. A C ] 1 7 Fe b 20 05 THE BETTI NUMBERS OF FORESTS

This paper produces a recursive formula of the Betti numbers of certain StanleyReisner ideals (graph ideals associated to forests). This gives a purely combinatorial definition of the projective dimension of these ideals, which turns out to be a new numerical invariant of forests. Finally, we propose a possible extension of this invariant to general graphs. 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 → ⊕ d R(−d)pd → · · · → ⊕ d R(−d)1d → R → R/I → 0 of R/I, in which R(−d) denotes the graded free module obtained by shifting the degrees of elements in R by d. The numbers βid, which we shall refer to as the ith Betti numbers of degree d of R/I, are independent of the choice of graded minimal finite free resolution. We set β0,d = δd,0 where δ is Kronecker’s delta, and by convention βi,d = 0 for all i < 0. We also define the total ith Betti number of I as βi := ∑ βid. We refer the reader to chapter 19 of [E] for an introduction to graded minimal resolutions. The aim of this paper is to exhibit an interesting combinatorial interpretation of Betti numbers of graph ideals, which we now define. Let G be any finite simple graph. We shall always denote the vertex set of G with V(G) and its edges with E(G). Fix an fieldK and letK[V(G)] be the polynomial ring on the vertices of G with coefficients in K. The graph ideal I(G) associated with G is the ideal of K[V(G)] generated by all degree-2 square-free monomials uv for which (u, v) ∈ E(G). It is not hard to see that every ideal in a polynomial ring generated by degree-2 square-free monomials is of the form I(G) for some graph G. The quotient K[V(G)]/I(G) is a always a Stanley-Reisner ring: define ∆(G) to be the simplicial complex on the vertices of G in which a face consists of a set of vertices, no two joined by an edge. It is easy to see that K[V(G)]/I(G) = K[∆(G)], the Stanley-Reisner ring associated with ∆(G). The simplicial complexes of the form ∆(G) are characterised by the fact that their minimal non-faces are one dimensional; these complexes are also known as flag complexes. Date: February 1, 2008. 1991 Mathematics Subject Classification. 13F55, 13D02, 05C05, 05E99. 1 2 SEAN JACQUES AND MORDECHAI KATZMAN Rather than attempting to describe the Betti numbers of graph ideals in terms of the combinatorial properties of the graph, we shall go the other way around. The main result in this paper (Theorem 4.8) establishes a new numerical combinatorial invariant of forests and which is shown to be well defined by the fact that it coincides with the projective dimension of the ideals associated with forests. To the best of our knowledge, this is a new invariant of forests. 1. Hochster’s formula Recall that for any fieldK and simplicial complex ∆ the Stanley-Reisner ring K[∆] is the quotient of the polynomial ring in the vertices of ∆ with coefficients in K by the monomial ideal generated by the product of vertices not in a face of ∆ (see chapter 5 in [BH] for a good introduction to Stanley-Reisner rings.) The main tool for investigating Betti numbers of a Stanley-Reisner ring K[∆] is the following theorem giving the Betti numbers as a sum of dimensions of the reduced homologies of sub-simplicial complexes of ∆. Theorem 1.1 (Hochster’s Formula (Theorem 5.1 in [H])). The ith Betti number of K[∆] of degree d is given by βi,d = ∑ W⊆V (∆),#W=d dimK H̃d−i−1(∆W ;K) where V (∆) is the set of vertices of ∆ and for any W ⊆ V (∆), ∆W denotes the simplicial complex with vertex set W and whose faces are the faces of ∆ containing only vertices in W . Notice that when ∆ = ∆(G) for some graph G, we can rewrite the formula above as