Approximations by maximal Cohen-Macaulay modules (Q498699)

Let \(R\) be a commutative Noetherian local Cohen-Macaulay ring with dualizing module \(\Omega\) and \(M\) a finitely generated \(R\)-module. A Cohen-Macaulay approximation for \(M\) is a short exact sequence \(0 \to I \to X \to M \to 0\) where \(X\) belongs to the class of maximal CM \(R\)- modules (MCM), and \(I\) has a finite injective dimension. In terms of relative homological algebra, having a CM approximation means that the homomorphism \( X \to M\) is a special MCM-precover of \(M\). In [Mém. Soc. Math. Fr., Nouv. Sér. 38, 5--37 (1989; Zbl 0697.13005)], \textit{M. Auslander} and \textit{R.-O. Buchweitz} showed the existence of a special MCM-precover for every finitely generated \(R\)-module. When \(R\) is Henselian, \textit{R. Takahashi} [Math. Z. 251, No. 2, 249--256 (2005; Zbl 1098.13014)] and \textit{Y. Yoshino}, ``Cohen-Macaulay approximations'', Proceedings of the \(4^{th}\) Symposium on Representation Theory of Algebras, (1993; in Japanese)] showed that every finitely generated \(R\)- module has an MCM-cover. In the paper, the author shows a ``dual'' theorem to the above theorems and gives examples. He proves Theorem A:{\parindent=6mm \begin{itemize} \item[(a)] Every finitely generated \(R\)- module has a special MCM-preenvelope (also called a special left MCM-approximation). \item [(b)] If \(R\) is henselian, the every finitely generated \(R\)- module has an MCM-envelope (also called a minimal left MCM-approximation). \item [(c)] Every special MCM-preenvelope (and hence every MCM-envelope) \(\mu: M \to X\) of a finitely generated \(R\)-module \(M\) has the property that \(\Hom_R(\text{Coker }\mu, \Omega)\) has finite injective dimension. \end{itemize}} Then in Theorem B, the author gives conditions for \(M\) to have a MCM-envelope with the unique lifting property. Finally in Theorem C, he shows that any cosyzygy of \(M\) with respect to MCM is maximal CM.