Relative Gorenstein dimensions over triangular matrix rings

Let $A$ and $B$ be rings, $U$ a $(B,A)$-bimodule and $T=\begin{pmatrix} A&0\\U&B \end{pmatrix}$ the triangular matrix ring. In this paper, several notions in relative Gorenstein algebra over a triangular matrix ring are investigated. We first study how to construct w-tilting (tilting, semidualizing) over $T$ using the corresponding ones over $A$ and $B$. We show that when $U$ is relative (weakly) compatible we are able to describe the structure of $G_C$-projective modules over $T$. As an application, we study when a morphism in $T$-Mod has a special $G_CP(T)$-precover and when the class $G_CP(T)$ is a special precovering class. In addition, we study the relative global dimension of $T$. In some cases, we show that it can be computed from the relative global dimensions of $A$ and $B$. We end the paper with a counterexample to a result that characterizes when a $T$-module has a finite projective dimension.


Introduction
Semidualizing modules were independently studied (under different names) by Foxby [12], Golod [14], and Vasconcelos [27] over a commutative Noetherian ring. Golod used these modules to study G C -dimension for finitely generated 1 INTRODUCTION modules. Motivated (in part) by Enochs and Jenda's extensions of the classical G-dimension given in [11], Holm and Jφrgensen, extended in [18] this notion to arbitrary modules. After that, several generalizations of semidualizing and G C -dimension have been made by several authors ( [29], [23], [3]).
As the authors mentioned in [6], to study the Gorenstein projective modules and dimension relative to a semidualizing (R, S)-bimodule C, the condition End S (C) ∼ = R, seems to be too restrictive and in some cases unnecessary. So the authors introduced weakly Wakamatsu tilting as a weakly notion of semidualizing which made the theory of relative Gorenstein homological algebra wider and less restrictive but still consistent. Weakly Wakamatsu tilting modules were subject of many publications which showed how important these modules could become in developing the theory of relative (Gorenstein) homological algebra ( [6], [7], [5]) Let A and B be rings and U be a (B, A)-bimodule. The ring T = A 0 U B is known as the formal triangular matrix ring with usual matrix addition and multiplication. Such rings play an important role in the representation theory of algebras. The modules over such rings can be described in a very concrete fashion. So, formal triangular matrix rings and modules over them have proved to be a rich source of examples and counterexamples. Some important Gorenstein notions over formal triangular matrix rings have been studied by many authors (see [30,10,26]). For example, Zhang [30] introduced compatible bimodules and explicitly described the Gorenstein projective modules over triangular matrix Artin algebra. Enochs, Izurdiaga, and Torrecillas [10] characterized when a left module over a triangular matrix ring is Gorenstein projective or Gorenstein injective under the "Gorenstein regular" condition. Under the same condition, Zhu, Liu, and Wang [26] investigated Gorenstein homological dimensions of modules over triangular matrix rings. Mao [25] studied Gorenstein flat modules over T (without the "Gorenstein regular" condition) and gave an estimate of the weak global Gorenstein dimension of T .
The main objective of the present paper is to study relative Gorenstein homological notions (w-tilting, relative Gorenstein projective modules, relative Gorenstein projective dimensions and relative global projective dimension) over triangular matrix rings. This article is organized as follows: In Section 2, we give some preliminary results.
In Section 3, we study how to construct w-tilting (tilting, semidualizing) over T using w-tilting (tilting, semidualizing) over A and B under the condition that U is relative (weakly) compatible. We introduce (weakly) C-compatible (B, A)bimodules for a T -module C (Definition 3.2). Given two w-tilting modules A C 1 and B C 2 , we prove in Proposition 3.6 that C = C 1 (U ⊗ A C 1 ) ⊕ C 2 is a w-tilting T -module when U is weakly C-compatible.
In Section 4, we first describe relative Gorenstein projective modules over T .
be a T -module. We prove in Theorem 4.3, that if As an application, we prove that the converse of Proposition 3.6 and refine in relative setting (Proposition 4.9), a result of when T is left (strongly) CM-free due to Enochs, Izurdiaga, and Torrecillas in [10]. Also when C is w-tilting, we characterize when a T -morphism is a special precover (see Proposition 4.10). Then in Theorem 4.11, we prove that class of G C -projective T -modules is special precovering if and only if so are the classes of G C1 -projective A-modules and G C2 -projective B-modules, respectively.
Finally, in Section 5, we give an estimate of G C -projective dimension of a left T -module and the left G C -projective global dimension of T . First, it is proven As an application, we prove that, if C = p(C 1 , C 2 ) is w-tilting and U is C- Some cases when this estimation becomes an exact formula are also given.
The authors in [2] establish a relationship between the projective dimension of modules over T and modules over A and B. Given an integer n ≥ 0 and ≤ n and the map related to the n-th syzygy of M is injective. We end the paper by giving a counterexample to this result (Example 5.11).

Preliminaries
Throughout this paper, all rings will be associative (not necessarily commutative) with identity, and all modules will be, unless otherwise specified, unitary left modules. For a ring R, We use Proj(R) (resp., Inj(R)) to denote the class of all projective (resp., injective) R-modules. The category of all left R-modules will be denoted by R-Mod. For an R-module C, we use Add R (C) to denote the class of all R-modules which are isomorphic to direct summands of direct sums of copies of C, and Prod R (C) will denote the class of all R-modules which are isomorphic to direct summands of direct products of copies of C.
Given a class of modules F (which will always be considered closed under isomorphisms), an F -precover of M ∈ R-Mod is a morphism ϕ : F → M (F ∈ F ) such that Hom R (F ′ , ϕ) is surjective for every F ′ ∈ F . If, in addition, any solution of the equation Hom R (F, ϕ)(g) = ϕ is an automorphism of F , then ϕ is said to be an F -cover. The F -precover ϕ is said to be special if it is surjective and Ext 1 (F, ker ϕ) = 0 for every F ∈ F . The class F is said to be special (pre)covering if every module has a special F -(pre)cover.
Given the class F , the class of all modules N such that Ext i R (F, N ) = 0 for every F ∈ F will be denoted by F ⊥i (similarly, ⊥i F = {N ; Ext i R (N, F ) = 0 ∀F ∈ F }). The right and left orthogonal classes F ⊥ and ⊥ F are defined as follows: Given a class of R-modules F , an exact sequence of R-modules is called Hom R (−, F )-exact (resp., Hom R (F , −)-exact) if the functor Hom R (−, F ) (resp., Hom R (F, −)) leaves the sequence exact whenever F ∈ F . If F = {F }, we simply say Hom R (−, F )-exact. Similarly, we can define F ⊗ R −exact sequences when F is a class of right R-modules.
We now recall some concepts needed throughout the paper.
both are ring isomorphisms.

([28, Section 3]) A Wakamatsu tilting module, simply tilting, is an R-
module C satisfying the following properties: (a) R C admits a degreewise finite projective resolution.
It was proved in [28,Corollary 3.2], that an (R, S)-bimodule C is semidualizing if and only if R C is tilting with S = End R (C). So the following notion, which will be crucial in this paper, generalizes both concepts.
Definition 2.2 ([6], Definition 2.1) An R-module C is weakly Wakamatsu tilting (w-tilting for short) if it has the following two properties: 1. Ext i≥1 R (C, C (I) ) = 0 for every set I.

2.
There exists a Hom R (−, Add R (C))-exact exact sequence of R-modules If C satisfies 1. but perhaps not 2. then C will be said to be Σ-self-orthogonal.

Definition 2.3 ([6], Definition 2.2)
Given any C ∈ R-Mod, an R-module M is said to be G C -projective if there exists a Hom R (−, Add R (C))-exact exact sequence of R-modules We use G C P (R) to denote the class of all G C -projective R-modules.
It is immediate from the definitions that w-tilting modules can be characterized as follows. is a B-morphism, and whose morphisms from satisfying that the following diagram is commutative.

Since we have the natural isomorphism
there is an alternative way of defining T -modules and T -homomorphisms in terms of maps ϕ M : , and for each morphism It is easy to see that q is exact. In particular, p preserves projective objects and h preserves injective objects. Note that the pairs of adjoint functors (p, q) and (q, h) are defined in [10]. In general, the three pairs of adjoint functors defined above can be found in [13].
For a future reference, we list these adjointness isomorphisms: Hom T (N, Now we recall the characterizations of projective, injective and finitely generated T -modules.

B
(U, N 2 ) = 0, then Ext n T (M, Proof. We prove only 1., since 2. is similar and 3. and 4. are dual. Assume that Tor A 1≤i≤n (U, M 1 ) = 0 and consider an exact sequence of A-modules where P 1 is projective. So, there exists an exact sequence of T -modules is projective by Lemma 2.5.
Let n = 1. By applying the functor Hom T (−, N ) to the above short exact sequence and since P 1 U ⊗ A P 1 and P 1 are projectives, we get a commutative diagram with exact rows where the first two columns are just the natural isomorphisms given by adjointeness and the last two horizontal rows are epimorphisms. Thus, the induced map is an isomorphism such that the above diagram is commutative.
Assume that n > 1. Using the long exact sequence, we get a commutative diagram with exact rows where σ is a natural isomorphism by induction, since Tor A k (U, K 1 ) = 0 for every k ∈ {1, · · · , n − 1} because of the exactness of the following sequence Thus, the composite map is a natural isomorphism, as desired.
Since T can be viewed as a trivial extension (see [13] and [4] for more details), the following Lemma can be easily deduced from ([4, Theorem 3.1 and Theorem 3.4]). For the convenience of the reader, we give a proof.
In this case, ϕ X is injective.
In this case, ϕ X is surjective.
for some (Y 1 , Y 2 ) ∈ A-Mod×B-Mod and some sets I 1 and I 2 . Without loss of generality, we may assume that I = I 1 = I 2 . Then Hence X ∈ Add T (p(C 1 , C 2 )).
which induces the following commutative diagram with exact rows and colm- where ϕ X , ϕ C and ϕ X are the canonical projections. Clearly, p 1 : C → X 2 are split epimorphisms. Then X 1 ∈ Add A (C 1 ) and X 2 ∈ Add B (C 2 ). It remains to prove that X ∼ = p(X 1 , X 2 ). For this, it suffices to prove that the short exact sequence denotes the canonical injection, then ϕ X p 2 ir 2 = p 2 ϕ C ir 2 = p 2 r 2 = 1 X2 and the proof is finished.

Remark 2.8
1. Since the class of projective modules over T is nothing but the class Add T (T ), when we take C 1 = A and C 2 = B in Lemma 2.7, we recover the characterization of projective and injective T -modules.

w-Tilting modules
In this section, we study when the functor p preserves w-tilting modules.
It is well known that the functor p preserves projective modules. However, the functor p does not preserve w-tilting modules in general, as the following example shows.
and let R = kQ be the path algebra over an algebraic closed field k. Put P 1 = Re 1 , P 2 = Re 2 , I 1 = Hom k (e 1 R, k) and I 2 = Hom k (e 2 R, k). Note that, C 1 and C 2 are projective and injective R-modules, respectively. By [3,Example 2.3], are semidualzing (R, R)-bimodules and then w-tilting R-modules. Now, consider the triangular matrix ring Motivated by the definition of compatible bimodules in [30, Definition 1.1], we introduce the following definition which will be crucial throughout the rest of this paper.
where the P i 2 's are all projective and C i 2 ∈ Add B (C 2 ) ∀i.
Moreover, U is called weakly C-compatible if it satisfies (b) and the following condition When C = T T = p(A, B), the bimodule U will be called simply (weakly) compatible.

Remark 3.3 1. It is clear by the definition that every C-compatible is weakly
C-compatible. [21]).

If A and B are Artin algebras and since
is easy to see that T T -compatible bimodules are nothing but compatible (B, A)-bimodules as defined in [30].
The following can be applied to produce examples of (weakly) C-compatible bimodules later on.

Assume that Ext
Proof. (3) is clear. We only prove (1), as (2) is similar. Consider an exact sequence of A-modules where the P i 1 's are all projective and C i 1 ∈ Add A (C 1 ) ∀i. We use induction on fd A U . If fd A U = 0, then the result is trivial. Now suppose that fd A U = n ≥ 1.
Then, there exists an exact sequence of right A-modules where fd A L = n − 1 and F is flat. Applying the functor − ⊗ X 1 to the above short exact sequence, we get the commutative diagram with exact rows : : : : 0 By induction hypothesis, the complexes L ⊗ A X 1 and F ⊗ A X 1 are exact. Thus Given a T -module C = p(C 1 , C 2 ), we have simple characterizations of conditions (a ′ ) and (b) if C 1 and C 2 are w-tilting.
1. If C 1 is w-tilting, then the following assertions are equivalent: In this case, Tor A i≥1 (U, C 1 ) = 0.
2. If C 2 is w-tilting, then the following assertions are equivalent: In this case, Ext i≥1 Proof. We only prove (1), since (2) is similar.
In the following proposition, we study when p preserves w-tilting (tilting) modules.
Proposition 3.6 Let C = p(C 1 , C 2 ) be a T -module and assume that U is weakly C-compatible. If C 1 and C 2 are w-tilting (tilting), then p(C 1 , C 2 ) is w-tilting (tilting).
Proof. By Lemma 2.5, the functor p preserves finitely generated modules, so we only need prove the statement for w-tilting. Assume that C 1 and C 2 are w-tilting and let I be a set. Then Ext i≥1 A (C 1 , C 1 ) = 0 and Ext i≥1 B (C 2 , C 2 ) = 0. By Proposition above, we have Ext i≥1 B (C 2 , U ⊗ A C (I) 1 ) = 0 and Tor A i≥1 (U, C 1 ) = 0. Using Lemma 2.6, for every n ≥ 1 we get that 2 ) = 0.
Moreover, there exist exact sequences which are Hom A (−, Add A (C 1 ))-exact and Hom B (−, Add B (C 2 ))-exact, respectively, and such that C i 1 ∈ Add A (C 1 ) and C i 2 ∈ Add B (C 2 ) for every i ∈ N. Since U is weakly C-compatible, the complex U ⊗ A X 1 is exact. So we construct in T -Mod the exact sequence . As a consequence of Lemma 2.7(1), X = p(X 1 , X 2 ) where X 1 ∈ Add A (C 1 ) and X 2 ∈ Add B (C 2 ). Using the adjoitness (p, q), we get an isomorphism of complexes But Hom A (X 1 , X 1 ) and Hom B (X 2 , X 2 ) are exact and the complex Hom B (X 2 , U ⊗ X 1 ) is also exact since U is weakly C-compatible. Then, Hom T (p(X 1 , X 2 ), X) is exact as well and the proof is finished. Now, we illustrate Proposition 3.6 with two applications.
Corollary 3.7 Let C = p(C 1 , C 2 ) be a T -module, A ′ and B ′ be two rings such that A C A ′ and A C B ′ are bimodules and assume that U is weakly C-compatible. If A C A ′ and A C B ′ are semidualizing bimodules, then p(C 1 , C 2 ) is a semidualizing (T, End T (C))-bimodule.
Corollary 3.8 Let R and S be rings, θ : R → S be a homomorphism with S R flat, and T = T (θ) =: is a w-tilting T (θ)-module.
Proof. 1. Let C 2 = S ⊗ R C 1 and note that C = p(C 1 , C 2 ) and that S S R is C-compatible. So, by Proposition 3.6, we only need to prove that C 2 is w-tilting S-module.
Since R C 1 is w-tilting, there exist Hom R (−, Add R (C 1 ))-exact exact sequences with each R P i projective and R C i ∈ Add R (C 1 ). Since S R is flat, we get an exact with each S ⊗ R P i a projective S-module and S ⊗ R C i ∈ Add R (C 2 ).
We prove now that S⊗ R P and S⊗ R X are Hom S −(, Add S (C 2 ))-exact . Let I be a set. Then, the complex Hom S (S⊗ R P, S⊗ R C 2. This assertion follows from Proposition 3.6. We only need to note that S is weakly C-compatible since S R is flat and S ⊗ R C 1 ∈ Add R (C 2 ).
We end this section with an example of a w-tilting module that is neither projective nor injective.
Example 3.9 Take R and C 2 as in example 3.1. So, by Corollary 3.8, C = C 2 C 2 ⊕ C 2 is a w-tilting T (R)-module. By Lemma 2.5, C is not projective since C 2 is not and it is not injective since the map ϕ C : C 2 → C 2 ⊕ C 2 is not surjective.

Relative Gorenstein projective modules
In this section, we describe G C -projective modules over T . Then we use this description to study when the class of G C -projective T -modules is a special precovering class.
Clearly the functor p preserves projective module. So we start by studying when the functor p also preserves relative Gorenstein projective modules. But first we need the following Lemma 4.1 Let C = p(C 1 , C 2 ) be a T -module and U be weakly C-compatible.
Proof. 1. Suppose that M 1 ∈ G C1 P (A). There exists a Hom A (−, Add A (C 1 ))exact exact sequence where the P i 1 's are all projective, C i 1 ∈ Add A (C 1 ) ∀i and M 1 ∼ = Im(P 0 1 → C 0 1 ). Using the fact that U is weakly C-compatible, we get that the complex U ⊗ A X 1 is exact in B-Mod, which implies that the complex p(X 1 , 0) : Lemma 2.5(1) and Lemma 2.7(1). If X ∈ Add T (C), then X 1 ∈ Add A (C 1 ) by Lemma 2.7(1) and using the adjointness, we get that the complex 2. Suppose that M 2 is G C2 -projective. There exists a Hom B (−, Add B (C 2 ))exact exact sequence where the P i 2 's are all projective, C i 2 ∈ Add B (C 2 ) ∀i and M 2 ∼ = Im(P 0 2 → C 0 2 ). Clearly the complex ∈ Proj(T ) and Add T (C) ∀i, by Lemma 2.5(1) and Lemma 2.7(1). Let X ∈ Add T (C). Then, by Lemma 2.7(1), X = p(X 1 , X 2 ) where X 1 ∈ Add A (C 1 ) and X 2 ∈ Add B (C 2 ). Using adjointness, we get that The complex Hom B (X 2 , X 2 ) is exact and since U is weakly C-compatible, the complex Hom B (X 2 , U ⊗ A X 1 ) is also exact. This means that Hom T (p(0, X 2 ), X) is exact as well and 0 M 2 is G C -projective. Proof. Note that So this direction follows from Lemma 4.1 and [6, Proposition 2.5]. Conversely, assume that C 1 and C 2 are w-tilting. By Proposition 3.5, it suffices to prove that Tor A 1 (U, G C1 P (A)) = 0 = Ext 1 B (G C2 P (B), U ⊗ A Add A (C 1 )). Let G 1 ∈ G C1 P (A). By [6, Corollary 2.13], there exits a Hom A (−, Add A (C 1 ))exact exact sequence 0 → L 1 ı → P 1 → G 1 → 0, where A P 1 is projective and L 1 is G C1 -projective. Note that A, C 1 ∈ G C1 P (A) and B, C 2 ∈ G C2 P (B) by Lemma 2.4. Then T T = p(A, B) and C = p(C 1 , C 2 ) are G C -projective, which imply by Lemma 2.4, that C is w-tilting. Moreover 0) is also G C -projective and by [6,Corollary 2.13] there exists a short exact sequence Since X 1 ∈ Add A (C 1 ), we have the following commutative diagram with exact rows: So if we apply the functor U ⊗ A − to the above diagram, we get the following commutative diagram with exact rows: The commutativity of this diagram implies that the map 1 U ⊗ ı injective, and since P 1 is projective, So, M is G C -projective by [6, Proposition 2.5].
1. ⇒ 2. There exists a Hom T (−, Add T (C))-exact sequence in T -Mod ∈ Proj(T ) ∀i ∈ N, and such that M ∼ = Im(P 0 → C 0 ). Then, we get the exact sequence where C i 1 ∈ Add A (C 1 ), P i 1 ∈ Proj(A) ∀i ∈ N by Lemma 2.7(1) and Lemma 2.5(1), and such that M 1 ∼ = Im(P 0 and ι 2 : M 2 → C 0 2 are the inclusions, then 1 U ⊗ ι 1 is injective and the following diagram commutes: By Lemma 2.7(1), ϕ C 0 is injective, then ϕ M is also injective. For every i ∈ N, ϕ P i and ϕ C i are injective by Lemma 2.5 and Lemma 2.7(1). Then the following diagram with exact columns is commutative. Since the first row and the second row are exact, we get the exact sequence of B-modules where P i 2 ∈ Proj(B), C i 2 ∈ Add B (C 2 ) by Lemma 2.5 and Lemma 2.7(1), and such that M 2 = Im(P 0 2 → C 0 2 ). It remains to see that X 1 and X 2 are Hom A (−, Add(C 1 ))-exact and Hom B (−, Add B (C 2 ))-exact, respectively. Let Add T (C) and p(0, X 2 ) = 0 X 2 ∈ Add T (C) by Lemma 2.7(1). So, by using adjointness, we get that Hom B (X 2 , X 2 ) ∼ = Hom T (X, 0 X 2 ) is exact. Using adjointness again we get that and Hom T (X, 6. So, if we apply the functor Hom T (X, −) to the sequence we get the following exact sequence of complexes Since U is C-compatible, it follows that Hom B (X 2 , U ⊗ A X 1 ) is exact and since C is w-tilting, Hom T (X, )is also exact. Thus Hom A (X 1 , X 1 ) is exact and the proof is finished.
The following consequence of the above theorem gives the converse of Proposition 3.6. Corollary 4.4 Let C = p(C 1 , C 2 ) and assume that U is C-compatible. Then C is w-tilting if and only if C 1 and C 2 are w-tilting.

Proof. An easy application of Proposition 2.4 and Theorem 4.3 on the T -
One would like to know if every w-tilting T -module has the form p(C 1 , C 2 ) where C 1 and C 2 are w-tilting. The following example gives a negative answer to this question.
Note that T (R) is noetherian ( [16,Proposition 1.7]) and then we can see that C := I 0 ⊕ I 1 is a w-tilting T (R)-module but does not have the form p(C 1 , C 2 ) where C 1 and C 2 are w-tilting by Lemma 2.7 since I 1 ∈ Add T (R) (C) and ϕ I 1 is not injective.
As an immediate consequence of Theorem 4.3, we have the following. An Artin algebra Λ is called Cohen-Macaulay free, or simply, CM-free if any finitely generated Gorenstein projective module is projective. The authors in [10], extended this definition to arbitrary rings and defined strongly CM-free as rings over which every Gorenstein projective module is projective. Now, we introduce a relative notion of these rings and give a characterization of when T is such rings.

Corollary 4.6 Let R be a ring and T
Definition 4.7 Let R be a ring. Given an R-module C, R is called CM-free (relative to C) if G C P (R) ∩ R-mod = add R (C) and it is called strongly CM-free (relative to C) if G C P (R) = Add R (C).
Remark 4.8 Let R be a ring and C a Σ-self-orthogonal R-module. Then Add R (C) ⊆ G C P (R) and add R (C) ⊆ G C P (R) ∩ R-mod by [6, Proposition 2,5, 2,6 and Corollary 2.10], then R is CM-free (relative to C) if and only if every finitely generated G C -projective is in add R (C) and it is strongly CM-free (relative to C) if every G C -projective is in Add R (C).
Using the above results we refine and extend [10, Theorem 4.1] to our setting.
Note that the condition B is left Gorenstein regular is not needed. Proposition 4.9 Let A C 1 and B C 2 be Σ-self-orthogonal, and C = p(C 1 , C 2 ). Assume that U is weakly C-compatible and consider the following assertions: 1. T is (strongly) CM-free relative to C.

2.
A and B are (strongly) CM-free relative to C 1 and C 2 , respectively.
Proof. We only prove the the result for relative strongly CM-free, since the the case of relative CM-free is similar.
1. ⇒ 2. By the remark above, we only need to prove that G C1 P (A) ⊆ Add A (C 1 ) and G C2 P (B) ⊆ Add B (C 2 ). Let M 1 be a G C1 -projective A-module and B M 2 a G C2 -projective B-module. By the assumption and Proposition Clearly, C is Σ-self-orthogonal, then by Remark above, we only need to prove that G C P (T ) ⊆ Add T (C). Let Our aim now is to study special G C P (T )-precovers in T -Mod. We start with the following result.
→ M 2 is surjective with its kernel lies in G C2 P (B) ⊥1 .
Let M 2 be a B-module and 0 f 2 : precover in T -Mod. By Proposition 4.10, G 1 → 0 is a special G C1 P (A)-precover.
Then Ext 1 A (G C1 P (A), G 1 ) = 0. On the other hand, by [6,Proposition 2.8], there exists an exact sequence of A-modules where X 1 ∈ Add A (C 1 ) and H 1 is G C1 -projective. But this sequence splits, since Using the snake lemma, there exists an exact sequence of B-modules where G 2 is G C2 -projective by Theorem 4.3. It remains to see that K 2 lies in . Then Ext 1 B (H 2 , K 2 ) = 0 by Proposition 4.10 and Ext i≥1 B (H 2 , U ⊗ A G 1 ) = 0 by Proposition 3.5(2). From the above diagram, ϕ K is injective. So, if we apply the functor Hom B (H 2 , −) to the short exact sequence we get an exact sequence

Relative global Gorenstein dimension
In this section, we investigate G C -projective dimension of T -modules and the left G C -projective global dimension of T . Let R be a ring. Recall ( [6]) that a module M is said to have G C -projective dimension less than or equal to n, with G i ∈ G C P (R) for every i ∈ {0, · · · , n}. If n is the least nonnegative integer for which such a sequence exists then G C −pd(M ) = n, and if there is no such The left G C -projective global dimension of R is defined as: Lemma 5.1 Let C = p(C 1 , C 2 ) be w-tilting and U C-compatible.
, and the equality holds if Proof. 1. Let n ∈ N and consider an exact sequence of B-modules Thus, there exists an exact sequence of Tmodules is G C -projective if and only if K n 2 is G C1 -projective which means that Hence 2. We may assume that n = G C −pd( there exists an exact sequence of T -modules Thus, there exists an exact sequence of A-modules . We may assume that m := G C1 −pd(M 1 ) < ∞. The hypothesis means that if is an exact sequence of A-modules where each P i 1 is projective, then the complex U ⊗ A X 1 is exact. Since C 1 is w-tilting, each P i is G C1 -projective by [6, Proposition 2.11] and then K m is G C1 -projective by [6,Theorem 3.8]. Thus, there exists and exact sequence of T -modules are G C -projectives by Theorem 4.3. Therefore, Given a T -module C = p(C 1 , C 2 ), we introduce a strong notion of G C2projective global dimension of B, which will be crucial when we estimate the G C -projective of a T -module and the left global G C -projective dimension of T . Set 2. Note that pd B (U ) = sup{pd B (U ⊗ A P ) | A P is projective }. Therefore, in the classical case, the strong left global dimension of B is nothing but the projective dimension of B U .
Proof. First of all, note that C 1 and C 2 are w-tilting by Proposition 4.4 and let k := SG C2 − P D(B).
Let us first prove that We may assume that n := G C −pd(M ) < ∞. Then, there exists an exact sequence of T -modules Thus, there exists an exact se-  Proposition 3.11]. So, using the exact sequence of B-modules and [6, Proposition 3.11(4)], we get that G C2 −pd(M 2 ) ≤ n + k.
Next we prove that We may assume that Then n 1 := G C1 −pd(M 1 ) < ∞ and n 2 : Then, we get an exact sequence of T -modules Similarly, there exists an exact sequence of B-modules G 1 2 g 1 2 → K 1 2 → 0 where G 1 2 is G C2 -projective and then, we get an exact sequence of T -modules repeat this process, we get the exact sequence of T -modules Note that G C2 −pd((U ⊗ A G i 1 ) ⊕ G i 2 ) = G C2 −pd(U ⊗ A G i 1 ) ≤ k, for every i ∈ {0, · · · , m − k − 1}. So, by [6, Proposition 3.11 (2)] and the exact sequence ) ≤ k. This means that, there exists an exact sequence of B-modules Thus, There exists an exact sequence of T -modules  The following theorem gives an estimate of the left G C -projective global dimension of T .
This shows that G C − P D(T ) = SG C2 − P D(B) + 1 and the proof is finished.
Corollary 5.9 Let R be a ring, T (R) = R 0 R R and C = p(C 1 , C 1 ) where C 1 is w-tilting. Then G C − P D(T (R)) = G C1 − P D(R) + 1.