Algebraic properties of the binomial edge ideal of complete bipartite graph

TL;DR

This paper investigates algebraic properties of the binomial edge ideal associated with complete bipartite graphs, providing explicit formulas and structural insights into its algebraic invariants.

Contribution

It offers a detailed analysis of the algebraic invariants of the binomial edge ideal for complete bipartite graphs, including explicit descriptions of modules and resolution properties.

Findings

Computed dimensions, depths, and regularities of the ideal.

Described modules of deficiencies and duals of local cohomology.

Proved the purity of the minimal free resolution.

Abstract

Let $J_G$ denote the binomial edge ideal of a connected undirected graph on $n$ vertices. This is the ideal generated by the binomials $x_iy_j - x_jy_i, 1\leq i < j \leq n,$ in the polynomial ring $S= K[x_1,...,x_n,y_1,...,y_n]$ where $\{i,j\}$ is an edge of $G$. We study the arithmetic properties of $S/J_G$ for $G$, the complete bipartite graph. In particular we compute dimensions, depths, Castelnuovo-Mumford regularities, Hilbert functions and multiplicities of them. As main results we give an explicit description of the modules of deficiencies, the duals of local cohomology modules, and prove the purity of the minimal free resolution of $S/J_G$.