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arXiv · 1201.3141

Ranks of indecomposable modules over rings of infinite Cohen-Macaulay type

A. Crabbe·S. Saccon
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Abstract

Let (R,m,k) be a one-dimensional analytically unramified local ring with minimal prime ideals P_1,...,P_s. 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,...,r_s) provided r_1 is greater than or equal to r_i for all i. This result allows us to describe the monoid C(R) when all analytic branches of R have infinite Cohen-Macaulay.

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A. Crabbe, S. Saccon. 2012-01-16. Ranks of indecomposable modules over rings of infinite Cohen-Macaulay type. https://arxiv.org/abs/1201.3141

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