Let R=K[x1,…,xn], a graded algebra S=R/I satisfies Nk,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 Nk,p. Eisenbud and Goto have shown that for any graded ring R/I, then R/I≥k, where I≥k=I∩Mk and M=(x1,…,xn), has a k-linear resolution (satisfies Nk,p for all p) if k≫0. For a squarefree monomial ideal I, we are here interested in the ideal Ik which is the squarefree part of I≥k. The ideal I is, via Stanley–Reisner correspondence, associated to a simplicial complex ΔI. In this case, all Betti numbers of R/Ik 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 ΔI. 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.
Lance Bryant noticed in his thesis (Bryant, 2009 [3]), that there was a flaw in our paper (Barucci and Froberg, 2006 (2]). It can be fixed by adding a condition, called the BF condition in Bryant (2009) [3]. We discuss some equivalent conditions, and show that they are fulfilled for some classes of rings, in particular for our motivating example of semigroup rings. Furthermore we discuss the connection to a similar result, stated in more generality, by Cortadellas and Zarzuela in [4]. Finally we use our result to conclude when a semigroup ring in embedding dimension at most three has an associated graded which is a complete intersection.
If S = < d(1),...,d(nu)> is a numerical semigroup, we call the ring C[S] = C[t(d1),...,t(d nu)] the semigroup ring of S. We study the ring of differential operators on C[S] , and its associated graded in the filtration induced by the order of the differential operators. We find that these are easy to describe if S is a so-called Art semigroup. If I is an ideal in C[S] that is generated by monomials, we also give some results on Der(I, I) (the set of derivations which map I into I).
For some numerical semigroup rings of small embedding dimension, namely those of embedding dimension 3, and symmetric or pseudosymmetric of embedding dimension 4, presentations has been determined in the literature. We extend these results by giving the whole graded minimal free resolutions explicitly. Then we use these resolutions to determine some invariants of the semigroups and certain interesting relations among them. Finally, we determine semigroups of small embedding dimensions which have strongly indispensable resolutions.
Given an ideal I = (f(1) ... , f(r)) in C[x(1), ... , x(n),] generated by forms of degree d, and an integer k > 1, how large can the ideal I-k be, i.e., how small can the Hilbert function of C[x(1), ... , x(n)] / I-k be? If r <= n the smallest Hilbert function is achieved by any complete intersection, but for r > n, the question is in general very hard to answer. We study the problem for r = n + 1, where the result is known for k = 1. We also study a closely related problem, the Weak Lefschetz property, for S/I-k, where I is the ideal generated by the d'th powers of the variables.
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.
Let k be a field and R= k[x1, … , xn] / 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, … , xn] / I, where I is generated by those xixj 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, … , xn] / I, where I is generated by the squarefree monomials xi1…xik for which { i1, … , ik} 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.
If k[x1,…,xn]/I=R=∑i≥0Ri, k a field, is a standard graded algebra, then the Hilbert series of R is the formal power series ∑i≥0dimkRiti. 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,…,fr), degfi=di,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.
Let R=k[t(n1),...t(ns)] = k[x(1),...x(s)]/P be a numerical semigroup ring and let P-(n) = (PRP)-R-n boolean AND R be the symbolic power of P and R-s(P) = circle plus(i >= 0)P((n))t(n) the symbolic Rees ring of P. It is hard to determine symbolic powers of P; there are even non-Noetherian symbolic Rees rings for 3-generated semigroups. We determine the primary decomposition of powers of P for some classes of 3-generated numerical semigroups.
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.
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.
For a graph G, we show a theorem that establishes a correspondence between the fine Hilbert series of the Stanley-Reisner ring of the clique complex for the complementary graph of G and the fine subgraph polynomial of G. We obtain from this theorem some corollaries regarding the independent set complex and the matching complex.
A hypergraph H = (V, E), where V = {x(1),...,x(n)} and E subset of 2(V) defines a hypergraph algebra R-H = k[x(1),...,x(n)]/(x(i1)...x(ik); {i(1),...,i(k)} is an element of E). All our hypergraphs are d-uniform, i.e., vertical bar e(i)vertical bar = d for all e(i) is an element of E. We determine the Poincare series P-RH (t) = Sigma(infinity)(i=1) dim(k) Tor(i)(RH) (k, k)(t)(i) for some hypergraphs generalizing lines, cycles, and stars. We finish by calculating the graded Betti numbers and the Poincare series of the graph algebra of the wheel graph.
Given an ideal of forms in an algebra (polynomial ring, tensor algebra, exterior algebra, Lie algebra, bigraded polynomial ring), we consider the Hilbert series of the factor ring. We concentrate on the minimal Hilbertseries, which is achieved when the forms are generic. In the polynomial ring we also consider the opposite case of maximal series. This is mainly a survey article, but we give a lot of problems and conjectures. The only novel results concern the maximal series in the polynomial ring.
In what follows, we present a large number of questions which were posed on the problem solving seminar in algebra at Stockholm University during the period Fall 2014—Spring 2017 along with a number of results related to these problems. Many of the results were obtained by participants of the latter seminar.
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.