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$$\Omega$$-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
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##### References:
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