Big Indecomposable Mixed Modules over Hypersurface Singularities

Abstract

IntroductionThis 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 sequencesin which T and X are indecomposable, T has finite length, and F is maximal CohenMacaulay of arbitrarily large constant rank. The main result (Theorem 2.3) on building indecomposables is quite general, and it is likely that there are other contexts where it will prove useful.In order to state our main application, we establish some terminology. Let k be a field. By a hypersurface singularity we mean a commutative Noetherian local ring (R, m, k) whose m-adic completion R is isomorphic to S/(f ), where (S, n, k) is a complete regular local ring and f is a non-zero element of n 2 . A Noetherian local ring (R, m, k) is Dedekind-like [KL1, Definition 2.5] provided R is one-dimensional and reduced, the integral closure R is generated by at most 2 elements as an R-module, and m is the Jacobson radical of R. (Examples include discrete valuation rings and rings such as k [[x, y]

Hypersurfaces of bounded Cohen–Macaulay type

“…In Theorems 1.1 and 3.1 (3), the form of the general hypersurface should be as in ( †) of [30], if the field is not algebraically closed.…”