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