G-Gorenstein modules (Q2909072)

The paper under review introduces and studies G-Gorenstein modules, the Gorenstein homological analogue of Gorenstein modules. The prefix ``G-'' stands for ``Gorenstein-'', similar to that in ``G-dimension''.NEWLINENEWLINELet \(R\) be a noetherian commutative ring with unit. Let us recall Gorenstein modules. R. Sharp found the following characterization [\textit{R. Sharp}, Math. Z. 112, 340--356 (1969); Zbl 0182.06103)] of Gorenstein rings: \(R\) is Gorenstein if and only if the associated Cousin complex is the injective resolution of the ring. He defined a finitely generated \(R\)-module \(M\) to be a Gorenstein module if the Cousin complex of \(M\) provides an injective resolutions for \(M\). Gorenstein modules can be characterized without looking at the Cousin complexes, see Theorem 3.5, 3.11, [\textit{R. Sharp}, Math. Z. 115, 117--139 (1970; Zbl 0186.07403)].NEWLINENEWLINETheorem. Suppose that \(R\) is a noetherian local ring and \(M\) a non-zero finite \(R\)-module. The following statements are equivalent:NEWLINENEWLINE(i) \(M\) is Gorenstein,NEWLINENEWLINE(ii) for every \(\mathfrak{p}\in V(M)\), \(M_{\mathfrak{p}}\) is Cohen-Macaulay of dimension depth \(R_{\mathfrak{p}}\) with finite injective dimension,NEWLINENEWLINE(iii) for every \(\mathfrak{p}\in V(M)\), \(\mu^i(\mathfrak{p},M)= 0\) if and only if \(i\neq \dim M_{\mathfrak{p}}\), where \(\mu^i(\mathfrak{p},M)\) is the \(i\)th Bass of number of \(M\) with respect to \(\mathfrak{p}\).NEWLINENEWLINE(iv) \(M\) is Cohen-Macaulay and \(M^d\) is injective, where \(d=\dim M\) and \(M^i\) is the \(i\)th module in the Cousin complex of \(M\).NEWLINENEWLINEThe Gorenstein homological dimensions were introduced by \textit{M. Auslander} and \textit{M. Bridger} [``Stable module theory'', Mem. Am. Math. Soc. 94, 146 p. (1969; Zbl 0204.36402)] for finitely generated modules, and more generally, for arbitrary modules by \textit{E. E. Enochs} and \textit{O. M. G. Jenda} [Math. Z. 220, No. 4, 611--633 (1995; Zbl 0845.16005)]. They support a vast generalization of homological algebra for noetherian rings and finitely generated modules. For the basic notions of Gorenstein homological algebra, see the books [\textit{E. E. Enochs} and \textit{O. M. G. Jenda}, Relative homological algebra. de Gruyter Expositions in Mathematics. 30. Berlin: Walter de Gruyter. (2000; Zbl 0952.13001)], [\textit{L.W. Christensen}, Gorenstein dimensions. Lecture Notes in Mathematics. 1747. Berlin: Springer. (2000; Zbl 0965.13010)] and the survey article [\textit{L.W. Christensen}, \textit{H.-B. Foxby} and \textit{H. Holm}, Fontana, Marco (ed.) et al., Commutative algebra. Noetherian and non-Noetherian perspectives. Springer 101--143 (2011; Zbl 1225.13019)].NEWLINENEWLINEDefinition: Let \(M\) be a non-zero finitely generated \(R\)-module. We say that \(M\) is a \textit{G-Gorenstein module} if the Cousin complex of \(M\) is a Gorenstein injective resolution of \(M\). NEWLINENEWLINENEWLINE In order to have a good theory of Gorenstein dimensions, normally we assume that \(R\) admits a dualizing complex. The above theorem of Sharp can be generalized to characterize G-Gorenstein modules when \(R\) has a dualizing complex. In particular, G-Gorenstein modules have the expected properties, e.g. if \(M\) is G-Gorenstein, then \(M\) localizes and \(M\) specializes modulo a sequence that is both \(R\)-regular and \(M\)-regular. It is proved that over a Gorenstein non-regular local ring, G-Gorenstein modules strictly contain Gorenstein modules. So the class of G-Gorenstein modules is larger than the class of Gorenstein modules in general. NEWLINENEWLINENEWLINE Finally, Gorenstein and regular local rings can be characterized via G-Gorenstein modules. For example, from Theorem 4.7, a Cohen-Macaulay local ring \(R\) is Gorenstein if and only if every maximal Cohen-Macaulay \(R\)-module is G-Gorenstein. From Theorem 4.8, a Gorenstein local ring \(R\) is regular if and only if every G-Gorenstein module is actually Gorenstein.