (assumption (iv)) we have as in Step 1 the following exact commutative diagram : and for all i ≥ 1 we have :

Then, as in
Step 1 we construct a resolution of Y by objects of the form (M ⊗ A P i ) ⊕ Q i and putting together these, we obtain the following exact sequence : Step 3 : We glue together the exact sequences ( * ) and ( * * ) and we derive the following sequence : We claim that the sequence T • is exact in mod-Λ (0,0) . First, since the following diagrams are commutative it follows that the maps are morphisms in mod-Λ (0,0) . Since the complexes ( * ) and ( * * ) are acyclic, it follows from Remark 2.1 (iii) that T • is an exact sequence in mod-Λ (0,0) . Moreover, the object (X, Y, f, g) arises as the kernel of the morphism (α 1 , β 1 ) and by Remark 2.1 (iv) we observe that f = (Id M ⊗π X ) • t and g = (Id N ⊗π Y ) • s.
Step 4 : The final step of the proof is devoted to show that the acyclic complex T • is totally acyclic. From Proposition 2.2, it is enough to show that the complexes Hom Λ (0,0) (T • , T A (P )) and Hom Λ (0,0) (T • , T B (Q)) are acyclic, where P is a projective A-module and Q is a projective B-module. From Lemma 3.8 we have the exact sequence 0 −→ Z B (M ⊗ A P ) −→ T A (P ) −→ Z A (P ) −→ 0 and since each term of the complex T • is a projective Λ (0,0) -module, it follows that the following sequence : is an exact sequence of complexes. Then, from Lemma 3.9 we have the isomorphism Hom Λ (0,0) (T • , Z A (P )) ∼ = Hom A (P • , P ) and since P • is totally acyclic we infer that Hom Λ (0,0) (T • , Z A (P )) is acyclic. Also, from Lemma 3.9 we have Hom   The next result is a consequence of Theorem 3.10 for the Morita ring ∆ (0,0) . Recall that mod-∆ (0,0) is the double morphism category DMor(mod-Λ) that we studied in subsection 2.2.
Remark 3.13. Let (X, Y, f, g) be a Λ (0,0) -module. Assume that for X and Y the conditions of Theorem 3.10 (α) are satisfied. Then, we cannot infer in general from Theorem 3.10 that (X, Y, f, g) lies in Gproj ∆ (0,0) . In other words, Theorem 3.10 does not provide us with sufficient conditions for a tuple (X, Y, f, g) to be Gorenstein-projective. We explain now where is the problem. Following the construction of Theorem 3.10, we conclude that the object (X, Y, From Remark 2.1 (v) we know that the maps (Id M ⊗π X ) • t and (Id N ⊗π Y ) • s are uniquely determined and satisfy the corresponding commutative diagrams (2.2). But since f and g are arbitrary maps, we don't know in general if they satisfy the diagrams (2.2). If f and g satisfy these diagrams, then from uniqueness it follows that f = (Id M ⊗π X ) • t, g = (Id N ⊗π Y ) • s and therefore (X, Y, f, g) is Gorenstein-projective. Hence, we cannot conclude from Theorem 3.10 that (X, Y, f, g) is Gorenstein-projective.
The next example shows how we can apply Corollary 3.12 to construct Gorenstein-projective modules over ∆ (0,0) from Gorenstein-projective modules over the underlying triangular matrix algebras. / such that the Λ-modules X and Coker f are Gorenstein-projective. Let (X, Y, f ) be a Gorensteinprojective Γ-module. Thus, we have the sequence (3.4) and we also form the split exact sequence : Then, Corollary 3.12 yields that the objects Consider now the upper triangular matrix algebra Σ = Λ Λ 0 Λ and let (Z, W, g) ∈ Gproj Σ. Then, from [48,Theorem 1.4] there is an exact sequence : such that the Λ-modules W and Coker g lie in Gproj Λ, and we also have the split exact sequence : Hence, by Corollary 3.12 it follows that the following objects : Z, Coker g ⊕ W, ρ • 1 0 , 0 1 • g and Coker g ⊕ W, Z, 0 1 • g, ρ • 1 0 are Gorenstein-projective ∆ (0,0) -modules. (ii) Let X be a Gorenstein-projective Λ-module. From (i) the objects (X, X, 0, Id X ) and (X, X, Id X , 0) are Gorenstein-projective ∆ (0,0) -modules. Note that this was also observed in Corollary 3.6.
The above example shows that using Theorem 3.10, we obtain non-trivial examples of Gorensteinprojective modules over the Morita ring ∆ (0,0) from Gorenstein-projective modules of the triangular matrix algebras Γ and Σ. It should be noted that we don't know if all Gorenstein-projective modules over ∆ (0,0) arises in this way, as well as how many objects from Gproj ∆ (0,0) we finally obtain. We recall that in the case where Λ is a Gorenstein Artin algebra, a tuple (X, Y, f, g) lies in Gproj ∆ (0,0) if and only if X and Y lie in Gproj Λ, see Proposition 3.4 (ii). But also in this case it seems to be a difficult problem to determine the Gorenstein-projectives ∆ (0,0) -modules. On the other hand, the above example shows the connections of the module categories of Γ and Σ with the module category of ∆ (0,0) . This fact will become clear in the next subsection, see Lemma 3.18. We close this subsection with the following consequence of Corollary 3.12 and an example. We mention that Example 3.16 provides an interesting connection between our main result (Theorem 3.10) and the class of strongly Gorenstein-projective modules.
be a totally acyclic complex of projective Λ-modules. Then, from Corollary 3.15 it follows that (P, P, f, f ) is a Gorenstein-projective ∆ (0,0) -module. In this case, the Λ-module Im f is called strongly Gorensteinprojective. We refer to [13] for more details on this class of modules. As a particular example, let K be a field, Λ = K[X]/(X 2 ) and consider the matrix algebra ∆ (0,0) . Denote by X the residue class of X in Λ. Then by [13, Example 2.5] the following sequence is a totally acyclic complex of projective Λ-modules and X = Im x = Ker x is a strongly Gorensteinprojective Λ-module. We infer that (Λ, Λ, x, x) is a Gorenstein-projective ∆ (0,0) -module. . Before we define functors between the module categories of the above rings, we recall the adjoint isomorphisms : π : Hom A (N, X)).
We define the following functors : is defined by G(X, Y, f, g) = (Coker g, Y, h) on the objects (X, Y, f, g) ∈ Mod-Λ (0,0) , where the morphism h : M ⊗ A Coker g −→ Y is obtained from the following commutative diagram : where ξ : Coker g −→ Coker g ′ is the unique morphism such that π • ξ = a • π ′ , where π : X −→ Coker g and π ′ : X ′ −→ Coker g ′ . (iii) The functor H : Mod-Λ (0,0) −→ Mod-A 0 B MA B is defined by H(X, Y, f, g) = (X, Ker ρ(g), π −1 (j)) on the objects (X, Y, f, g) ∈ Mod-Λ (0,0) , where the morphism π −1 (j) : M ⊗ A X −→ Ker ρ(g) is obtained from the following commutative diagram : then H(a, b) = (a, ζ) where ζ : Ker ρ(g) −→ Ker ρ(g ′ ) is the unique morphism which makes the next diagram commutative : Similarly we have the functor Proof. We leave to the reader to check that the above functors provide adjoint pairs. However, we give another direct proof following [25,Theorem 4.6]. Since the bimodule homomorphisms φ and ψ are zero, there are surjective ring homomorphisms Λ (0,0) −→ We close this subsection with the next result, which shows that the left adjoints of the full embeddings F and F ′ preserve the Gorenstein-projective modules constructed in Theorem 3.10. In the following result we keep the assumptions of Theorem 3.10.
Proof. We show that the functor G preserves the Gorenstein-projective objects of Theorem 3.10 (α). The other cases are treated with the same way. Let (X, Y, f, g) be a Gorenstein-projective Λ (0,0) -module of the form (3.1). This means that we have the following exact sequences : We claim that the object G(X, Y, f, g) = (Coker g, Y, h) is Gorenstein-projective. Thus, from [48, Theorem 1.4] we have to show that the map h : M ⊗ A Coker g −→ Y is a monomorphism, Coker g ∈ Gproj A and Coker h ∈ Gproj B. Since the map Id N ⊗π Y : N ⊗ B Y −→ N ⊗ B Z is an epimorphism, it follows that Coker g = Coker s and therefore Coker g lies in Gproj A. Also, we have π = π X since g = (Id N ⊗π Y ) • s. Then, from the relations f = (Id M ⊗π X ) • t and f = (Id M ⊗π X ) • h it follows that h = t. This implies that h is a monomorphism and Coker h = Coker t = Z is Gorenstein-projective. We infer that G(X, Y, f, g) lies in Gproj A 0 B MA B .

Homological Embeddings and Gorenstein Artin Algebras
Our purpose in this section is to provide a method for constructing Morita rings Λ ( In this subsection we investigate when the ideals e = ΛeΛ and f = Λf Λ are stratifying. We recall first the notion of stratifying ideals due to Cline-Parshall-Scott [20].
Let R be a ring and e an idempotent element of R. Then we have the exact sequence where Im µ R = ReR and Ker µ R lies in Mod-R/ReR. The ideal e = ReR is called stratifying, if the following two conditions hold : (i) The multiplication map Re ⊗ eRe eR −→ ReR is an isomorphism. (ii) Tor i eRe (Re, eR) = 0, for all i > 0. The surjective ring homomorphism R −→ R/ReR induces a fully faithful functor I R : Mod-R/ReR −→ Mod-R. Then it is known from [20] that the ideal e is stratifying if and only if the functor I R is a homological embedding [36], i.e. the exact functor I R induces an isomorphism between the extension groups : for all X, Y in Mod-R/ReR and n ≥ 0. For more details on homological embeddings between abelian categories we refer to [36]. We now characterize when the ideals e and f are stratifying, or equivalently when the functors I B : Mod-B/ Im φ −→ Mod-Λ (φ,ψ) and I A : Mod-A/ Im ψ −→ Mod-Λ (φ,ψ) are homological embeddings.  Proof. We only prove (i) since part (ii) is dual. For simplicity we write Λ for the Morita ring Λ (φ,ψ) . An easy computation shows that f Λe = M and eΛf = N . Then, since Λe = eΛe ⊕ f Λe and eΛ = eΛe ⊕ eΛf , we have the following isomorphisms :    [25]. This means that we have projective  . . , S n } be the corresponding simple Λ-modules. Assume that S := S 1 is localizable, i.e. pd Λ S ≤ 1 and Ext 1 Λ (S, S) = 0. If we consider the idempotent element α = e 2 + · · · + e n , then it is easy to see that α(S 1 ) = 0. This shows that add S is the kernel of the exact functor αΛ ⊗ Λ − : mod-Λ −→ mod-αΛα, in particular the category mod-Λ/ΛαΛ is precisely the additive closure add S of S. From the short exact sequence 0 −→ ΛαΛ −→ Λ −→ Λ/ΛαΛ −→ 0 and since pd Λ Λ/ΛαΛ ≤ 1, it follows that ΛαΛ is a projective Λ-module. Then by [30,Remark 3.2] we get that αΛ is a projective left αΛα-module and the map Λα ⊗ αΛα αΛ −→ ΛαΛ is an isomorphism. This implies that the map e 1 Λα ⊗ αΛα αΛe 1 −→ e 1 Λe 1 is a monomorphism and Tor i αΛα (e 1 Λα, αΛe 1 ) = 0 for all i > 0. Note that we view Λ as the Morita ring with A = αΛα, B = e 1 Λe 1 , N = αΛe 1 and M = e 1 Λα, see Example 2.3 (i). Hence, from Proposition 4.1 we infer that the ideal ΛαΛ is stratifying. The above claim, that ΛαΛ being projective implies that ΛαΛ is a stratifying ideal, can be proved in a different way. We refer to [36,Example 3.14] for more details.
We restrict now to the case where the bimodule homomorphisms φ and ψ are zero, that is Λ (0,0)   (i) For every A-modules X, X ′ and n ≥ 0 there are the following isomorphisms : (a) For n = 0, 1 : Ext n (c) For n = 2k + 1 : For every B-modules Y, Y ′ and n ≥ 0 there are the following isomorphisms : (a) For n = 0, 1 : (c) For n = 2k + 1 : Proof. We only sketch the proof of (ii), statement (i) follows similarly. . We refer to [22] for more details on trivial extensions of abelian categories. We compute only Ext 2 ). Using the description of Mod-Λ (0,0) as a trivial extension and [9,Corollary 4.4], it follows that Ext 2 ). The latter extension group is isomorphic with The rest isomorphisms follow in the same way, the details are left to the reader.
As a consequence of Lemma 4.5 we have the next result. Note that it also follows from Proposition 4.1.
) for every A-module X, X ′ and n ≥ 0.
(ii) This follows as in (i) using Lemma 4.5 (ii).
The next result provides another reason for our interest in stratifying ideals. It is a consequence of Proposition 4.1 and the well known result of Cline-Parshal-Scott [20] which relates stratifying ideals and recollements of derived module categories. For the notion of recollement of triangulated categories see [7], and for more details on deriving recollements of abelian categories we refer to [38].
We start with the next result which, under some conditions, gives isomorphisms between the extension groups induced from the adjoint pairs (T A , U A ) and (T B , U B ). It follows from [36, Theorem 3.10], but for completeness we give a direct proof.
Lemma 4.8. Let Λ (φ,ψ) be a Morita ring. Let X be an A-module and let Y be a B-module.
(i) Assume that the module M A is projective. Then for every Λ (φ,ψ) -module (X ′ , Y ′ , f ′ , g ′ ) and n ≥ 0 we have an isomorphism : Assume that the module N B is projective. Then for every Λ (φ,ψ) -module (X ′ , Y ′ , f ′ , g ′ ) and n ≥ 0 we have an isomorphism : Proof. (i) Let X be an A-module and let · · · −→ P 1 −→ P 0 −→ X −→ 0 be a projective resolution of X.
Since the functor M ⊗ A − is exact, it follows from Proposition 2.2 and Remark 2.1 (iv) that the sequence Then, using the adjoint pair (T A , U A ) we have the following commutative diagram : This implies that Ext n Λ ( φ,ψ) (T A (X), (X ′ , Y ′ , f ′ , g ′ )) ∼ = Ext n A (X, X ′ ) for every n ≥ 0. (ii) This follows similarly as in (i). Lemma 4.9. Let Λ (φ,ψ) be a Morita ring which is an Artin algebra.
(i) Assume that M A and N B are projective modules. If id Λ (φ,ψ) Λ (φ,ψ) < ∞, then : Proof. (i) From Proposition 2.2 we have id Λ (φ,ψ) T A (A) < ∞ and id Λ (φ,ψ) T B (B) < ∞. Then, from Lemma 4.8 (i) we have the following isomorphisms for every A-module X and n ≥ 0 : Similarly, for every B-module Y and n ≥ 0 we have from Lemma 4.8 (ii) the following isomorphisms : (ii) In this part we use right modules. If X is a right A-module, then T A (X) = (X, X ⊗ A N, Id X⊗N , Ψ X ) and since A N is projective it follows that the functor − ⊗ A N : mod-A −→ mod-B is exact and therefore T A is exact. Similarly, since B M is projective we obtain that the functor T B is exact. Then using the isomorphisms of Lemma 4.8, but for right modules now, the result follows as in case (i).
It should be clear from the proof of the above result, that the assumption of M A , resp. N B , being projective, implies that id A A < ∞ and id A N < ∞, resp. id B B < ∞ and id B M < ∞. The same separation property also holds for part (ii). We continue with the next consequence of Lemma 4.9. Let Λ be an Artin algebra and consider the Morita ring ∆ (φ,φ) = Λ Λ Λ Λ . If ∆ (φ,φ) is Gorenstein, then by Corollary 4.10 the algebra Λ is also Gorenstein. We mention that this was observed in Proposition 3.4 (i), where the the converse also holds in this case. Hence, Corollary 4.10 generalizes the one direction of Proposition 3.4 (i). We give an example to show that the conditions in Corollary 4.10 are only sufficient. Example 4.11. Let Λ be a bimodule d-Calabi-Yau noetherian algebra over a field k, where d ≥ 2 is an integer. Let e be a non-trivial idempotent element of Λ such that Λ/ΛeΛ is a finite dimensional kalgebra. By [1, Theorem 2.2] and it's proof, the algebra eΛe is Gorenstein and the eΛe-module eΛ is a non-projective Gorenstein-projectve. Note that from [1, Proposition 2.4] the algebra Λ has finite global dimension and therefore Λ is Gorenstein.
In the rest of the subsection our aim is to consider the converse of Corollary 4.10, that is how the Gorensteinness of A and B should be inherited to the whole Morita ring. We first need the following preliminary result. As usual we denote by D : mod-Λ −→ mod-Λ op the duality for Artin algebras, see [5].
be a Morita ring which is an Artin algebra. The following is the main result of this section which provides sufficient conditions for Morita rings Λ (0,0) with zero bimodule homomorphisms such that silp Λ (0,0) < ∞ and spli Λ (0,0) < ∞. This result constitutes the second part of Theorem A presented in the Introduction.
We continue with the following result which gives us a class of Morita rings, in particular a class of trivial extension rings (Example 2.3 (iv)), where Theorem 4.13 can be applied. If A is Gorenstein then the ring Λ (0,0) is Gorenstein. In particular, if gl. dim A < ∞ then gl. dim Λ (0,0) < ∞.
Proof. The condition N ⊗ A N = 0 implies that the functor Z A : mod-A −→ mod-Λ (0,0) is a homological embedding, see Corollary 4.6. The second functor we have to check that is a homological embedding is Z ′ A : mod-A −→ mod-Λ (0,0) given by Z ′ A (X) = (0, X, 0, 0). Note that this is the functor Z 2 in the notation of subsection 2.2. Exactly in the same way with Remark 2.6 (iii), we show that the two recollements of mod-Λ (0,0) are equivalent. Both of them are of the form (mod-A, mod-Λ (0,0) , mod-A), see Proposition 2.4. Using this equivalence it follows that Z A is a homological embedding if and only if Z ′ A is a homological embedding. Then the result follows from Theorem 4.13 and Corollary 4.14.
The above method for constructing Gorenstein algebras is illustrated in the next example. Example 4.16. Let A be a finite dimensional Gorenstein k-algebra, where k is a field, and let e and f be two idempotents elements of A such that f Ae = 0. Consider the A-A-bimodule N := Ae ⊗ k f A. Then it follows easily that N ⊗ A N = 0 and therefore from Corollary 4.15 we get the Gorenstein algebra : Note that Λ (0,0) is the trivial extension algebra (A × A) ⋉ N ⊕ N , see Example 2.3 (iv).
We close this section with an example of a Morita ring which is a Gorenstein algebra and the conditions of Theorem 4.13 are not satisfied.  [15], defined by the pair (A, M ). If M A is a finitely generated projective right A-module, then from [15, Proposition 2.6, Corollary 1.10] it follows that the rings A and Λ (φ,ψ) are Morita equivalent, and therefore A is Gorenstein if and only if Λ (φ,ψ) is Gorenstein. Hence, if A is a Gorenstein algebra and M A is a finitely generated projective module, then the Morita ring Λ (φ,ψ) is Gorenstein. By Example 2.3 (ii), the bimodule homomorphisms of this Morita ring are not zero, and also the rest assumptions of Theorem 4.13 are not satisfied in general.

Gorenstein Subcategories and Coherent Functors
In this section we study the monomorphism category mono(Λ), see equation (2.6) in subsection 2.3. In particular, we investigate the full subcategory C of mono(Λ) consisting of all monomorphisms f : X −→ Y such that the projective dimension of X is finite. In the first subsection, we show that it is a Gorenstein subcategory of mono(Λ) when Λ is a Gorenstein Artin algebra. In the second subsection, we prove that the category of coherent functors over the stable category of C is a Gorenstein abelian category. 5.1. The Gorenstein subcategory of mono(Λ). Let A be an abelian category with enough projective and injective objects and let n be a non-negative integer. Recall from [12, Theorem 2.2, Chapter VII] that A is n-Gorenstein if and only if every object has Gorenstein-projective dimension at most n. For our purpose, we need the notion of a Gorenstein subcategory but now in the context of exact categories (see subsection 2.3). Before that, we define Gorenstein-projective objects for exact categories.
Definition 5.1. Let A = (A , E ) be an exact category with enough projective objects. An object X in A is called Gorenstein-projective if there is an E -acyclic complex of projective objects in A : such that Hom A (P • , P ) is acyclic for every object P in Proj A and d 0 = λ • κ, where κ : P 0 −→ X is a deflation and λ : X −→ P 1 is an inflation. We denote by GProj A the full subcategory of Gorensteinprojective objects of A .
For a complex being acyclic in an exact category we refer to [16,Definition 10.1]. From now on, when we write A for an exact category we fix a class E of exact pairs. Definition 5.2. Let A be an exact category with enough projective objects. Then A is n-Gorenstein for some non-negative integer n if every object has Gorenstein-projective dimension at most n. Let B be an exact subcategory of A . We call B an n-Gorenstein subcategory of A , if for all X in B there exists an exact sequence 0 −→ G n −→ · · · −→ G 0 −→ X −→ 0 in B such that G j ∈ GProj A for all 0 ≤ j ≤ n.
Let Λ be an Artin algebra and T 2 (Λ) = Λ 0 Λ Λ be the lower triangular matrix algebra. Let ∆ (0,0) = Λ Λ Λ Λ be the Morita ring which is an Artin algebra, see Example 2.3 (iv) and Example 2.7. The following results provides a description of the Gorenstein-projective objects of mono(Λ). Recall that the singularity category D sg (Λ) of Λ is defined to be the Verdier quotient D b (mod-Λ)/K b (proj Λ), see [14,34].
Proof. From Proposition 5.3 (i) and since Λ is selfinjective, it follows that the category Gproj(mono(Λ)) is precisely mono(Λ). Then the result follows immediately from Proposition 5.3 (ii).
Consider now the following subcategory of mono(Λ) : C := (X, Y, f, 0) ∈ mono(Λ) | pd Λ X < ∞ . (5.2) We denote by P <∞ (Λ) the full subcategory of mod-Λ consisting of all Λ-modules of finite projective dimension. Then P <∞ (Λ) is an exact subcategory of mod-Λ, since it is extension closed, and this implies that C is also an exact subcategory of mono(Λ). The first main result on the structure of C is as follows. This result constitutes the first part of Theorem B presented in the Introduction.
Theorem 5.5. Let Λ be an n-Gorenstein algebra for some non-negative integer n. Then C is an n-Gorenstein subcategory of mono(Λ).
kernels, that is, for each morphism f : X −→ Y in A there exists a morphism g : Z −→ X in A such that the sequence Hom A (−, Z) −→ Hom A (−, X) −→ Hom A (−, Y ) is exact. We denote by mod-A the category of coherent functors over A , i.e. functors F : A op −→ Ab such that there is an exact sequence Hom A (−, X) −→ Hom A (−, Y ) −→ F −→ 0 with X and Y in A . It is known that mod-A is an abelian category with enough projective objects. We refer to [2,3] for more details on coherent functors. Then, Matsui and Takahashi [32] considered the Verdier quotient D sg (mod-A ) := D b (mod-A )/K b (proj(mod-A )) and call it the singularity category of mod-A . We remark that this triangulated category is included in the general framework of the stabilization of an abelian or exact category studied by Beligiannis [8].
In what follows, we show that the singularity category of mono(Λ) is trivial. We write mod-mono(Λ) for the category of coherent functors over the monomorphism category mono(Λ).
Proposition 5.6. Let Λ be an Artin algebra. Then the following hold (i) The category mod-mono(Λ) is abelian.