Ascent of finite Cohen-Macaulay type
Let \(A\) be a Noetherian local ring. A finitely generated \(A\)-module~\(M\) is said to be maximal Cohen-Macaulay if \(\operatorname {depth} M = \dim A\). We say that \(A\) has finite Cohen-Macaulay (representation) type if there are only finitely many indecomposable maximal Cohen-Macaulay modules up to isomorphism. There are many studies on complete Cohen-Macaulay rings having finite Cohen-Macaulay type. In the present paper, the authors study the relation between the ring~\(A\) and its completion~\(\widehat A\). They show that an excellent Cohen-Macaulay ring~\(A\) has finite Cohen-Macaulay type if and only if \(\widehat A\) has finite Cohen-Macaulay type. The second author already showed the if part. The only if part is new. The key of the proof is the existence of the canonical module. The authors use Hinich's result: An excellent Henselian local ring has a canonical module [\textit{V. Hinich}, Math. Nachr. 163, 289-296 (1993; Zbl 0799.13003)].
finite Cohen-Macaulay representation type
scientific article; zbMATH DE number 1471836
scientific article