Relative Homological Algebra
Relative homological algebra was conceived by Auslander and Bridger [14, 15] and was formed by Enochs, Jenda, and Torrecillas (for example, [54, 55, 57]).
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Relative Homological Algebra
Relative homological algebra was conceived by Auslander and Bridger [14, 15] and was formed by Enochs, Jenda, and Torrecillas (for example, [54, 55, 57]). During 1993–1995, Enochs and Jenda introduced the notions of Gorenstein projective modules and Gorenstein injective modules and also introduced, together with Torrecillas, the concept of Gorenstein flat modules. After 2004, Holm, Bennis, Mahdou, et al. introduced various kinds of Gorenstein homological dimensions such that relevant theories and methods of classical homological algebra may be applied to relative homological algebra, despite that the proofs of the corresponding propositions are very complex and difficult, and some require various extra conditions. Our aim in this chapter is to introduce relative homological algebra, especially where the related concepts of integral domains which appear in classical ideal theory are defined and investigated by using the class of Gorenstein projective modules.
11.1 Gorenstein Projective Modules and Strongly Gorenstein Projective Modules 11.1.1 Gorenstein Projective Modules Definition 11.1.1 A module M is said to be Gorenstein projective (G-projective for short) if M has a complete projective resolution, that is, there is an exact sequence P = · · · → Pn → · · · → P1 → P0 → P 0 → P 1 → · · · ,
(11.1.1)
The original version of this chapter was revised: For detailed information please see Erratum. The erratum to this chapter is available at DOI 10.1007/978-981-10-3337-7_12 © Springer Nature Singapore Pte Ltd. 2016 F. Wang and H. Kim, Foundations of Commutative Rings and Their Modules, Algebra and Applications 22, DOI 10.1007/978-981-10-3337-7_11
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11 Relative Homological Algebra
where each of Pi , P j is a projective module such that M ∼ = Im(P0 → P 0 ) and for any projective module Q, the complex Hom R (P, Q) = · · · → Hom R (P 0 , Q) → Hom R (P0 , Q) → Hom R (P1 , Q) → · · · remains an exact sequence. Remark 11.1.1 Since every projective module is a direct summand of a free module, in Definition 11.1.1 projective modules can be chosen free modules. Also, considering the unification, we may write a complete projective resolution of M as follows: P = · · · → Pn → · · · → P1 → P0 → P−1 → P−2 → · · · ,
(11.1.2)
where M ∼ = Im(P0 → P1 ). For a complete projective resolution (11.1.2) of M, define L i = Ker(Pi → Pi−1 ), i ∈ Z. This L i is called the syzygy of (11.1.2) or L i is called the i-th syzygy of (11.1.2). Note that M = L −1 . Clearly each G-projective module is a w-module. Theorem 11.1.2 The following statements are equivalent for an R-module M: (1) M is G-projective. (2) M has a complete projective resolution (11.1.2), and for any projective module Q and any syzygy L of (11.1.2), Ext 1R (L , Q) = 0. (3) M has a complete projective resolution (11.1.2), and for any projective module Q, any k 1, and any syzygy L of (11.1.2), Ext kR (L , Q) = 0. (4) M has a complete projective resolution (11.1.2), and for any free module F, any k 1, and any syzygy L of (11.1.2), Ext kR (
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