The facet ideals of chessboard complexes

TL;DR

This paper analyzes the algebraic structure of facet ideals associated with chessboard complexes, providing their irreducible decompositions and bounds on algebraic invariants like depth and regularity.

Contribution

It offers the first detailed description of the irreducible decomposition of facet ideals of chessboard complexes and establishes bounds for their algebraic properties.

Findings

Irreducible decomposition of facet ideals for $ abla_{m,n}$ with $n \\geq m$

Lower bounds for depth and regularity of these ideals

Exact bounds obtained for cases where $m \\leq 3$

Abstract

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.