Indecomposable modules of large rank over Cohen-Macaulay local rings

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 $i$.