Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Explicit Betti Numbers for Skeletons of Chordal Clique Complexes and Their Alexander Duals

Mohammed Rafiq Namiq

Abstract

We study the homological properties of $Δ_{\mathbf{r}}(n_1, \dots, n_e)$, a simplicial complex formed by sequentially gluing complete graphs along $(r_i-1)$-simplices. This construction generates precisely the chordal clique complexes, whose Stanley-Reisner ideals admit 2-linear resolutions. By computing the $f$-vector and evaluating the Hilbert series, we establish explicit graded Betti numbers for all $k$-skeletons. We show that the regularity of these skeletons is $k+1$ and the projective dimension stabilizes at $N_{\mathbf{r}} - r_{\min} - 1$ for $k \ge r_{\min}$, providing a complete classification of when the complex is Cohen-Macaulay, sequentially Cohen-Macaulay, or initially Cohen-Macaulay. We also obtain explicit formulas for the ring multiplicity and reduced Euler characteristic. Applying Alexander duality, we derive the $f$-vector, rational $h$-polynomial, and exact graded Betti numbers of the dual and its skeletons. Furthermore, analyzing these dual skeletons yields a family of complexes that resolve recent open bounds on regularity. Finally, equating the topological and rational evaluations of the Hilbert series produces a new family of combinatorial binomial identities.

Explicit Betti Numbers for Skeletons of Chordal Clique Complexes and Their Alexander Duals

Abstract

We study the homological properties of , a simplicial complex formed by sequentially gluing complete graphs along -simplices. This construction generates precisely the chordal clique complexes, whose Stanley-Reisner ideals admit 2-linear resolutions. By computing the -vector and evaluating the Hilbert series, we establish explicit graded Betti numbers for all -skeletons. We show that the regularity of these skeletons is and the projective dimension stabilizes at for , providing a complete classification of when the complex is Cohen-Macaulay, sequentially Cohen-Macaulay, or initially Cohen-Macaulay. We also obtain explicit formulas for the ring multiplicity and reduced Euler characteristic. Applying Alexander duality, we derive the -vector, rational -polynomial, and exact graded Betti numbers of the dual and its skeletons. Furthermore, analyzing these dual skeletons yields a family of complexes that resolve recent open bounds on regularity. Finally, equating the topological and rational evaluations of the Hilbert series produces a new family of combinatorial binomial identities.
Paper Structure (12 sections, 35 theorems, 62 equations, 1 table)

This paper contains 12 sections, 35 theorems, 62 equations, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Simplicial Complexes, Skeletons, and Duality
  4. Stanley-Reisner Rings and Homological Invariants
  5. The Hilbert Series, Multiplicity, and Euler characteristic
  6. Cohen-Macaulay and Initially Cohen-Macaulay Properties
  7. The Glued Complex $\Delta_{\mathbf{r}}$
  8. Homological Invariants of the Skeletons
  9. The Alexander Dual of Glued Complexes
  10. Homological Invariants of Dual Skeletons
  11. Explicit Computational Examples
  12. Combinatorial Identities via Hilbert Series Evaluations

Key Result

Proposition 3.1

For any $-1 \le k < \dim \Delta_{\mathbf{r}}$, the Krull dimension of the Stanley-Reisner ring of the $k$-skeleton is $\dim \mathbb{K}[(\Delta_{\mathbf{r}})_{(k)}] = k+1$. Its algebraic multiplicity is given by

Theorems & Definitions (73)

  • Proposition 3.1
  • proof
  • Proposition 3.2
  • proof
  • Remark 3.3
  • Lemma 4.1
  • proof
  • Theorem 4.2
  • Theorem 4.3
  • proof
  • ...and 63 more