All modules and linear maps considered here are defined over a Cohen-Macaulay local ring of dimension \(d\) admitting a dualizing module \(D\). The notion of \(G\)-dimension of a finitely generated module was introduced by M. Auslander [in: Séminaire Algèbre commutative Paris, 1966/67, Éc. Norm. Supér. Jeunes Filles (1968)]. Foxby established a duality between two full subcategories in the category of modules and proved that the finitely generated modules in one category are precisely those of finite \(G\)-dimension [H.-B. Foxby, in: Commutative algebra, internat. Conf., Vechta 1994, Vechtaer Univ.-Schnitten 13, 59-63 (1994; Zbl 0834.13014)]. The main purposes of the paper are to generalize the notion of \(G\)-dimension and to extend Foxby’s result.
A modulo \(M\) is said to be Gorenstein- \((G\)- for short) projective if there is an exact sequence \(\cdots \to P_2\to P_1 \to P_0 @>d_0>> P_{-1}\to \cdots\) of projective modules such that \(M= \text{Ker} (d_0)\) and such that \(\operatorname{Hom} (\;,P)\) leaves the sequence exact whenever \(P\) is a projective module. A module \(N\) is said to be \(G\)-injective if there is an exact sequence \(\cdots \to E^{-1} \to E^0 @>d^0>> E^1\to E^2\to \cdots\) of injective modules such that \(N=\text{Ker} (d^0)\) and such that \(\operatorname{Hom} (E,\;)\) leaves the sequence exact whenever \(E\) is an injective module. The class \({\mathcal G}_0\) is defined to consist of those modules \(M\) such that \(\text{Tor}_i (D,M)=0\) and \(\text{Ext}^i (D,D \otimes M)=0\) for all \(i\geq 1\) and such that the natural map \(M\to \operatorname{Hom} (D,D \otimes M)\) is an isomorphism. The class \({\mathcal J}_0\) consists of those \(N\) such that \(\text{Ext}^i (D,N) =0\) and \(\text{Tor}_i (D,\operatorname{Hom} (D,N))=0\) for all \(i\geq 1\) and such that \(D\otimes \operatorname{Hom} (D,N)\to N\) is an isomorphism. The functor \(D\otimes-\) from \({\mathcal G}_0\) to \({\mathcal J}_0\) gives an equivalence between these two categories, and similarly \(\operatorname{Hom} (D,\;): {\mathcal J}_0 \to {\mathcal G}_0\) is an equivalence. In section two, the “main theorems”, the authors prove are: For a module \(M\), \[ M\in {\mathcal G}_0 \Leftrightarrow G\text{-proj} \dim M< \infty \Leftrightarrow G \text{-proj} \dim M\leq d; \] and for \(N\), \[ N\in {\mathcal J}_0 \Leftrightarrow G \text{-inj} \dim N<\infty \Leftrightarrow G\text{-inj} \dim N\leq d. \] \((G\)-projective and injective dimensions are defined in the usual way.)
In section three the notion of a \(G\)-flat module is introduced and a characterization of a \(G\)-flat module is given. For a class \({\mathcal F}\) of modules, \({\mathcal F}\)-(pre)covers and \({\mathcal F}\)-(pre)envelopes are defined. Some results concerning \(G\)-projective precovers, \(G\)-injective preenvelopes and \(G\)-flat covers are given.