Super-stretched and graded countable Cohen-Macaulay type

TL;DR

This paper introduces the concept of super-stretched Cohen-Macaulay rings and demonstrates that rings with graded countable Cohen-Macaulay type possess specific structural properties, including restrictions on their h-vectors and classifications in certain cases.

Contribution

It defines super-stretched rings and proves that graded countable Cohen-Macaulay type rings are super-stretched, providing new classifications for these rings based on their dimension and Gorenstein property.

Findings

Rings of graded countable Cohen-Macaulay type have restricted h-vectors: (1), (1,n), or (1,n,1).

One-dimensional Gorenstein rings of graded countable type are hypersurfaces.

Higher-dimensional non-Gorenstein rings of graded countable Cohen-Macaulay type have minimal multiplicity.

Abstract

We define what it means for a Cohen-Macaulay ring to to be super-stretched and show that Cohen-Macaulay rings of graded countable Cohen-Macaulay type are super-stretched. We use this result to show that rings of graded countable Cohen-Macaulay type, and positive dimension, have possible h-vectors (1), (1,n), or (1,n,1). Further, one dimensional standard graded Gorenstein rings of graded countable type are shown to be hypersurfaces; this result is not known in higher dimensions. In the non-Gorenstein case, rings of graded countable Cohen-Macaulay type of dimension larger than 2 are shown to be of minimal multiplicity.