Bounds for one-dimensional rings of finite Cohen-Macaulay type
Let \(R\) be a one-dimensional reduced noetherian ring such that the integral closure \(R^ -\) of \(R\) is a finitely generated \(R\)-module. The rank of an MCM \(R\)-module (i.e., a torsion-free \(R\)-module) \(M\) is the sequence \(\{\text{rank} (M_ p) | p\) is a minimal prime ideal of \(R\}\). The ring \(R\) is said to be of finite CM type if \(R\) has only finitely many indecomposable MCM modules up to isomorphism and is of bounded CM type if there exists a bound on the ranks of MCM \(R\)-modules. Using the notion of modules over ``Artin pairs'', considerations of the case of local rings, and some glueing techniques, the authors show the following: Theorem. (1) If \(R\) is of bounded CM type and any residue field of \(R^ -\) is not purely inseparable of degree 2 over \(R\), then every MCM \(R\)- module of constant rank has a direct summand of modules of constant rank 1, 2, 3, 4, 5, 6, 8, 9, or 12. (2) Let \(k\) be a field admitting a separable non-Galois extension of degree 3. Then there exists a one-dimensional integral domain of finite CM type which is essentially of finite type over \(k\), has exactly two maximal ideals having residue field \(k\), and has indecomposable MCM modules of rank 1, 2, 3, 4, 5, 6, 8, 9, and 12.
scientific article; zbMATH DE number 584669
scientific article