Almost Gorenstein homogeneous rings and their $h$-vectors

TL;DR

This paper explores the relationship between almost Gorenstein properties of Cohen--Macaulay homogeneous rings and their $h$-vectors, providing conditions, characterizations, and examples involving lattice polytopes.

Contribution

It offers new criteria linking $h$-vectors to almost Gorensteinness and characterizes such rings with small socle degrees, including lattice polytope examples.

Findings

Provided a sufficient condition for almost Gorensteinness based on $h$-vectors.

Characterized almost Gorenstein homogeneous domains with small socle degrees.

Presented examples of almost Gorenstein domains from lattice polytopes.

Abstract

In this paper, for the development of the study of almost Gorenstein graded rings, we discuss some relations between almost Gorensteinness of Cohen--Macaulay homogeneous rings and their $h$-vectors. Concretely, for a Cohen--Macaulay homogeneous ring $R$, we give a sufficient condition for $R$ to be almost Gorenstein in terms of the $h$-vector of $R$ and we also characterize almost Gorenstein homogeneous domains with small socle degrees in terms of the $h$-vector of $R$. Moreover, we also provide the examples of almost Gorenstein homogeneous domains arising from lattice polytopes.