×

zbMATH — the first resource for mathematics

\(\Omega\)-Gorenstein projective and flat covers and \(\Omega\)-Gorenstein injective envelopes. (English) Zbl 1092.13031
From the paper: \(R\) denotes a local Cohen-Macaulay ring of Krull dimension \(d\) admitting a dualizing module \(\Omega\). Let \({\mathcal W}\) denote the class of \(R\)-modules \(W\) such that \(W\cong\Omega\otimes P\) for some projective \(R\)-module \(P\), \({\mathfrak X}\) denote the class of \(R\)-modules \(X\) such that \(X\cong\Omega \otimes F\) for some flat \(R\)-module \(F\), and \({\mathcal V}\) denote the class of all \(R\)-modules \(V\) such that \(V\cong\operatorname{Hom}(\Omega,E)\) for some injective \(R\)-module \(E\). We say that an \(R\)-module \(M\) is \(\Omega\)-Gorenstein injective if there exists an exact sequence \[ \dots\to V_1\to V_0\to V^0\to V^1\to \cdots \] of modules in \({\mathcal V}\) with \(M=\text{Ker}(V^0\to V^1)\) such that \(\text{Hom}(V,-)\) and \(\operatorname{Hom}(-,V)\) leave the sequence exact whenever \(V\in{\mathcal V}\). Similarly, we say that an \(R\)-module \(M\) is \(\Omega\)-Gorenstein projective if there is an exact sequence \[ \cdots\to W_1\to W_0\to W^0\to W^1\to\cdots \] with \(M=\text{Ker} (W^0\to W^1)\) such that \(\operatorname{Hom}(W,-)\) and \(\operatorname{Hom}(-,W)\) leave the sequence exact for all \(W\in{\mathcal W}\). Furthermore, we say that an \(R\)-module \(M\) is \(\Omega\)-Gorenstein flat if there exists an exact sequence \[ \cdots\to X_1\to X_0\to X^0 \to X^1\to\cdots \] with \(M=\text{Ker}(X^0\to X^1)\) such that \(\operatorname{Hom}(W,-)\) and \(V \otimes-\) leave the sequence exact whenever \(V\in{\mathcal V}\) and \(W\in{\mathcal W}\). In the paper under review, it is shown that if \(R\) is a local Cohen-Macaulay ring admitting a dualizing module \(\Omega\), then \(\Omega\)-Gorenstein projective and flat covers and \(\Omega\)-Gorenstein injective envelopes exist for certain modules. These results generalize the well known results for local Gorenstein rings.

MSC:
13H10 Special types (Cohen-Macaulay, Gorenstein, Buchsbaum, etc.)
13C11 Injective and flat modules and ideals in commutative rings
13C12 Torsion modules and ideals in commutative rings
13C14 Cohen-Macaulay modules
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] DOI: 10.1007/BF02572634 · Zbl 0845.16005
[2] Enochs E. E., Interactions Between Ring Theory and Representatives of Algebras pp 159– (2000)
[3] DOI: 10.1515/9783110803662
[4] DOI: 10.1090/S0002-9947-96-01624-8 · Zbl 0862.13004
[5] Foxby H.-B., Gorenstein dimensions over Cohen–Macaulay rings (1994) · Zbl 0834.13014
[6] Salce L., Symposia Math. 23 pp 11– (1979)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.