Non-Gorenstein isolated singularities of graded countable Cohen-Macaulay type

TL;DR

This paper investigates whether standard graded Cohen-Macaulay rings with isolated singularities and countable Cohen-Macaulay type are necessarily of finite type, providing partial answers and classifications.

Contribution

It offers partial answers to a question about the Cohen-Macaulay representation type of graded rings with isolated singularities and classifies certain Cohen-Macaulay rings of countable type.

Findings

Affirmative answer for non-Gorenstein rings with isolated singularities.

Affirmative answer for Gorenstein rings of minimal multiplicity.

Partial classification of graded Cohen-Macaulay rings of countable type.

Abstract

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.