We determine the Hilbert series of some classes of ideals generated by generic forms of degree two and three, and investigate the difference to the Hilbert series of ideals generated by powers of linear generic forms of the corresponding degrees.

For a graph G, Bayer–Denker–Milutinović–Rowlands–Sundaram–Xue study in [1] a new graph complex Δkt(G), namely the simplicial complex with facets that are complements to independent sets of size k in G. They are interested in topological properties such as shellability, vertex decomposability, homotopy type, and homology of these complexes. In this paper we study more algebraic properties, such as Cohen-Macaulayness, Betti numbers, and linear resolutions of the Stanley-Reisner ring of these complexes and their Alexander duals.

For a graph G=(V,E) the edge ring k[G] is k[x1,…,xn]/I(G), where n=|V| and I(G) is generated by {xixj;{i,j}∈E}. The conjecture we treat is the following.

Conjecture 1. If k[G] has a 2-linear resolution, then the projective dimension of K[G], pd (k[G]), equals the maximal degree of a vertex in G.

As far as we know, this conjecture is first mentioned in a paper by Gitler and Valencia, and there it is called the Eliahou-Villarreal conjecture. The conjecture is treated in a recent paper by Ahmed, Mafi, and Namiq. That there are counterexamples was noted already by Moradi and Kiani. By interpreting k[G] as a Stanley-Reisner ring, we are able to characterize those graphs for which the conjecture holds.

Let R=K[x₁,…,x_n], a graded algebra S=R/I satisfies N_k,p if I is generated in degree k, and the graded minimal resolution is linear the first p steps, and the k-index of S is the largest p such that S satisfies N_k,p. Eisenbud and Goto have shown that for any graded ring R/I, then R/I_≥k, where I_≥k=I∩M^k and M=(x₁,…,x_n), has a k-linear resolution (satisfies N_k,p for all p) if k≫0. For a squarefree monomial ideal I, we are here interested in the ideal I_k which is the squarefree part of I_≥k. The ideal I is, via Stanley–Reisner correspondence, associated to a simplicial complex Δ_I. In this case, all Betti numbers of R/I_k for k>min{deg(u)∣u∈I}, which of course are a much finer invariant than the index, can be determined from the Betti diagram of R/I and the f-vector of Δ_I. We compare our results with the corresponding statements for I_≥k. (Here I is an arbitrary graded ideal.) In this case, we show that the Betti numbers of R/I_≥k can be determined from the Betti numbers of R/I and the Hilbert series of R/I_≥k.

Let k be a field and R= k[x₁, … , x_n] / I= S/ I a graded ring. Then R has a t-linear resolution if I is generated by homogeneous elements of degree t, and all higher syzygies are linear. Thus, R has a t-linear resolution if Tor(S/I,k)=0 if j≠ i+ t- 1. For a graph G on { 1 , … , n} , the edge algebra is k[x₁, … , x_n] / I, where I is generated by those x_ix_j for which { i, j} is an edge in G. We want to determine the Betti numbers of edge rings with 2-linear resolution. But we want to do that by looking at the edge ring as a Stanley–Reisner ring. For a simplicial complex Δ on [n] = { 1 , … , n} and a field k, the Stanley–Reisner ring k[Δ] is k[x₁, … , x_n] / I, where I is generated by the squarefree monomials x_i1…x_ik for which { i₁, … , i_k} does not belong to Δ. Which Stanley–Reisner rings that are edge rings with 2-linear resolution are known. Their associated complexes has had different names in the literature. We call them fat forests here. We determine the Betti numbers of many fat forests and compare our result with what is known. We also consider Betti numbers of Alexander duals of fat forests. 

We characterize all Gorenstein rings generated by strongly stable sets of monomials of degree two. We compute their Hilbert series in several cases, which also provides an answer to a question by Migliore and Nagel. 

If k[x₁,…,x_n]/I=R=∑_i≥0R_i, k a field, is a standard graded algebra, then the Hilbert series of R is the formal power series ∑_i≥0dim_kR_itⁱ. It is known already since Macaulay which power series are Hilbert series of graded algebras. A much harder question is which series are Hilbert series if we fix the number of generators of I and their degrees, say for ideals I=(f₁,…,f_r), degf_i=d_i,i=1,…,r. In some sense “most” ideals with fixed degrees of their generators have the same Hilbert series. There is a conjecture for the Hilbert series of those “generic” ideals, see below. In this article we make a conjecture, and prove it in some cases, in the case of generic ideals of fixed degrees in the coordinate ring of P¹×P¹, which might be easier to prove.

In this paper we describe the algebraic relations satisfied by the harmonic and anti-harmonic moments of simply connected, but not necessarily convex planar polygons with a given number of vertices.

Moreno studies the following question. Let I be an ideal in k[x1,…,xn] generated minimally by elements of degree d, d + 1, d + 2,…. How long can such a sequence of generators be? Later he also studies the opposite question.

This article is a try to describe the algebraic side of the story on Stanley-Reisner rings.