The First Syzygy of Hibi Rings Associated with Planar distributive lattices

Abstract

Let $\mathcal{L}$ be a finite distributive lattice and $S=K[x_α: α\in \mathcal{L}]$ be a polynomial ring over a field $K$ and $I=\langle x_αx_β- x_{α\vee β} x_{α\wedgeβ} : α\nsim β,α,β\in {\mathcal{L}} \rangle$ an ideal of $S$. In this article we describe the first syzygy of the Hibi ring $R[\mathcal{L}]=S/I$, for a planar distributive lattice $\mathcal{L}$. We also derive an exact formula for the first Betti number of a planar distributive lattice. We give a characterization of planar distributive lattices for which the first syzygy is linear.