The facet ideals of chessboard complexes

In this paper we describe the irreducible decomposition of the facet ideal $\text{F}({\mathrm{\Delta }}_{m,n})$ of the chessboard complex ${\mathrm{\Delta }}_{m,n}$ with $n\ge m$. We also provide some lower bounds for depth and regularity of the facet ideal $\text{F}({\mathrm{\Delta }}_{m,n})$. When $m\le 3$, we prove that these lower bounds can be obtained.