Let $\mathcal{ L}$ be a finite distributive lattice and $S=K[x_\alpha: \alpha \in \mathcal{ L}]$ be a polynomial ring over a field $K$ and $I=\langle x_\alpha x_\beta - x_{ \alpha\vee \beta} x_{ \alpha\wedge\beta} : \alpha \nsim \beta,\alpha,\beta \in { \mathcal{ L}} \rangle$ 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.
Categories: math.AC