[Paper Review] A note on the edge ideals of Ferrers graphs

Margherita Barile|ArXiv.org|Jun 15, 2006

Commutative Algebra and Its Applications5 references18 citations

TL;DR

This paper determines the arithmetical rank of edge ideals of Ferrers graphs, showing it equals both the cohomological dimension and projective dimension of the ideal. The arithmetical rank is computed as the maximum value of $\lambda_j + j - 1$ over all $j$, and the ideal is a set-theoretic complete intersection if and only if $\lambda = (m, m-1, \dots, 1)$, corresponding to a rectangular Ferrers diagram with decreasing row lengths.

ABSTRACT

We determine the arithmetical rank of every edge ideal of a Ferrers graph.

Motivation & Objective

• To determine the arithmetical rank of edge ideals associated with Ferrers graphs.
• To establish the equality of arithmetical rank, cohomological dimension, and projective dimension for these ideals.
• To characterize when such ideals are set-theoretic complete intersections.
• To provide a combinatorial interpretation of the arithmetical rank via diagonal sums in the Ferrers diagram.

Proposed method

• Uses Schmitt-Vogel's lemma to construct a generating set of radicals with $\mu = \max_j(\lambda_j + j - 1)$ elements.
• Defines $q_i$ as the sum of all generators $x_r y_s$ with $r + s = i + 1$, forming a minimal radical generating set.
• Applies the minimal prime decomposition of Ferrers ideals from Corso and Nagel's work.
• Leverages the equality $\mathrm{cd}(I) = \mathrm{pd}(I)$ for squarefree monomial ideals, established in [4].
• Uses closed-formula Betti numbers from [2] to compute $\mathrm{pd}(I(G)) = \max_j(\lambda_j + j - 1)$.
• Applies the characterization of set-theoretic complete intersections via equality of height and arithmetical rank.

Experimental results

Research questions

• RQ1 What is the arithmetical rank of the edge ideal of a Ferrers graph?
• RQ2 Does the arithmetical rank of a Ferrers ideal equal its cohomological dimension and projective dimension?
• RQ3 Under what conditions is a Ferrers ideal a set-theoretic complete intersection?
• RQ4 Can the arithmetical rank be described combinatorially via diagonals in the Ferrers diagram?

Key findings

• The arithmetical rank of the edge ideal $I(G)$ of a Ferrers graph equals $\max_{j=1,\dots,n}(\lambda_j + j - 1)$.
• The arithmetical rank equals both the cohomological dimension and the projective dimension of $I(G)$, i.e., $\mathrm{ara}\,I(G) = \mathrm{cd}\,I(G) = \mathrm{pd}\,I(G)$.
• The ideal $I(G)$ is a set-theoretic complete intersection if and only if $\lambda = (m, m-1, \dots, 1)$, i.e., the Ferrers diagram is a staircase shape.
• The arithmetical rank is realized by taking the sum of generators along each ascending diagonal in the Ferrers diagram, with $\mu$ such sums forming a minimal radical generating set.
• The construction of the radical generators $q_i$ ensures that $\mathrm{Rad}(q_1, \dots, q_\mu) = \mathrm{Rad}(I(G))$, satisfying the conditions of Schmitt-Vogel's lemma.
• In Example 1, $\mathrm{ara}\,I(G) = 6$, $\mathrm{ht}\,I(G) = 5$, and the radical is generated by six diagonal sums, confirming the theoretical bound.