Characteristic-independence of Betti numbers of graph ideals

In this paper we study the Betti numbers of Stanley-Reisner ideals generated in degree 2. We show that the first six 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. 0. Introduction Throughout this paper K will denote a field. For any homogeneous ideal I of a polynomial ring R = K [x1, . . . , xn] there exists a graded minimal finite free resolution 0 → ⊕ j R(−j)pj → · · · → ⊕ j R(−j)1j → R → R/I → 0 of R/I, in which R(−j) denotes the graded free module obtained by shifting the degrees of elements in R by j. The numbers βij , which we shall refer to as the ith Betti numbers of degree j of R/I, are independent of the choice of graded minimal finite free resolution. We also define the ith Betti number of I as βi := ∑ βij . One of the central problems in Commutative Algebra is the description of minimal resolutions of ideals. Even when one restricts one’s attention to ideals of polynomial rings generated by monomials, the structure of the resulting resolutions is very poorly understood. There have two main approaches to this problem. The first is to describe non-minimal free resolutions of these ideals, e.g., the Taylor resolutions (cf. [T]) and its generalization, cellular resolutions (cf. [BS]). The other approach, which we follow here, has been to describe the Betti numbers of these minimal resolutions. It has been known for quite some time that the Betti numbers of monomial ideals may depend on the characteristic of the ground field (e.g., see §5.4 in [BH1] and section 4 below.) The aim of this paper is to investigate this dependence for Stanley-Reisner rings which are quotients by monomial ideals generated in degree 2. In [TH] Naoki Terai and Takayuki Hibi have shown that the third and fourth Betti numbers of these Stanley-Reisner rings do not depend on the ground field– this paper extends this result to show that the fifth and sixth Betti numbers are also independent of the ground Date: October 18, 2007. 1991 Mathematics Subject Classification. Primary 13F55, 13D02.