A generalization of Auslander's last theorem
Let \(R\) be a local Cohen-Macaulay ring admitting a dualizing module. An \(R\)-module \(N\) is called Gorenstein-injective if there is an exact sequence \(\cdots\to E^{-2}\to E^{-1}\to E^0\to E^1\to E^2\to\cdots\) of injective modules with \(N=\text{Ker}(E^0\to E^1)\) and such that \(\Hom(E,-)\) leaves the sequence exact when \(E\) is an injective module. Working in the category of finitely generated \(R\)-modules one has, dually, the notion of (finitely generated) Gorenstein-projective modules. These are exactly the Auslander modules of Gorenstein dimension 0 [see \textit{M. Auslander} and \textit{M. Bridger}, ``Stable module theory'', Mem. Am. Math. Soc. 94 (1969; Zbl 0204.36402)]. When \(R\) is Gorenstein, Gorenstein-projective = maximal Cohen-Macaulay. Let \({\mathcal F}\) be a class of \(R\)-modules. An \({\mathcal F}\)-cover of an \(R\)-module \(M\) is a morphism \(\varphi:F\to M\) with \(F\) in \({\mathcal F}\) such that: (1) any morphism \(\psi:G\to M\) with \(G\) in \({\mathcal F}\) factorizes through \(\varphi\), and (2) any endomorphism \(f:F\to F\) such that \(\varphi\circ f= \varphi\) is an automorphism. An \({\mathcal F}\)-envelope is defined dually. The main result of the paper is that every \(R\)-module \(N\) of finite Gorenstein-injective dimension has a Gorenstein-injective envelope \(\psi:N\to G\). Furthermore, \(\psi\) is a monomorphism and \(\text{Coker} \psi\) has finite injective dimension. Using some Matlis duality arguments, the authors deduce that, when \(R\) is complete, every finitely generated \(R\)-module \(M\) of finite Gorenstein-projective dimension has a Gorenstein-projective cover \(\varphi:C\to M\). Furthermore, \(\varphi\) is surjective and \(\text{Ker} \varphi\) has finite projective dimension. In the last paragraph, using some arguments of Auslander, the authors drop the hypothesis of completeness, obtaining a generalization of a result of Auslander for Gorenstein rings.
scientific article; zbMATH DE number 1385252
scientific article