Non-Gorenstein Isolated Singularities of Graded Countable Cohen–Macaulay Type

In this article we show a partial answer to a question of Huneke and Leuschke (Proc. Am. Math. Soc. 131(10):3003–3007, 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.