GORENSTEIN PROJECTIVE MODULES

In this paper, we study the pair (GP(R),GP(R)⊥) where GP(R) is the class of all Gorenstein projective modules. We prove that it is a complete hereditary cotorsion theory, provided l.Ggldim(R) < ∞. We discuss also, when every Gorenstein projective module is Gorenstein flat. AMS Subject Classification: 13D05, 13D02.


Introduction
Throughout the paper, all rings are associative with identity, and an R-module will mean left R-module unless explicitly stated otherwise. Let R be a ring, and let M be an R-module. As usual, we use pd R (M), id R (M), and fd R (M) to denote, respectively, the classical projective dimension, injective dimension, and flat dimension of M. We denote by M + = Hom Z (M, Q/Z) the character module of M.
For a two-sided Noetherian ring R, Auslander and Bridger [1] introduced the G-dimension, Gdim R (M), for every finitely generated R-module M. They showed that Gdim R (M) ≤ pd R (M) for all finitely generated R-modules M, and equality holds if pd R (M) is finite.
Several decades later, Enochs and Jenda [5,6] introduced the notion of Gorenstein projective dimension (G-projective dimension for short), as an extension of G-dimension to modules that are not necessarily finitely generated, and the Gorenstein injective dimension (G-injective dimension for short) as a dual notion of Gorenstein projective dimension. Then, to complete the analogy with the classical homological dimension, Enochs, Jenda, and Torrecillas [8] introduced the Gorenstein flat dimension. Some references are [2,3,4,5,6,8,12]. * E-mail address: tamekkante@yahoo.fr The orthogonal complement relative to the functor extension of ...

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Recall that an R-module M is called Gorenstein projective, if there exists an exact sequence of projective R-modules: such that M ∼ = Im(P 0 → P 1 ) and such that the functor Hom R (−, Q) leaves P exact whenever Q is a projective R-module. The complex P is called a complete projective resolution.
The Gorenstein injective R-modules is defined dually. An R-module M is called Gorenstein flat, if there exists an exact sequence of flat Rmodules: such that M ∼ = Im(F 0 → F 1 ) and such that the functor I ⊗ R − leaves F exact whenever I is a right injective R-module. The complex F is called a complete flat resolution. The Gorenstein projective, injective, and flat dimensions are defined in terms of resolutions and denoted by Gpd(−), Gid(−), and Gfd(−), respectively (see [3,7,12]).
Notation. By P (R) and I (R) we denote the classes of all projective and injective R-modules, respectively, and by P (R) and I (R) we denote the classes of all modules with finite projective dimensions and injective dimensions, respectively. Furthermore, we let GP (R) and GI (R) denote the classes of all Gorenstein projective and injective R-modules, respectively.
In [2], the authors proved the equality Given a class X of R-modules we set: The class X ⊥ (resp., ⊥ X) is usually called the right (resp., left) orthogonal complement relative to the functor Ext 1 R (−, −) of the class X.

Definition 1.1 (Precovers and Preenvelopes)
. Let X be any class of R-modules and let M be an R-module.
• An X-precover of M is an R-homomorphism ϕ : X → M where X ∈ X and such that the sequence is exact for every X ∈ X. An X-precover is called special, if ϕ is surjective and ker(ϕ) ∈ X ⊥ .

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Tamekkante Mohammed • An X-preenvelope of M is an R-homomorphism ϕ : M → X where X ∈ X and such that the sequence, is exact for every X ∈ X. An X-preenvelope is called special, if ϕ is injective and coker(ϕ) ∈ ⊥ X.
For more details about precovers (and preenvelopes), the reader may consult [7, Chapters 5 and 6]. • We call X projectively resolving, if P (R) ⊆ X and for every short exact sequence 0 −→ X −→ X −→ X" −→ 0 with X" ∈ X the conditions X ∈ X and X ∈ X are equivalent.
• We call X injectively resolving, if I (R) ⊆ X and for every short exact sequence 0 −→ X −→ X −→ X" −→ 0 with X ∈ X the conditions X" ∈ X and X ∈ X are equivalent.
A pair (X, Y) of classes of R-modules is called a cotorsion theory [7], if X ⊥ = Y and ⊥ Y = X. In this case, we call X ∩ Y the kernel of (X, Y). Note that each element K of the kernel is a splitter in the sense of [11], i.e., Ext 1 R (K, K) = 0. If C is any class of modules, then ( ⊥ C, ( ⊥ C) ⊥ ) is easily seen be a cotorsion theory, called a cotorsion theory generated by C (see [13, Definition 1.10]). A cotorsion theory (X, Y) is called complete [13], if every R-module has a special Y-preenvelope (or equivalently every R-module has a special X-precover; see [13, Lemma 1.13]). A cotorsion theory (X, Y) is said to be hereditary The aim of this paper is the study of the pair (GP(R), GP(R) ⊥ ).
Note: Below, we have only proved the results concerning the Gorenstein projective modules. The proofs of the Gorenstein injective ones are dual, and we can find a dual of the results using in the proofs in [12].

Lemmas
In this section, we recall some fundamental results about Gorenstein projective modules and dimensions. These results are extracted from the work of Holm in [12].
The first lemma shows that the class of Gorenstein projective modules is projectively resolving: In the the following lemma, Holm gave a functorial description of the finite Gorenstein projective dimension of modules. 4. For every exact sequence of R-module 0 → K n → G n−1 → · · · → G 0 → M → 0 where G 0 , ..., G n−1 are Gorenstein projectives, K n is also Gorenstein projective.
Recall that the finitistic projective dimension of R is defined as:

Main results
We begin with the following theorem: (2) By a dual argument to (1).
(3) We claim that GP(R) ⊥ is projectively resolving. Using the long exact sequence in homology, we conclude that GP(R) ⊥ is closed by extension, i.e., arbitrary Gorenstein projective R-module G, consider a short exact sequence 0 → G → P → G → 0 where P is projective and G is Gorenstein projective (such a sequence exists by the definition of Gorenstein projective modules). From the long exact sequence of homology, we have: From the above theorem, we conclude the following two corollary. Consequently, M is a direct summand of P, and then projective.
Proof. The injective case is dual. Dually, we can prove that every module with finite projective dimension is an element of ⊥ GI(R). Consequently, by Corollary 3.2, every Gorenstein injective module with finite projective dimension is injective.
The main result of this paper is the following theorem:    Proof. From [13, Theorem 2.2], every R-module admits a special GP(R) ⊥ -preenvelope. On the other hand, by hypothesis, (GP(R), GP(R) ⊥ ) is the cotorsion theory generated by P (R). Then, (GP(R), P (R)) is a complete cotorsion theory. Therefore, every R-module M has a special GP(R)-precover.
From the above propositions, we conclude the following characterization of the left Gorenstein global dimension of a ring R, provided FPD(R) < ∞. 0. Consequently, I + ∈ GP(R) ⊥ .
(2) ⇒ (1) Consider a complete projective resolution P: · · · → P −2 We decompose it into a short exact sequences 0 → G i → P i → G i → 0 where G i = ker( f i ) and G i = Im( f i ). From [12, Observation 2.2], G i and G i are Gorenstein projectives. Now, let I be a right injective R-module. By hypothesis, we have (Tor 1 R (I, G i )) + = Ext 1 R (G i , I + ) = 0. Then, Tor 1 R (I, G i ) = 0. Therefore, is exact. Thus, I ⊗ R − keeps the exactness of P. Then, P is a complete flat resolution. Consequently, every Gorenstein projective module is Gorenstein flat.
(3) ⇒ (2) Let I be a right injective R-module. There exists a flat R-module F such that F → I + → 0 is exact. Then, 0 → (I + ) + → F + is exact. However, 0 → I → (I + ) + is exact (by [9, Proposition 3.52]). Thus, 0 → I → F + is exact, and then I is a direct summand of F + . Hence, I + is a direct summand of (F + ) + . On the other hand, it is easy to see that GP(R) ⊥ is closed under direct summands. Consequently, I + ∈ GP(R) ⊥ , as desired. Proof. Note that if Gfd R (M) ≤ n, then we have Tor i R (I, M) = 0 for all i > n. Indeed, the case n = 0 follows directly from the definition of the Gorenstein flats modules, whereas the case n > 0 is deduced from the first case by an n-step projective resolution of M. Suppose that sup{Gfd R (M) | M is Gorenstein projective} = n < ∞. Then, Ext n+1 R (G, I + ) = (Tor n+1 R (I, G)) + = 0 for every right injective module I and every Gorenstein projective module G. However, for every Gorenstein projective module G we can find an exact sequence 0 → G → P n−1 → ... → P 0 → G → 0 where all P i are projective and G is Gorenstein projective. Thus, Ext 1 R (G, I + ) = Ext n+1 R (G , I + ) = 0. So, I + ∈ GP(R) ⊥ for every right injective module I. Then, by Proposition 3.8, every Gorenstein projective module is Gorenstein flat. Consequently, sup{Gfd R (M) | M is Gorenstein projective} = 0, as desired. A direct consequence of the above proposition is the following corollary: Corollary 3.10. If l.wGgldim(R) < ∞, then every Gorenstein projective R-module is Gorenstein flat.