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Stability of Gorenstein categories. (English) Zbl 1140.18010
In this paper, the definition of Gorenstein projective (injective) modules in the sense of E. E. Enochs and O. M. G. Jenda [Math. Z. 220, No. 4, 611–633 (1995; Zbl 0845.16005)] is examined from a quite general and useful point of view: For a subcategory \(\mathcal{X}\) of an abelian category \(\mathcal{A}\), an object \(M \in \mathcal{A}\) is defined to be in \(\mathcal{G}(\mathcal{X})\) if there exists an exact, \(\mathrm{Hom}_{\mathcal{A}}(\mathcal{X},-)\)-exact, and \(\mathrm{Hom}_{\mathcal{A}}(-,\mathcal{X})\)-exact complex \(X\), consisting of objects in \(\mathcal{X}\), such that \(M \cong \mathrm{Coker}(\partial^X_1)\). The authors prove several interesting properties for the category \(\mathcal{G}(\mathcal{X})\), and under the assumption that \(\mathrm{Ext}^{n}_{\mathcal{A}}(X,X')=0\) for all \(X, X' \in \mathcal{X}\) and all \(n>0\), their results are very satisfying: In this case, \(\mathcal{G}(\mathcal{G}(\mathcal{X}))=\mathcal{G}(\mathcal{X})\) is closed under direct summands. If, in addition, \(\mathcal{X}\) is closed under kernels of epimorphisms (cokernels of monomorphisms) then so is \(\mathcal{G}(\mathcal{X})\). The paper also contains many nice examples, e.g. if \(C\) is a semidualizing module over a commutative ring \(R\) then \(\mathcal{G}(\mathcal{P}_C(R))=\mathcal{GP}_C(R) \cap \mathcal{B}_C(R)\). Here \(\mathcal{P}_C(R)\) and \(\mathcal{GP}_C(R)\) are the classes of \(C\)-projective and \(C\)-Gorenstein projective \(R\)-modules, respectively, in the sense of P. Jørgensen and the reviewer [J. Pure Appl. Algebra 205, No. 2, 423–445 (2006; Zbl 1094.13021)]; and \(\mathcal{B}_C(R)\) is the Bass category with respect to \(C\) in the sense of H.-B. Foxby [Math. Scand. 31, 267–284 (1973; Zbl 0272.13009)].

18G10 Resolutions; derived functors (category-theoretic aspects)
13C05 Structure, classification theorems for modules and ideals in commutative rings
13D02 Syzygies, resolutions, complexes and commutative rings
13D07 Homological functors on modules of commutative rings (Tor, Ext, etc.)
18G15 Ext and Tor, generalizations, Künneth formula (category-theoretic aspects)
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