Betti Numbers Of Ideals Research Articles

Betti numbers under small perturbations

We study how Betti numbers of ideals in a local ring change under small perturbations. Given p∈N and given an ideal I of a Noetherian local ring (R,m), our main result states that there exists N>0 such that if J is an ideal with I≡JmodmN and with the same Hilbert function as I, then the Betti numbers βiR(R/I) and βiR(R/J) coincide for 0≤i≤p. Moreover, we present several cases in which an ideal J such that I≡JmodmN is forced to have the same Hilbert function as I, and therefore the same Betti numbers.

The type defect of a simplicial complex

Fix a field k. When Δ is a simplicial complex on n vertices with Stanley–Reisner ideal IΔ, we define and study an invariant called the type defect of Δ. Except when Δ is a single simplex, the type defect of Δ, td(Δ), is the difference dim⁡TorcS(S/IΔ,k)−c, where c is the codimension of Δ and S=k[x1,…xn]. We show that this invariant admits surprisingly nice properties. For example, it is well-behaved when one glues two complexes together along a face. Furthermore, Δ is Cohen–Macaulay if td(Δ)≤0. On the other hand, if Δ is a simple graph (viewed as a one-dimensional complex), then td(Δ′)≥0 for every induced subgraph Δ′ of Δ if and only if Δ is chordal. Requiring connected induced subgraphs to have type defect zero allows us to define a class of graphs that we call treeish, and which we generalize to simplicial complexes. We then extend some of our chordality results to higher dimensions, proving sharp lower bounds for most Betti numbers of ideals with linear resolution, and classifying when equalities occur. As an application, we prove sharp lower bounds for Betti numbers of graded ideals (not necessarily monomial) with linear resolution.

A note on resolutions mod i

In this note we give new examples of Golod pairs, extending results of Shamash, Avramov, Gover and Salmon. We also prove a result, relating Betti numbers of ideals I and I + (x) in a local ring R where the ideal I has a linear protective resolution and x is a non zero divisor in R.

Pascal type properties of Betti numbers

In this paper, we will describe the Pascal Type properties of Betti numbers of ideals associated to n‐gons. These are quite similar to the properties enjoyed by the Pascal′s Triangle, concerning the binomial coefficients. By definition, the Betti numbers βt(n) of an ideal I associated to an n‐gon are the ranks of the modules in a free minimal resolution of the R‐module R/I, where R is the polynomial ring k[x1, x2, …, xn]. Here k is any field and x1, x2, …, xn are indeterminates. We will prove those properties using a specific formula for the Betti numbers.