The facet ideals of chessboard complexes

Abstract

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