Ranks of Indecomposable Modules over Rings of Infinite Cohen-Macaulay Type

Abstract

Let (R, m, k) be a one-dimensional analytically unramified local ring with minimal prime ideals P 1 , . . . , Ps. Our ultimate goal is to study the direct-sum behavior of maximal Cohen-Macaulay modules over R. Such behavior is encoded by the monoid C(R) of isomorphism classes of maximal Cohen-Macaulay R-modules: the structure of this monoid reveals, for example, whether or not every maximal Cohen-Macaulay module is uniquely a direct sum of indecomposable modules; when uniqueness does not hold, invariants of this monoid give a measure of how badly this property fails. The key to understanding the monoid C(R) is determining the ranks of indecomposable maximal Cohen-Macaulay modules. Our main technical result shows that if R/P 1 has infinite Cohen-Macaulay type and the residue field k is infinite, then there exist |k| pairwise non-isomorphic indecomposable maximal Cohen-Macaulay R-modules of rank (r 1 , . . . , rs) provided r 1 ≥ r i for all i ∈ {1, . . . , s}. This result allows us to describe the monoid C(R) when R/Q has infinite Cohen-Macaulay type for every minimal prime ideal Q of the m-adic completion R.

Brauer–Thrall Theory for Maximal Cohen–Macaulay Modules

“…Crabbe and Saccon [12] have recently proved the following: Theorem 2.7. Let (R, m, k) be an analytically unramified local ring of dimension one, with minimal prime ideals p 1 , .…”

“…If .R; m; k/ is a one-dimensional, analytically unramified local ring with minimal prime ideals p 1 ; : : : ; p s , we define the rank of a module to be the s-tuple .r 1 ; : : : ; r s /, where r i is the dimension of .M p i / as a vector space over the field R p i . Crabbe and Saccon [12] have recently proved the following: …”