Levelness versus nearly Gorensteinness of homogeneous domains

TL;DR

This paper explores the relationship between levelness and nearly Gorensteinness in graded rings, providing conditions and characterizations, especially for Cohen-Macaulay affine semigroup rings and Stanley-Reisner rings.

Contribution

It establishes necessary conditions for nearly Gorensteinness, shows that nearly Gorenstein rings of type 2 are level, and characterizes nearly Gorenstein Stanley-Reisner rings.

Findings

Nearly Gorenstein rings of type 2 are level.

Necessary conditions relate h-vectors to Gorensteinness.

Counterexamples exist for higher Cohen-Macaulay types.

Abstract

Levelness and nearly Gorensteinness are well-studied properties of graded rings as a generalized notion of Gorensteinness. In this paper, we compare the strength of these properties. For any Cohen-Macaulay homogeneous affine semigroup ring R, we give a necessary condition for R to be non-Gorenstein and nearly Gorenstein in terms of the h-vector of R and we show that if R is nearly Gorenstein with Cohen-Macaulay type 2, then it is level. We also show that if Cohen-Macaulay type is more than 2, there are 2-dimensional counterexamples. Moreover, we characterize nearly Gorensteinness of Stanley-Reisner rings of low-dimensional simplicial complexes.