Finitely generated cotorsion modules (Q2716984)

Let \(R\) be a commutative noetherian ring, and let \(M\) be a finitely generated \(R\)-module. Then \(M\) is cotorsion (i.e., \(\text{Ext}^1 (F,M) =0\) for all flat \(R\)-modules \(F)\) if and only if every linear \(R\to M\) has an \(R\)-linear extension \(\widehat R\to M\), where \(\widehat R\) is the completion of \(R\) with respect to the topology having finite products of maximal ideals as a fundamental system of neighborhoods of zero. When \(M\) is cotorsion, it also has a unique \(\widehat R\)-module structure that extends the original \(R\)-module structure of \(M\). If \(R\) has finite Krull dimension and \(I\) is an ideal, then \(R/I\) is cotorsion if and only if \(R/I\) is a complete semilocal ring. Also, \(M\) always has a flat cotorsion cover. If \(M'\) is finitely generated and \(f:M \to M'\), then \(f\) is a covering morphism in the sense of \textit{E. F. Enochs}, \textit{J. R. García Rozas}, and \textit{L. Oyonarte} [Commun. Algebra 28, No. 8, 3823-3835 (2000; Zbl 0958.16006)] if and only if \(f\) is surjective, \(\ker f\) is cotorsion, and \((\ker f)/D\) does not contain a non-zero direct summand of \(M/D\) for any non-zero submodule \(D\) of \(\ker f\).