Relative Gorenstein objects in abelian categories

Abstract

Let A be an abelian category. For a pair (X , Y) of classes of objects in A, we define the weak and the (X , Y)-Gorenstein relative projective objects in A. We point out that such objects generalize the usual Gorenstein projective objects and others generalizations appearing in the literature as Ding-projective, Ding-injective, X -Gorenstein projective, Gorenstein ACprojective and G C -projective modules and Cohen-Macaulay objects in abelian categories. We show that the principal results on Gorenstein projective modules remains true for the weak and the (X , Y)-Gorenstein relative objects. Furthermore, by using Auslander-Buchweitz approximation theory, a relative version of Gorenstein homological dimension is developed. Finally, we introduce the notion of W-cotilting pair in the abelian category A, which is very strong connected with the cotorsion pairs related with relative Gorenstein objects in A. It is worth mentioning that the W-cotilting pairs generalize the notion of cotilting objects in the sense of L. Angeleri Hügel and F. Coelho [3]. Contents 1. Introduction. 1 2. Auslander-Buchweitz approximation theory 3 3. General properties of the relative Gorenstein objects 8 4. Relative Gorenstein homological dimensions 22 5. Cotorsion pairs and relative Gorenstein projective objects 34 6. Cotilting objects and W-cotilting pairs 41 References 46

Cut cotorsion pairs

“…We have by hypothesis that Y ∧ is closed under direct summands. On the other hand, by [4,Corollary 3.33] we also have that GP (X ,Y) is closed under extensions and direct summands. We then have that conditions (1) and ( 2) in Proposition 2.22 are valid.…”

“…Indeed, consider the pair (GP (X ,Y) , (GP (X ,Y) ) ⊥ ). The closure under direct summands of (GP (X ,Y) ) ⊥ is clear, and the same property holds for GP (X ,Y) due to [4,Corollary 3.33]. Also, the following containments are clear:…”

“…Following [4], let us denote by GP (X ,Y) and GI (X ,Y) the classes of (X , Y)-Gorenstein projective and (X , Y)-Gorenstein injective objects of C, respectively. For example, GP (P,P) and GI (I,I) are precisely the classes of Gorenstein projective and Gorenstein injective objects of C, which we shall write as GP and GI, for simplicity.…”

“…Proof. It follows after applying Proposition 2.7 to the pair (GP (X ,Y) , ω), which is left Frobenius by [4,Corollary 4.10].…”

“…The most important result in this section is Theorem 3.18, where it is shown that it is not possible to obtain non-trivial weak AB contexts from Gorenstein objects relative to hereditary complete cotorsion pairs. This points out the need of a cut version of weak AB contexts, more suitable for Gorenstein objects relative to GP-admissible pairs [4].…”