[Paper Review] Extremal Betti numbers of some classes of binomial edge ideals

Ahmet Dokuyucu|arXiv (Cornell University)|Oct 10, 2013

Commutative Algebra and Its Applications14 references16 citations

TL;DR

This paper confirms a conjecture by Herzog et al. for binomial edge ideals of cycles and complete bipartite graphs by proving that the extremal Betti numbers of the ideal $J_G$ and its lexicographic initial ideal $ ext{in}_<(J_G)$ coincide. Using known resolutions and Gröbner basis theory, the authors show that both ideals have the same projective dimension and regularity, and compute the unique extremal Betti number explicitly as $\binom{n-1}{2} - 1$ for cycles and $m-1$ or $n-1$ for complete bipartite graphs.

ABSTRACT

Let $G$ be a cycle or a complete bipartite graph. We show that the binomial edge ideal $J_{G}$ and its initial ideal with respect to the lexicographic order have the same extremal Betti number.

Motivation & Objective

• To verify a conjecture stating that the extremal Betti numbers of a binomial edge ideal $J_G$ and its lexicographic initial ideal $ ext{in}_<(J_G)$ are equal for all graphs $G$.
• To establish this coincidence specifically for cycles and complete bipartite graphs, which are known to have unique extremal Betti numbers.
• To leverage known resolutions of $J_G$ for these graph classes to analyze the homological invariants of $ ext{in}_<(J_G)$.
• To demonstrate that $ ext{in}_<(J_G)$ preserves projective dimension and regularity, ensuring the same extremal Betti number location.
• To compute the exact value of the extremal Betti number for both graph classes using Tor modules and recursive exact sequences.

Proposed method

• Use the reduced Gröbner basis of $J_G$ with respect to the lexicographic order, as characterized by the HHHKR theorem, to determine the monomial generators of $ ext{in}_<(J_G)$.
• Apply the definition of admissible paths to compute the initial ideal for complete bipartite graphs and cycles, showing that $ ext{in}_<(J_G)$ is generated in degrees 2 and 3.
• Use the exact sequence $0 \to S/I_{k-1}:(v_k)(-\deg v_k) \to S/I_{k-1} \to S/I_k \to 0$ to inductively compute Betti numbers of $S/\text{in}_<(J_G)$.
• Leverage the fact that $\text{in}_<(J_G)$ for complete bipartite graphs has linear quotients, enabling componentwise linearity and easy Betti number computation.
• For cycles, order the generators of $\text{in}_<(J_G)$ strategically to compute the extremal Betti number via the long exact sequence of Tor modules.
• Use the regularity and projective dimension bounds from $S/J_G$ and $S/\text{in}_<(J_G)$ to show equality in both invariants, confirming the extremal Betti number is preserved.

Experimental results

Research questions

• RQ1 Do the extremal Betti numbers of $J_G$ and $\text{in}_<(J_G)$ coincide for all graphs $G$?
• RQ2 Is the extremal Betti number preserved under lexicographic initial ideal formation for cycles and complete bipartite graphs?
• RQ3 Can the projective dimension and regularity of $\text{in}_<(J_G)$ be shown to equal those of $J_G$ for these graph classes?
• RQ4 What is the exact value of the extremal Betti number for $\text{in}_<(J_G)$ when $G$ is a cycle or a complete bipartite graph?
• RQ5 Does the Lyubeznik resolution yield a minimal resolution for $\text{in}_<(J_G)$ in the case of cycles?

Key findings

• The extremal Betti number of $J_G$ and its lexicographic initial ideal $\text{in}_<(J_G)$ are equal for cycles and complete bipartite graphs, confirming a partial case of the conjecture.
• For a cycle on $n$ vertices, the unique extremal Betti number is $\beta_{n,2n-2}(S/J_G) = \binom{n-1}{2} - 1$.
• For a complete bipartite graph $K_{m,n}$ with $n > 1$, the extremal Betti number is $\beta_{p,p+2} = n-1$, where $p = 2m + n - 2$.
• For $K_{m,1}$, the extremal Betti number is $\beta_{m,m+2} = m-1$.
• The projective dimension and regularity of $\text{in}_<(J_G)$ match those of $J_G$ for both graph classes, implying the same extremal Betti number location.
• The initial ideal $\text{in}_<(J_G)$ for $K_{m,n}$ is componentwise linear and generated in degrees 2 and 3, enabling explicit Betti number computation.