Almost Gorenstein homogeneous rings and their h-vectors

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 ( Theorem 3.1) and we also characterize almost Gorenstein homogeneous domains with small socle degrees in terms of the h-vector of R ( Theorem 4.4). Moreover, we also provide the examples of almost Gorenstein homogeneous domains arising from lattice polytopes.