  A few years ago, Huneke and Leuschke proved a theorem which solved a
conjecture of Schreyer. It asserts that an excellent Cohen-Macaulay local ring
of countable Cohen-Macaulay type which is complete or has uncountable residue
field has at most a one-dimensional singular locus. In this paper, it is
verified that the assumption of the excellent property can be removed, and the
theorem is considered over an arbitrary local ring. The main purpose of this
paper is to prove that the existence of a certain prime ideal and a certain
totally reflexive module implies the existence of an uncountably infinite
number of isomorphism classes of indecomposable totally reflexive modules.
