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-Gorenstein Projective Modules
We introduce and study the -Gorenstein projective modules, where is a projective class. These modules are a generalization of the Gorenstein projective modules.
Throughout the paper, all rings are associative with identity, and an -module will mean right -module. As usual, we use to denote, respectively, the classical projective dimension of .
Theorem A. Let be a commutative Noetherian local ring with residue field . The following conditions are equivalent: (1) is regular; (2) has finite projective dimension; (3)every -module has finite projective dimension.
This result opened the door to the solution of two long-standing conjectures of Krull. Moreover, it introduced the theme that finiteness of a homological dimension for all modules characterizes rings with special properties. Later work has shown that modules of finite projective dimension over a general ring share many properties with modules over a regular ring. This is an incitement to study homological dimensions of individual modules.
In line with these ideas, Auslander and Bridger  introduced in 1969 the -dimension. It is a homological dimension for finitely generated modules over a Noetherian ring, and it gives a characterization of the Gorenstein local rings [4, Section 3.2]. Namely, is Gorenstein if has finite -dimension, and only if every finitely generated -module has finite -dimension.
In Section 2, we recall the notion of the Gorenstein projective modules, and we put the point on its place in the theory of homological dimensions as a generalization of the classical projective modules. So, we recall some fundamental results about the Gorenstein projective modules and dimensions.
In Section 3, which is the main section of this paper, we show that every time we choose a projective class (Definition 3.1), we can consider a generalization of (Gorenstein) projective modules via . In the general case these generalizations are different.
2. The Gorenstein Projective Modules
In the early 1990s, the -dimension was extended beyond the realm of finitely generated modules over a Noetherian ring. This was done by Enochs and Jenda who introduced the notion of the Gorenstein projective modules . The same authors and their collaborators, studied these modules in several subsequent papers. The associated dimension was studied by Christensen  and Holm .
Definition 2.1. An -module is called Gorenstein projective if there exists an exact sequence of projective -modules such that and such that the functor leaves exact whenever is a projective -module. The resolution is called a complete projective resolution.
It is evident that every projective module is Gorenstein projective. While, the converse is not true [8, Example 2.5].
Basic categorical properties are recorded in [7, Section 2]. Recall that a class of -modules is called projectively resolving  if and for every short exact sequence with , the conditions and are equivalent.
Proposition 2.2 (see [7, Theorem 2.5]). The class of Gorenstein projective -modules is closed under direct sums and summands.
In , the authors define a particular subclass of the class of the Gorenstein projective modules.
Definition 2.3. A module is said to be special Gorenstein projective if there exists an exact sequence of free modules of the form such that and such that leaves the sequence exact whenever is projective. The resolution is called a complete free resolution.
Every projective module is a direct summand of a free one. A parallel result for the Gorenstein projective modules holds.
Proposition 2.4 (see [9, Corollary 2.4]). A module is Gorenstein projective if and only if it is a direct summand of a special Gorenstein projective module.
An (augmented) Gorenstein projective resolution of a module is an exact sequence , where each module is Gorenstein projective. Note that every module has a Gorenstein projective resolution, as a free resolution is trivially a Gorenstein projective one.
Definition 2.5. The Gorenstein projective dimension of a module , denoted by , is the least integer such that there exists a Gorenstein projective resolution of with for all . If no such exists, then is infinite. By convention, set .
In , Holm gave the following fundamental functorial description of the Gorenstein dimension.
Theorem 2.6 (see [7, Theorem 2.22]). Let be an -module of finite Gorenstein projective dimension. For every integer , the following conditions are equivalent: (1);(2) for all and all projective module ;(3) for all and all -modules with finite ;(4) for every exact sequence , if are Gorenstein projective, then also is Gorenstein projective.
The Gorenstein projective dimension is a refinement of the projective dimension; this follows from [7, Proposition 2.27].
Proposition 2.7. For every -module , with equality if is finite.
3. -Gorenstein Projective Modules
Notation 3. By and we denote the classes of all projective and Gorenstein projective -modules, respectively. Given a class of -modules, we set:
We define the projective class as follows.
Definition 3.1. Let be a ring. A class of -modules is a projective class, if it is projectively resolving and closed under direct sum.
Remark 3.2. For any ring , any projective class is closed under direct summands (by [7, Proposition 1.4]).
The main purpose of this section, which is the main section of this paper, is to see that every time we chose a projective class , we can consider a generalization of (Gorenstein) projective modules via . In the general case these generalizations are different.
Definition 3.3. Let be a ring, and let be a projective class over . The -projective dimension of an -module , , is defined by declaring that if and only if has an -resolution of length .
Note that, by the definition of the -projective dimension, for each module we have . On the other hand, we have the following.
Proposition 3.4. Let be a ring and a projective class over . Then, if and only if for each exact sequence where for each , the module belongs in .
Proof. The condition “if” is clear. So, we have to prove the “only if” condition. Assume that , and let be an exact sequence, where for each . Since , there exists an exact sequence where for each . Since the class is projectively resolving and closed under arbitrary sums and under direct summands, by using [3, Lemma 3.12], since .
We introduce the -Gorenstein projective modules as follows.
Definition 3.5. Let be a ring, and let be a projective class over . An -module is called -Gorenstein projective if there exists an exact sequence of projective modules such that and such that the functor leaves exact whenever . The complex is called an -complete projective resolution.
Remark 3.6. Let be a projective class over a ring . Then, we have the following:(1)every projective module is -Gorenstein projective;(2)every -Gorenstein projective module is Gorenstein projective; (3)the class of all -Gorenstein projective module is closed under direct sums by definition.(4)if is an -complete projective resolution then, by symmetry, all the images, all the kernels, and all the cokernels of are -Gorenstein projective;(5)if then, the -Gorenstein projective modules are just the Gorenstein projective modules.
The next example shows that there exists a Gorenstein projective module which is not -Gorenstein projective for a given projective class .
Example 3.7. Consider the local quasi-Frobenius ring , where is a field. Let be the residue class of in . Let be any projective class which contain . Then, is a Gorenstein projective module which is not -Gorenstein projective.
Proof. First, note that there is always a projective class which contain any module . A trivial case is .
Since is quasi-Frobenius, is Gorenstein projective [10, Theorem 2.2]. Moreover, the short sequence , where is the multiplication by is exact. Then, if we suppose that is -Gorenstein projective, then . Thus, is a direct summand of and so projective. Then, is free since is local. However, . Then, cannot be free.
The next result is a direct consequence of the definition of -Gorenstein projective modules.
Proposition 3.8. Given a projective class , an -module is -Gorenstein projective if and only if (1) for all and all (i.e., ) and(2) there exists an exact sequence , where all are projectives and leaves this sequence exact whenever .
The next result shows that an -projective module with finite is projective.
Proposition 3.9. Let be a projective class over , and let be an -Gorenstein projective module. Then, (1) for all module with and all .(2)either is projective or .
Proof. (1) Since is -Gorenstein projective, for all and all . Thus, by dimension shifting,we obtain the desired result.
(2) Suppose that and consider a short exact sequence where is projective. It is clear that . Then, by (1), . Hence, this short exact sequence splits, and then is a direct summand of . Hence, it is projective.
Proposition 3.10. Let be a ring. If , then an -module is -Gorenstein projective if and only if it is projective.
Proof. First note that is a projective class [7, Theorem 2.5] and it clear that every projective module is -Gorenstein projective. Now, suppose that is an -Gorenstein projective module. It is trivial that is also a Gorenstein projective module. Now, consider an exact sequence , where is projective. Since is resolving, is also Gorenstein projective. Then, . Thus, . So, the short exact sequence splits and so, is a direct summand of and therefore projective.
The converse implication is immediate.
Next we set out to investigate how -Gorenstein projective modules behave in short exact sequences.
Theorem 3.11. Let be a projective class over a ring . The class of all -Gorenstein projective modules is projectively resolving. Furthermore, it is closed under arbitrary direct sums and under direct summands.
Proof. It is clear that every projective module is -Gorenstein projective. So, consider any short exact sequence of -modules , where is -Gorenstein projective.
First suppose that is -Gorenstein projective. We claim that is also -Gorenstein projective. Since is projectively resolving (by [11, Lemma 2.2.9]) and by Proposition 3.8, we get that belongs to . Thus, to show that is -Gorenstein projective, we only have to prove the existence of an exact sequence , where all are projectives and leaves this sequence exact whenever (by Proposition 3.8). By assumption, there exist exact projective resolutions where keeps the exactness of these sequences whenever and all the cokernels of and -Gorenstein projectives (such a sequences exists by the definition of -Gorenstein projective modules). Consider the following diagram: (3.7)Since is -Gorenstein projective, we have . Hence, the following sequence is exact. (3.8)Thus, there exists an -morphism such that .
It is easy to check that the morphism defined by setting for each completes the above diagram and makes it commutative. Then, using the snake lemma, we get the following commutative diagram:(3.9) Since and are -Gorenstein projectives, they belong to which is projectively resolving (by [11, Lemma ]). Then, belongs also to. Accordingly, keeps the exactness of the short exact sequence whenever . By induction, we can construct a commutative diagram with the form: (3.10)such that leaves exact whenever . Consequently, is -Gorenstein projective.
Now suppose that is -Gorenstein projective, and we claim that is -Gorenstein projective. As above belongs to . Hence, we have to prove that satisfies condition (2) of Proposition 3.8. To do it, pick a short exact sequence where is projective and is -Gorenstein projective (such a sequence exists by Remark 3.6(4)), and consider the following push-out diagram: (3.11)
The first part of this proof, applying to the short exact sequence , shows that is -Gorenstein projective. Hence, it admits a right projective resolution which remains exact by whenever . In addition, the short exact sequence remains exact by whenever since is -Gorenstein projective. Finally, it is easy to check that: (3.12)is also exact by whenever .
The closing of the class of -Gorenstein projective modules under direct sums is clear by the definition of these modules, while its closing under direct summands is deduced from [7, Proposition 1.4].
Corollary 3.12. Let be a projective class over a ring . Let be an exact sequence where and are -Gorenstein projective modules and where for all projective modules . Then, is -Gorenstein projective.
Proof. Pick a short exact sequence where is projective and is -Gorenstein projective. Consider the following push-out diagram.(3.13)
Using Theorem 3.11, is -Gorenstein projective. On the other hand, since , the short exact sequence splits. Thus, is a direct summand of . Consequently, is -Gorenstein projective, as a direct summand of an -Gorenstein projective module (by Theorem 3.11).
The author would like to thank the referees for the valuable suggestions and comments.
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