Upper Bounds for the Number of Facets of a Simplicial Complex

Here we study the maximal dimension of the annihilator ideals 0 :A m j of artinian graded rings A = P/(I, x1, x 2 2, . . . , x 2 v) with a given Hilbert function, where P is the polynomial ring in the variables x1, x2, . . . , xv over a field K with each deg xi = 1, I is a graded ideal of P , and m is the graded maximal ideal of A. As an application to combinatorics, we introduce the notion of j-facets and obtain some informations on the number of j-facets of simplicial complexes with a given f -vector. Let P = K[x1, x2, . . . , xv] denote the polynomial ring in v variables over a field K with the standard grading, i.e., each deg xi = 1, and write K{Γ} for the quotient algebra P/(x1, x 2 2, . . . , x 2 v). We are interested in the dimensions of the annihilator ideals 0 :K{Γ}/I m of K{Γ}/I, where m is the graded maximal ideal of K{Γ}/I. In particular, among all graded ideals I of K{Γ} with a given Hilbert function, we determine the maximal dimension of the socles 0 :K{Γ}/I m of K{Γ}/I. The graded ring K{Γ}/I is studied in [A–H–H] when I is generated by (squarefree) monomials. First, we recall some standard notation and terminology on graded rings and modules. When M is a Z-graded module, where Z is the set of integers, we write Mi, i ∈ Z, for the i-th graded component of M . Moreover, for every a ∈ Z, we define M(a) to be the Z-graded module with graded components M(a)i = Ma+i for all i ∈ Z. If M is a finitely generated Z-graded module over the polynomial ring P = K[x1, x2, . . . , xv], then the modules Tor K i (K,M) are finite-dimensional graded K-vector spaces. Then we say that βij(M) := dimK Tor K i (K,M)j is the (i, j)-th graded Betti number of M . Finally, when A is a graded ring over K and J is a graded ideal of A, we denote by 0 :A J the annihilator of J in A. Let ( V q ) denote the set of all squarefree monomials of degree q ≥ 1 in the variables V = {x1, x2, . . . , xv}. We write ≤lex for the lexicographic order on ( V q ) , i.e., if S = xi1xi2 · · ·xiq and T = xj1xj2 · · ·xjq are squarefree monomials belonging to ( V q ) with 1 ≤ i1 < i2 < · · · < iq ≤ v and 1 ≤ j1 < j2 < · · · < jq ≤ v, then S jk for some 1 ≤ k ≤ q. A nonempty set M ⊂ (Vq ) is called a squarefree lexsegment set of degree q if T ∈ M, S ∈ (Vq ) and T ≤lex S imply S ∈M. An ideal I of K{Γ} generated by squarefree monomials is Received by the editors August 28, 1995 and, in revised form, October 26, 1995. 1991 Mathematics Subject Classification. Primary 05D05; Secondary 13D40. This paper was written while the authors were staying at the Mathematische Forschungsinstitut Oberwolfach in the frame of the RiP program which is financed by Volkswagen–Stiftung. c ©1997 American Mathematical Society