Betti Numbers Of Monomial Ideals Research Articles

Powers of monomial ideals with characteristic-dependent Betti numbers

We explore the dependence of the Betti numbers of monomial ideals on the characteristic of the field. A first observation is that for a fixed prime p either the i-th Betti number of all high enough powers of a monomial ideal differs in characteristic 0 and in characteristic p or it is the same for all high enough powers. In our main results, we provide constructions and explicit examples of monomial ideals all of whose powers have some characteristic-dependent Betti numbers or whose asymptotic regularity depends on the field. We prove that, adding a monomial on new variables to a monomial ideal allows to spread the characteristic dependence to all powers. For any given prime p, this produces an edge ideal such that all its powers have some Betti numbers that are different over mathbb {Q} and over mathbb {Z}_p. Moreover, we show that, for every r ge 0 and i ge 3 there is a monomial ideal I such that some coefficient in a degree ge r of the Kodiyalam polynomials {mathfrak {P}}_3(I),ldots ,{mathfrak {P}}_{i+r}(I) depends on the characteristic. We also provide a summary of related results and speculate about the behavior of other combinatorially defined ideals.

Betti numbers of monomial ideals in four variables

We express the multigraded Betti numbers of monomial ideals in four variables in terms of the multigraded Betti numbers of 66 squarefree monomial ideals, also in four variables. We use this class of 66 ideals to prove that monomial resolutions in four variables are independent of the base field. In addition, we give a formula for the Betti numbers of an arbitrary monomial ideal in four variables.

An equivariant Hochster's formula for Sn$\mathfrak {S}_n$‐invariant monomial ideals

Let $R=\Bbbk[x_1,\dots,x_n]$ be a polynomial ring over a field $\Bbbk$ and let $I\subset R$ be a monomial ideal preserved by the natural action of the symmetric group $\mathfrak S_n$ on $R$. We give a combinatorial method to determine the $\mathfrak S_n$-module structure of $\mathrm{Tor}_i(I,\Bbbk)$. Our formula shows that $\mathrm{Tor}_i(I,\Bbbk)$ is built from induced representations of tensor products of Specht modules associated to hook partitions, and their multiplicities are determined by topological Betti numbers of certain simplicial complexes. This result can be viewed as an $\mathfrak S_n$-equivariant analogue of Hochster's formula for Betti numbers of monomial ideals. We apply our results to determine extremal Betti numbers of $\mathfrak S_n$-invariant monomial ideals, and in particular recover formulas for their Castelnuovo--Mumford regularity and projective dimension. We also give a concrete recipe for how the Betti numbers change as we increase the number of variables, and in characteristic zero (or $>n$) we compute the $\mathfrak S_n$-invariant part of $\mathrm{Tor}_i(I,\Bbbk)$ in terms of $\mathrm{Tor}$ groups of the unsymmetrization of $I$.

Lower bounds for Betti numbers of monomial ideals

Let I be a monomial ideal of height c in a polynomial ring S over a field k. If I is not generated by a regular sequence, then we show that the sum of the betti numbers of S/I is at least 2c+2c−1 and characterize when equality holds. Lower bounds for the individual betti numbers are given as well.

Betti numbers of monomial ideals via facet covers

We give a combinatorial condition that ensures a monomial ideal has a nonzero Betti number in a given multidegree. As a result, we combinatorially characterize all multigraded Betti numbers, projective dimension and regularity of facet ideals of simplicial forests. Our condition is expressed in terms of minimal facet covers of simplicial complexes.

Bounding Betti numbers of monomial ideals in the exterior algebra

Let $K$ be a field, $V$ a $K$-vector space with basis $e_1,\ldots,e_n$, and $E$ the exterior algebra of $V$. To a given monomial ideal $I\subsetneq E$ we associate a special monomial ideal $J$ with generators in the same degrees as those of $I$ and such that the number of the minimal monomial generators in each degree of $I$ and $J$ coincide. We call $J$ the colexsegment ideal associated to $I$. We prove that the class of strongly stable ideals in $E$ generated in one degree satisfies the colex lower bound, that is, the total Betti numbers of the colexsegment ideal associated to a strongly stable ideal $I\subsetneq E$ generated in one degree are smaller than or equal to those of $I$.

Dependence of Betti Numbers on Characteristic

We study the dependence of graded Betti numbers of monomial ideals on the characteristic of the base field. The examples we describe include bipartite ideals and Stanley–Reisner ideals of vertex-decomposable complexes. We give a description of bipartite graphs and, using discrete Morse theory, provide a way of looking at the homology of arbitrary simplicial complexes through bipartite ideals.

Multigraded Betti numbers without computing minimal free resolutions

We use Mayer-Vietoris trees to obtain the multigraded Betti numbers of monomial ideals without computing their minimal free resolutions. This method provides not only a competitive algorithm for such computations but also a new tool for the analysis of the homological structure of monomial ideals. Using Mayer-Vietoris trees we obtain new results for several families of monomial ideals and new proofs of known results for other families.

Betti Numbers of Monomial Ideals and Shifted Skew Shapes

We present two new problems on lower bounds for Betti numbers of the minimal free resolution for monomial ideals generated in a fixed degree. The first concerns any such ideal and bounds the total Betti numbers, while the second concerns ideals that are quadratic and bihomogeneous with respect to two variable sets, but gives a more finely graded lower bound. These problems are solved for certain classes of ideals that generalize (in two different directions) the edge ideals of threshold graphs and Ferrers graphs. In the process, we produce particularly simple cellular linear resolutions for strongly stable and squarefree strongly stable ideals generated in a fixed degree, and combinatorial interpretations for the Betti numbers of other classes of ideals, all of which are independent of the coefficient field.

Betti numbers of some monomial ideals

In this paper we compute the graded Betti numbers of certain monomial ideals that are not stable. As a consequence we prove a conjecture, stated by G. Fatabbi, on the graded Betti numbers of two general fat points in P 3 . \mathbf {P}^3.

Computation of betti numbers of monomial ideals associated with stacked polytopes

LetP(v, d) be a stackedd-polytope withv vertices, δ(P(v, d)) the boundary complex ofP(v, d), andk[Δ(P(v, d))] =A/I Δ(P(v,d)) the Stanley-Reisner ring of Δ(P(v,d)) over a fieldk. We compute the Betti numbers which appear in a minimal free resolution ofk[Δ(P(v,d))] overA, and show that every Betti number depends only onv andd and is independent of the base fieldk. We also show that the Betti number sequences above are unimodal.

Computation of betti numbers of monomial ideals associated with cyclic polytopes

We give a combinatorial formula for the Betti numbers which appear in a minimal free resolution of the Stanley-Reisner ringk[Δ(P)]=A/IΔ(P) of the boundary complex Δ(P) of an odd-dimensional cyclic polytopePover a fieldk. A corollary to the formula is that the Betti number sequence ofk[Δ(P)] is unimodal and does not depend on the base fieldk.