One-dimensional rings of finite Cohen-Macaulay type Original Research Article

Let R be a commutative one-dimensional reduced local Noetherian ring whose integral closure image (in its total quotient ring) is a finitely generated R-module. We settle the last remaining unkown case of the following theorem by proving it for the case that some residue field of image is purely inseparable of degree 2 over the residue field of R. Theorem. Let R be a ring as above. R has, up to isomorphism, only finitely many indecomposable finitely generated maximal Cohen-Macaulay modules if and only if 1. (1) R is generated by 3 elements as an R-module; and 2. (2) the intersection of the maximal R-submodules of image is a cyclic R-module. Moreover, over such a ring, the rank of every indecomposable maximal Cohen-Macaulay module of constant rank is 1, 2, 3, 4, 5, 6, 8, 9 or 12.