One-Dimensional Rings Of Finite Cohen-Macaulay Type

Abstract

Let R be a commutative one-dimensional reduced local Noetherian ring whose integral closure (R) over tilde (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 (R) over tilde 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) is generated by 3 elements as art R-module; and (2) the intersection of the maximal R-submodules of (R) over tilde/R 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. (C) 1998 Elsevier Science B.V. All rights reserved.