Curve singularities of finite Buchsbaum-representation type (Q1320190)

Let \((R,m)\) be a Noetherian local ring, and let \(M\) denote a finitely generated \(R\)-module. A Buchsbaum \(R\)-module \(M\) is maximal if \(M\) has the same dimension as \(R\), and \(R\) has finite Buchsbaum representation type if there are only finitely many isomorphism classes of indecomposable maximal Buchsbaum \(R\)-modules. In this paper after a short overview on the subject the author studies the case of curves; he proves under the assumptions that \(R\) is complete and \(R/m\) is infinite that the following conditions are equivalent: (i) \(R\) has finite Buchsbaum representation type. (ii) \(e(R) \leq 2\), \(\nu (R) \leq 2\) and the ring \(R/H^ 0_ m (R)\) is reduced \((\nu (R)\) is the embedding dimension). The author obtains this theorem after the proof of the following: (same hypothesis) \(R\) is a Cohen-Macaulay ring of finite Buchsbaum representation type if and only if \(R\) is a reduced ring of \(e(R) \leq 2\).