Poset embeddings of Hilbert functions for two hypersurface rings

Abstract

Let $R=\mathbb{𝕂}\left[a,b,c,d\right]∕\left(ad-bc\right)$ or $\mathbb{𝕂}\left[a,b,c\right]∕\left(ac-b^2\right)$, where $\mathbb{𝕂}$ is a field of arbitrary characteristic and $a,b,c,d$ are indeterminates. In this paper we will show that there is an order preserving embedding from the poset of Hilbert functions of homogeneous ideals of $R$ to the poset of homogeneous ideals of $R$. We also obtain related results for Betti numbers when the characteristic of $\mathbb{𝕂}$ is $0$.