Characteristic-independence of Betti numbers of graph ideals

Abstract

In this paper, we study the Betti numbers of Stanley-Reisner ideals generated in degree 2. We show that the first 6 Betti numbers do not depend on the characteristic of the ground field. We also show that, if the number of variables n is at most 10, all Betti numbers are independent of the ground field. For n = 11, there exists precisely 4 examples in which the Betti numbers depend on the ground field. This is equivalent to the statement that the homology of flag complexes with at most 10 vertices is torsion free and that there exists precisely 4 non-isomorphic flag complexes with 11 vertices whose homology has torsion.In each of the 4 examples mentioned above the 8th Betti numbers depend on the ground field and so we conclude that the highest Betti number which is always independent of the ground field is either 6 or 7; if the former is true then we show that there must exist a graph with 12 vertices whose 7th Betti number depends on the ground field.

Regularity, depth and arithmetic rank of bipartite edge ideals

“…Additionally, see [5,Corollary 6.9], for the same conclusion for the edge ideals of chordal graphs. For arbitrary graphs, the maximum size of a pairwise disconnected set of edges is a lower bound for reg R/I ; this follows essentially from [6,Lemma 2.2].…”

A Beginner’s Guide to Edge and Cover Ideals

“…Katzman [43] studied how the characteristic of the field affects the graded Betti numbers of I(G). In particular, he showed that the graph G with vertex set V (G) = {x 1 , x 2 , .…”

The reconstruction conjecture and edge ideals

“…In [5] Katzman constructed a graph G for which the Betti diagram of G depends on the characteristic of the ground field. If H is a smallest such graph, then the Betti diagrams of H \{v} are all characteristic independent, while the Betti diagram of H itself depends on the characteristic of the ground field.…”