Big Indecomposable Mixed Modules over Hypersurface Singularities

This research began as an effort to determine exactly which one-dimensional local rings have indecomposable finitely generated modules of arbitrarily large constant rank. The approach, which uses a new construction of indecomposable modules via the bimodule structure on certain Ext groups, turned out to be effective mainly for hypersurface singularities. The argument was eventually replaced by a direct, computational approach [HKKW], which applies to all one-dimensional Cohen-Macaulay local rings. In this paper we resurrect the Ext argument to build indecomposable modules of large rank over hypersurface singularities of any dimension d ≥ 1. The main point of the construction is that, modulo an indecomposable finite-length part, the modules constructed are maximal Cohen-Macaulay modules. Thus, even when there are no indecomposable maximal Cohen-Macaulay modules of large rank, we can build short exact sequences