G-Gorenstein modules
type: Item
instance of: scholarly article
zbMATH Open document ID: 1250.13015
DOI: 10.1142/S1005386712000296
author: Mohsen Aghajani
author: Hossein Zakeri
published in: Algebra Colloquium
publication date: 2012-08-29T00:00:00Z
full work available at URL: https://arxiv.org/abs/1106.4972
Mathematics Subject Classification ID: 13C14
Mathematics Subject Classification ID: 13D02
Mathematics Subject Classification ID: 13H10
zbMATH DE Number: 6073844
zbMATH Keywords: balanced big Cohen-Macaulay module
zbMATH Keywords: Cousin complex
zbMATH Keywords: dualizing complex
zbMATH Keywords: G-Gorenstein module
zbMATH Keywords: Gorenstein injective module
MaRDI profile type: Publication
arXiv ID: 1106.4972
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title: G-Gorenstein modules
review text: 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)]…
reviewed by: Dang Hop Nguyen
description: scientific article; zbMATH DE number 6073844