A classification of van der Waerden complexes with linear resolution

TL;DR

The paper classifies van der Waerden complexes $vdW(n,k)$ with linear resolution by showing the Stanley-Reisner ideal has a linear resolution exactly when $k=1$ or $\frac{n}{2}\le k < n$, using leaf orders and chordal-graph theory to establish sufficiency and a minimal-nonface analysis to rule out other cases; it also proves that the Stanley-Reisner rings of all Cohen–Macaulay $vdW(n,k)$ are level, with $vdW(5,2)$ being Gorenstein and $vdW(6,2)$ level but not Gorenstein, and shows that aside from $(5,2)$ and $(6,2)$, CM cases have linear resolution. The results bridge combinatorial topology, chordal graphs, and commutative algebra, clarifying when van der Waerden complexes admit linear strands and how their CM instances exhibit levelness, complementing prior classifications. The work sharpens understanding of the algebraic properties tied to the arithmetic-sequence facets of $vdW(n,k)$ and provides concrete Betti-table evidence for the level vs linear-resolution distinctions.

Abstract

In 2017, Ehrenborg, Govindaiah, Park, and Readdy defined the van der Waerden complex ${\tt vdW}(n,k)$ to be the simplicial complex whose facets correspond to all the arithmetic sequences on the set $\{1,\ldots,n\}$ of a fixed length $k$. To complement a classification of the Cohen--Macaulay van der Waerden complexes obtained by Hooper and Van Tuyl in 2019, a classification of van der Waerden complexes with linear resolution is presented. Furthermore, we show that the Stanley--Reisner ring of a Cohen--Macaulay van der Waerden complex is level.

Figures (1)

• Figure 1: ${\tt vdW}(5,2)$ and ${\tt vdW}(6,2)$

Theorems & Definitions (8)

• Theorem 1.1
• proof
• Lemma 1.2
• proof
• Theorem 2.1
• Example 2.2
• Theorem 2.3
• proof