Indecomposable modules of large rank over Cohen-Macaulay local rings

Abstract:

A commutative Noetherian local ring $(R,\mathfrak m,k)$ is called Dedekind-like provided $R$ is one-dimensional and reduced, the integral closure $\overline {R}$ is generated by at most 2 elements as an $R$-module, and $\mathfrak m$ is the Jacobson radical of $\overline {R}$. If $M$ is an indecomposable finitely generated module over a Dedekind-like ring $R$, and if $P$ is a minimal prime ideal of $R$, it follows from a classification theorem due to L. Klingler and L. Levy that $M_P$ must be free of rank 0, 1 or 2. Now suppose $(R,\mathfrak m,k)$ is a one-dimensional Cohen-Macaulay local ring that is not Dedekind-like, and let $P_1,\dotsc ,P_t$ be the minimal prime ideals of $R$. The main theorem in the paper asserts that, for each non-zero $t$-tuple $(n_1,\dotsc ,n_t)$ of non-negative integers, there is an infinite family of pairwise non-isomorphic indecomposable finitely generated $R$-modules $M$ satisfying $M_{P_i}\cong (R_{P_i})^{(n_i)}$ for each $i$.