2 Sean Jacques And

This paper produces a recursive formula of the Betti numbers of certain Stanley-Reisner 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, any by convention βi,d = 0 for all i < 0. We also define the total ith Betti number of I as βi := ∑ βid. 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 field K and let K[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: April 15, 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 field K 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 ringK[∆] 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 βi,d = ∑ H⊆G induced # V(H)=d dimK H̃d−i−1(∆(H);K) We shall henceforth write β i,d(G), β K i (G), and pd (G) for the ith Betti number of degree d, the ith Betti number and the projective dimension of K[∆(G)], respectively. When K is irrelevant or clear from the context, we shall omit the superscript. As mentioned in the introduction, this paper aims to discover interesting combinatorial interpretations of Betti numbers of graphs. We would like to mention that these Betti numbers always have some combinatorial significance, as counters (with appropriate weights) of numbers of induced subgraphs in a certain list. Specifically, if we wanted to interpret β i,d(G) in these terms, we would compile a (finite!) list Li,d of all graphs H with d vertices and such that dH := dimK H̃d−i−1(∆(H);K) > 0 (this list would depend on K; cf. [K], for example) and we could write β i,d(G) = ∑ H∈Li,d n(H)dH THE BETTI NUMBERS OF FORESTS 3 where n(H) is the number of induced subgraphs G which are isomorphic to H . For example, for any field K β 2 (G) = n (