  In this paper we show a partial answer the a question of C. Huneke and G.
Leuschke (2003): Let R be a standard graded Cohen-Macaulay ring of graded
countable Cohen-Macaulay representation type, and assume that R has an isolated
singularity. Is R then necessarily of graded finite Cohen-Macaulay
representation type? In particular, this question has an affirmative answer for
standard graded non-Gorenstein rings as well as for standard graded Gorenstein
rings of minimal multiplicity. Along the way, we obtain a partial
classification of graded Cohen-Macaulay rings of graded countable
Cohen-Macaulay type.
