GORENSTEIN HOMOLOGICAL DIMENSIONS WITH RESPECT TO A SEMIDUALIZING MODULE

In this paper, let R be a commutative ring and C a semidualizing module. We investigate the (weak) C-Gorenstein global dimension of R and we get a simple formula to compute the C-Gorenstein global dimension. Moreover, we compare it with the classical (weak) global dimension of R and get the relations between them. At last, we compare the weak C-Gorenstein global dimension with the C-Gorenstein global dimension and we get that they are equal when R is Noetherian. Mathematics Subject Classification (2010): 13D02, 13D05, 13D07


Introduction
The notion of semidualizing module was studied more than 27 years ago under other names by, e.g., Foxby [6] (PG-modules of rank 1), Golod [7] (suitable modules) and Vasconcelos [12] (spherical modules), which can be viewed as a generalization of dualizing module and free module of rank one.Relative algebra with respect to a semidualizing module has caught many authors' attention.Let C be a semidualizing module over commutative Noetherian ring R, Holm and Jørgensen [9, Definition 2.7] introduced the notions of C-Gorenstein projective (injective and flat) modules, which are build from projective (injective and flat) and C-projective (injective and flat) modules, respectively.White [14] defined the C-Gorenstein projective (injective) modules over any commutative ring.In this field, projective (injective, flat) modules are generalized to C-projective (injective, flat) modules and Gorenstein projective (injective, flat) modules are generalized to C-Gorenstein projective (injective, flat) modules, etc., and the classical homological algebra is generalized to the Gorenstein homological algebra induced by a semidualizing module C. For this topic, we refer the reader to [9,11,14].Throughout this paper, R is a commutative ring and M odR is the category of all R-modules.

Preliminaries
In this section, we recall a number of definitions, notions and results which will be used throughout the paper.For the definitions of Gorenstein projective (injective, flat) modules we refer the readers to see [2,8].

ZHEN ZHANG AND JIAQUN WEI
Note that White [14] extended the definition of C-Gorenstein projective modules to the non-Noetherian ring, where she called G C -projective modules, we refer the reader to [9,14].
(2) There exist injective R-modules I 0 , I 1 , • • • together with an exact sequence: such that it stays exact when we apply the functor Hom R (Hom R (C, J), −) such that it stays exact when we apply the functor Hom R (−, (1) T or R i≥1 (Hom R (C, I), M ) = 0 for all injective R-modules I.

Gorenstein global dimensions induced by C
In this section, we investigate the (weak) C-Gorenstein global dimension of R.
Firstly, we prove an important lemma, which makes [9, Theorem 2.16(1),( 2)] hold true over any commutative ring, not necessarily Noetherian ring.Hence we can show our main theorems over any commutative ring.
Lemma 3.1.Let R be any commutative ring.For any R-module M and integer n, we have: , where E is any injective R-module; Proof.We only prove (1) and the proof of ( 2) is similar.By Definition 2.5, there exists an Consider the projective resolution of the R-module Hom R (C, E), Thus we get another exact sequence after applying the functor (R C) ⊗ R − to P: By [9, Lemma 1.5], (R C) ⊗ R P i is a projective R C-module for any i ≥ 0. So the above exact sequence is a projective resolution of the R C-module (R C) ⊗ R Hom R (C, E).Hence we have that: where the second isomorphism is a Hom-tensor adjointness.
Remark 3.2.By Lemma 3.1, we know [9, Proposition 2.13 and Theorem 2.16(1), ] hold true over any commutative ring R, which can be easily seen from the proof process in [9].Now, we show the C-Gorenstein global dimension of R is definable.And we use a different method from [2].
Lemma 3.3.Let E be any injective and Q any projective R-module.Then we have (2) Let P be a projective resolution of the R-module Hom R (C, E).Following from the proof of Lemma 3.1, (R C) Lemma 3.4.Let E be an injective and Q a projective R-module.For any nonnegative integer n, we have In the classical homological algebra, the global dimension of a ring R, denoted by gldim(R), can be computed via the following formula: We will show the C-Gorenstein global dimension of R can also be computed via a similar formula.Lemma 3.8.Let R be a commutative ring with C-Ggldim(R) < ∞.Denoted by P rojR the class of projective R-modules, we have On the other hand, assume that C-Ggldim(R) = n and Q is any projective 132 ZHEN ZHANG AND JIAQUN WEI The homological dimension which arises by resolving a given module by C-Gorenstein projective (flat) or C-Gorenstein injective modules is known as the C-Gorenstein projective (flat) or C-Gorenstein injective dimension of the module.Bennis and Mahdou investigated the global Gorenstein dimension and weak global Gorenstein dimension of an associative ring R.They showed that the supremum of the Gorenstein projective dimensions of all the R-modules is equal to the supremum of the Gorenstein injective dimensions over an associative ring R and that the supremum of the Gorenstein flat dimensions is smaller than the common value of the terms of this equality, cf.[2, Theorem 1.1].It is natural to ask whether the global Gorenstein projective dimension with respect to a semidualizing module is equal to the global Gorenstein injective dimension of R. On the other hand, Holm and Jørgensen [9, Theorem 2.16] studied the trivial extension of R by C, denoted by R C. They showed that the C-Gorenstein projective, injective and flat R-module is in fact the Gorenstein projective, injective and flat R C-module over commutative Noetherian ring R, respectively.However, their conclusions only applies to R-modules which are viewed as R C-modules via the natural surjection (R C → R).We are not sure whether they hold true for R C-modules.Hence it is not trivial to show that the global Gorenstein projective dimension with respect to a semidualizing module C of a ring R is equal to the global Gorenstein injective dimension of R. In this paper, we use a new technique to show the global Gorenstein dimension induced by C is definable.Obviously, it is not a trivial extension of [2, Theorem 1.1].Moreover, we showed the following theorems over any commutative ring R. Theorem.Let C-Ggldim(R) denote the Gorenstein global dimension of R induced by C. If C-Ggldim(R) < ∞, then C-Ggldim(R)=sup{C-Gpd(R/I) | I is an ideal of R}, where C-Gpd(R/I) is the C-Gorenstein projective dimension of R/I.Compared with the classical global dimension of R, denoted by gldim(R), we get that C-Ggldim(R) ≤ gldim(R) in general and when gldim(R) < ∞, they are equal.Enochs and Jenda [4, Proposition 10.3.2]proved that every finitely presented Gorenstein projective R-module is Gorenstein flat over a left and right coherent ring.In this paper, we get the C-Gorenstein projective R-module is C-Gorenstein flat R-module when C-Ggldim(R) < ∞.Moreover, we have: Theorem.Let C-wGgldim(R) denote the supremum of the C-Gorenstein flat dimension of all R-modules.We have C-wGgldim(R) ≤ C-Ggldim(R).If R is Noetherian, they are equal.If we let C = R, we get wGgldim(R) = Ggldim(R) over Noetherian ring R, which extends [2, Corollary 1.2].
The X -injective dimension of M , denoted by X -id(M ) is defined dually.Particularly, pd(M ), id(M ), and f d(M ) is, respectively, the classical projective, injective, and flat dimension of R-module M .And we use Gpd(M ), Gid(M ), and Gf d(M ) to denote, respectively, the Gorenstein projective, injective, and flat dimension of M .Definition 2.2.[14, 1.8]An R-module C is called semidualizing if (1) C admits a degreewise finitely generated projective resolution; (2) the natural homothety map R −→ Hom R (C, C) is an isomorphism; (3) Ext ≥1 R (C, C) = 0. Let C be a semidualizing R-module.The class of C-projective (flat) R-modules, denoted by F C (P C ) and C-injective R-modules, denoted by I C , consists of modules which have the form C ⊗ R F , F is projective (flat) R-modules and Hom R (C, I), I is injective R-module, cf.[10, Definition 5.1].By C-flat (projective, injective) R-modules, Holm and Jørgensen defined the C-Gorenstein flat, projective and injective modules in commutative ring R, which are clearly the generalization of Gorenstein flat, projective and injective modules.
such that it stays exact when we apply the functor Hom R (C, I) ⊗ R − for any injective R-module I. Remark 2.4.By [9, Example 2.8], projective modules are C-Gorenstein projective, injective modules are C-Gorenstein injective and flat modules are C-Gorenstein flat over commutative Noetherian ring R.However, the condition of R being Noetherian is not needed, which can be easily seen from the proof process in [9].Hence every R-module M admits C-Gorenstein projective (injective and flat) resolution and the C-Gorenstein projective (injective, flat) dimension of the R-module M is definable over any commutative ring R. By [9, Definition 9], let C-Gpd(M ), C-Gid(M ) and C-Gf d(M ), denote the C-Gorenstein projective, injective and flat dimension of M , respectively.At last, we recall the definition of trivial extension: Definition 2.5.Let R be a ring and C a semidualizing module.The direct sum R ⊕ C can be equipped with the product: (r, c) • (r , c ) = (rr , rc + r c).This turns R ⊕ C into a ring which is called the trivial extension of R by C and denoted by R C. There are canonical ring homomorphisms, R R C, which enable us to view R-modules as R C-modules, and vice versa.
for any R-module M and i > n by [14, Proposition 2.12] and Proposition 3.7.Hence id R