# 𝔛-Gorenstein Projective Modules

Authors: Mohammed Tamekkante
Journal: ISRN Algebra
 (2011)

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## Abstract

We introduce and study the𝔛-Gorenstein projective modules, where 𝔛 is a projective class. These modules are a generalization of the Gorenstein projective
modules.


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## Body

## 1. Introduction

Throughout the paper, all rings are associative with identity, and anR-module will mean right R-module. As usual, we use pdR(M) to denote, respectively, the classical projective dimension of M.An important motivation for the study of homological dimensions dates back to 1956, when Auslander and Buchsbaum [1] and Serre [2] proved the following.Theorem A.
LetR be a commutative Noetherian local ring R with residue field k. The following conditions are equivalent:(1)
R is regular;(2)
k has finite projective dimension;(3)
everyR-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 [3] introduced in 1969 the G-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, R is Gorenstein if k has finite G-dimension, and only if every finitely generated R-module has finite G-dimension.In Section2, 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 Section3, which is the main section of this paper, we show that every time we choose a projective class 𝔛 (Definition 3.1), we can consider a generalization of (Gorenstein) projective modules via 𝔛. In the general case these generalizations are different.


## 2. The Gorenstein Projective Modules

In the early 1990s, theG-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 [5]. The same authors and their collaborators, studied these modules in several subsequent papers. The associated dimension was studied by Christensen [6] and Holm [7].Definition 2.1.
AnR-module M is called Gorenstein projective if there exists an exact sequence of projective R-modules
(2.1)P:⋯⟶P1⟶P0⟶P0⟶P1⟶⋯
such that M≅Im(P0→P0) and such that the functor HomR(-,Q) leaves P exact whenever Q is a projective R-module. The resolution P 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 𝔛 of R-modules is called projectively resolving [7] 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.Proposition 2.2 (see [7, Theorem 2.5]).
The class of Gorenstein projectiveR-modules is closed under direct sums and summands.In [9], the authors define a particular subclass of the class of the Gorenstein projective modules.Definition 2.3.
A moduleM is said to be special Gorenstein projective if there exists an exact sequence of free modules of the form
(2.2)F:⋯⟶F⟶fF⟶fF⟶fF⟶⋯
such that M≅Im(f) and such that HomR(-,P) leaves the sequence F exact whenever P is projective. The resolution F 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 moduleM 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 moduleM is an exact sequence ⋯→Gi→Gi-1→⋯→G0→M→0, where each module Gi 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 moduleM≠0, denoted by GpdR(M), is the least integer n≥0 such that there exists a Gorenstein projective resolution of M with Gi=0 for all i>n. If no such n exists, then GpdR(M) is infinite. By convention, set GpdR(M)=∞.In [7], Holm gave the following fundamental functorial description of the Gorenstein dimension.Theorem 2.6 (see [7, Theorem 2.22]).
LetM be an R-module of finite Gorenstein projective dimension. For every integer n≥0, the following conditions are equivalent:(1)
GpdR(M)≤n;(2)
ExtRi(M,P)=0 for all i>0 and all projective module P;(3)
ExtRi(M,Q)=0 for all i>0 and all R-modules Q with finite pdR(Q);(4)
for every exact sequence0→Kn→Gn-1→⋯→G0→M→0, if G0;…;Gn-1 are Gorenstein projective, then also Kn 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 everyR-module M, GpdR(M)≤pdR(M) with equality if pdR(M) is finite.


## 3.𝔛-Gorenstein Projective Modules

Notation 3.
By𝒫(R) and 𝒢𝒫(R) we denote the classes of all projective and Gorenstein projective R-modules, respectively. Given a class 𝔛 of R-modules, we set:
(3.1)⊥∞X={M∣Ext⁡Ri(M,X)=0∀X∈Xandalli>0}.We define the projective class as follows.Definition 3.1.
LetR be a ring. A class 𝔛 of R-modules is a projective class, if it is projectively resolving and closed under direct sum.Remark 3.2.
For any ringR, 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 class𝔛, we can consider a generalization of (Gorenstein) projective modules via 𝔛. In the general case these generalizations are different.Definition 3.3.
LetR be a ring, and let 𝔛 be a projective class over R. The 𝔛-projective dimension of an R-module M, 𝔛-pdR(M), is defined by declaring that 𝔛-pdR(M)≤n if and only if M has an 𝔛-resolution of length n.Note that, by the definition of the𝔛-projective dimension, for each module M we have 𝔛-pdR(M)≤pdR(M). On the other hand, we have the following.Proposition 3.4.
LetR be a ring and 𝔛 a projective class over R. Then, 𝔛-pdR(M)≤n if and only if for each exact sequence
(3.2)0⟶Xn⟶Xn-1⟶⋯⟶X0⟶M⟶0,
where Xi∈𝔛 for each i=0,…,n-1, the module Xn belongs in 𝔛.Proof.
The condition “if” is clear. So, we have to prove the “only if” condition. Assume that𝔛-pdR(M)≤n, and let
(3.3)0⟶Xn⟶Xn-1⟶⋯⟶X0⟶M⟶0
be an exact sequence, where Xi∈𝔛 for each i=0,…,n-1. Since 𝔛-pdR(M)≤n, there exists an exact sequence
(3.4)0⟶Xn′⟶Xn-1′⟶⋯⟶X0′⟶M⟶0,
where Xi′∈𝔛 for each i=0,…,n. Since the class 𝔛 is projectively resolving and closed under arbitrary sums and under direct summands, by using [3, Lemma 3.12], Xn∈𝔛 since 𝔛n′∈𝔛.We introduce the𝔛-Gorenstein projective modules as follows.Definition 3.5.
LetR be a ring, and let 𝔛 be a projective class over R. An R-module M is called 𝔛-Gorenstein projective if there exists an exact sequence of projective modules
(3.5)P:⋯⟶P1⟶P0⟶P0⟶P1⟶⋯
such that M≅Im(P0→P0) and such that the functor HomR(-,X) leaves P exact whenever X∈𝔛. The complex P is called an 𝔛-complete projective resolution.Remark 3.6.
Let𝔛 be a projective class over a ring R. Then, we have the following:(1)
every projective module is𝔛-Gorenstein projective;(2)
every𝔛-Gorenstein projective module is Gorenstein projective;(3)
the class of all𝔛-Gorenstein projective module is closed under direct sums by definition.(4)
ifP:⋯→P1→P0→P0→P1→⋯ is an 𝔛-complete projective resolution then, by symmetry, all the images, all the kernels, and all the cokernels of P are 𝔛-Gorenstein projective;(5)
if𝔛=𝒫(R) then, the 𝔛-Gorenstein projective modules are just the Gorenstein projective modules.The next example shows that there exists a Gorenstein projective module which is not𝔛-Gorenstein projective for a given projective class 𝔛.Example 3.7.
Consider the local quasi-Frobenius ringR=k[X]/(X2), where k is a field. Let (X¯) be the residue class of X in R. Let 𝔛 be any projective class which contain (X¯). Then, (X¯) is a Gorenstein projective module which is not 𝔛-Gorenstein projective.Proof.
First, note that there is always a projective class which contain any moduleM. A trivial case is 𝔛=RMod.
SinceR is quasi-Frobenius, (X¯) is Gorenstein projective [10, Theorem 2.2]. Moreover, the short sequence 0→(X¯)↪R→f(X¯)→0, where f is the multiplication by X¯ is exact. Then, if we suppose that (X¯) is 𝔛-Gorenstein projective, then ExtR1((X¯),(X¯))=0. Thus, (X¯) is a direct summand of R and so projective. Then, (X¯) is free since R is local. However, X¯2=0. Then, (X¯) cannot be free.The next result is a direct consequence of the definition of𝔛-Gorenstein projective modules.Proposition 3.8.
Given a projective class𝔛, an R-module M is 𝔛-Gorenstein projective if and only if  (1)
ExtRi(M,X)=0 for all X∈𝔛 and all i>0 (i.e., M∈⊥∞𝔛) and(2)
there exists an exact sequence0→M→P0→P1→⋯, where all Pi are projectives and HomR(-,X) leaves this sequence exact whenever X∈𝔛.The next result shows that an𝔛-projective module with finite 𝔛-pdR(-) is projective.Proposition 3.9.
Let𝔛 be a projective class over R, and let M be an 𝔛-Gorenstein projective module. Then,(1)
ExtRi(M,G)=0 for all module G with 𝔛-pdR(G)<∞ and all i>0.(2)
eitherM is projective or 𝔛-pdR(M)=∞.Proof.
(1) SinceM is 𝔛-Gorenstein projective, ExtRi(M,X)=0 for all X∈𝔛 and all i>0. Thus, by dimension shifting,we obtain the desired result.
(2) Suppose that𝔛-pdR(M)<∞ and consider a short exact sequence 0→K→P→M→0 where P is projective. It is clear that 𝔛-pdR(K)<∞. Then, by (1), ExtR1(M,K)=0. Hence, this short exact sequence splits, and then M is a direct summand of P. Hence, it is projective.Proposition 3.10.
LetR be a ring. If 𝔛=𝒢𝒫(R), then an R-module M is 𝔛-Gorenstein projective if and only if it is projective.Proof.
First note that𝔛=𝒢𝒫(R) is a projective class [7, Theorem 2.5] and it clear that every projective module is 𝔛-Gorenstein projective. Now, suppose that M is an 𝔛-Gorenstein projective module. It is trivial that M is also a Gorenstein projective module. Now, consider an exact sequence 0→M′→P→M→0, where P is projective. Since 𝒢𝒫(R) is resolving, M′ is also Gorenstein projective. Then, M′∈𝔛. Thus, ExtR1(M,M′)=0. So, the short exact sequence splits and so, M is a direct summand of P and therefore projective.
The converse implication is immediate.Next we set out to investigate how𝔛-Gorenstein projective modules behave in short exact sequences.Theorem 3.11.
Let𝔛 be a projective class over a ring R. The class of all 𝔛-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 is𝔛-Gorenstein projective. So, consider any short exact sequence of R-modules 0→M′→M→M′′→0, where M′′ is 𝔛-Gorenstein projective.
First suppose thatM′ is 𝔛-Gorenstein projective. We claim that M is also 𝔛-Gorenstein projective. Since ⊥∞𝔛 is projectively resolving (by [11, Lemma 2.2.9]) and by Proposition 3.8, we get that M belongs to ⊥∞𝔛. Thus, to show that M is 𝔛-Gorenstein projective, we only have to prove the existence of an exact sequence 0→M→P0→P1→⋯, where all Pi are projectives and HomR(-,X) leaves this sequence exact whenever X∈𝔛 (by Proposition 3.8). By assumption, there exist exact projective resolutions(3.6)M′=0⟶M′⟶P0′→P1′⟶⋯,M′′=0⟶M′′⟶P0′′⟶P1′′⟶⋯,
where Hom(-,X) keeps the exactness of these sequences whenever X∈𝔛 and all the cokernels of M′ and M′′𝔛-Gorenstein projectives (such a sequences exists by the definition of 𝔛-Gorenstein projective modules). Consider the following diagram:
(3.7)Since M′′ is 𝔛-Gorenstein projective, we have ExtR1(M′′,P0′)=0. Hence, the following sequence is exact.
(3.8)Thus, there exists an R-morphism γ:M→P0′ such that f′=γ∘α.
It is easy to check that the morphismf:M→P0′⊕P0′′ defined by setting f(m)=(γ(m),f′′∘β(m)) for each m∈M completes the above diagram and makes it commutative. Then, using the snake lemma, we get the following commutative diagram:(3.9)
Since coker(f′) and coker(f′′) are 𝔛-Gorenstein projectives, they belong to⊥∞𝔛 which is projectively resolving (by [11, Lemma 2.2.9]). Then, coker(f) belongs also to⊥∞𝔛. Accordingly, HomR(-,X) keeps the exactness of the short exact sequence 0→M→P0′⊕P0′′→coker(f)→0 whenever X∈𝔛. By induction, we can construct a commutative diagram with the form:
(3.10)such that HomR(-,X) leaves M exact whenever X∈𝔛. Consequently, M is 𝔛-Gorenstein projective.
Now suppose thatM is 𝔛-Gorenstein projective, and we claim that M′ is 𝔛-Gorenstein projective. As above M belongs to ⊥∞𝔛. Hence, we have to prove that M satisfies condition (2) of Proposition 3.8. To do it, pick a short exact sequence 0→M→P→X→0 where P is projective and X is 𝔛-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 sequence0→M′′→Y→X→0, shows that Y is 𝔛-Gorenstein projective. Hence, it admits a right projective resolution Y:0→Y→F0→F1→⋯ which remains exact by HomR(-,X) whenever X∈𝔛. In addition, the short exact sequence 0→M′→P→Y→0 remains exact by HomR(-,X) whenever X∈𝔛 since Y is 𝔛-Gorenstein projective. Finally, it is easy to check that:
(3.12)is also exact by HomR(-,X) whenever X∈𝔛.
The closing of the class of𝔛-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.
Let𝔛 be a projective class over a ring R. Let 0→G′→G→M→0 be an exact sequence where G′ and G are 𝔛-Gorenstein projective modules and where ExtR1(M,P)=0 for all projective modules P. Then, M is 𝔛-Gorenstein projective.Proof.
Pick a short exact sequence0→G′→P→G′′→0 where P is projective and G′′ is 𝔛-Gorenstein projective. Consider the following push-out diagram.(3.13)Using Theorem3.11, X is 𝔛-Gorenstein projective. On the other hand, since ExtR1(M,P)=0, the short exact sequence 0→P→X→M→0 splits. Thus, M is a direct summand of X. Consequently, M is 𝔛-Gorenstein projective, as a direct summand of an X-Gorenstein projective module (by Theorem 3.11).



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