Let C be a clutter and let I(C) be its edge ideal. This is a survey paper on the algebraic and combinatorial properties of R/I(C) and C, respectively. We give a criterion to estimate the regularity of R/I(C) and apply this criterion to give new proofs of some formulas for the regularity. If C is a clutter and R/I(C) is sequentially Cohen-Macaulay, we present a formula for the regularity of the ideal of vertex covers of C and give a formula for the projective dimension of R/I(C). We also examine the associated primes of powers of edge ideals, and show that for a graph with a leaf, these sets form an ascending chain.

Edge Ideals: Algebraic and Combinatorial Properties

Asymptotic multiplicities of graded families of ideals and linear series

We prove that limits of multiplicities associated to graded families of ideals exist under very general conditions. Most of our results hold for analytically unramified equicharacteristic local rings, with perfect residue fields. We give a number of applications, including a "volume = multiplicity" formula, generalizing the formula of Lazarsfeld and Mustata, and a proof that the epsilon multiplicity of Ulrich and Validashti exists as a limit for ideals in rather general rings, including analytic local domains. We also prove an asymptotic "additivity formula" for limits of multiplicities, and a formula on limiting growth of valuations, which answers a question posed by the author, Kia Dalili and Olga Kashcheyeva. Our proofs are inspired by a philosophy of Okounkov, for computing limits of multiplicities as the volume of a slice of an appropriate cone generated by a semigroup determined by an appropriate filtration on a family of algebraic objects.

/pdf/multiplicities-associated-to-graded-families-of-ideals-46s5jrfdqg.pdf

Multiplicities associated to graded families of ideals

Monomial ideals, although intrinsically interesting, play an important role in studying the connections between commutative algebra and combinatorics. Broadly speaking, problems in combinatorics are encoded into monomial ideals, which then allow us to use techniques and methods in commutative algebra to solve the original question. Stanley’s proof of the Upper Bound Conjecture [180] for simplicial spheres is seen as one of the early highlights of exploiting this connection between two fields. To bridge these two areas of mathematics, Stanley used square-free monomial ideals.

/pdf/a-beginner-s-guide-to-edge-and-cover-ideals-45rn5rcnpg.pdf

A Beginner’s Guide to Edge and Cover Ideals

We focus in this paper on edge ideals associated to bipartite graphs and give a combinatorial characterization of those having regularity 3. When the regularity is strictly bigger than 3, we determine the first step i in the minimal graded free resolution where there exists a minimal generator of degree >i+3, show that at this step the highest degree of a minimal generator is i+4, and determine the corresponding graded Betti number β i,i+4 in terms of the combinatorics of the graph. The results are then extended to the non-square-free case through polarization. We also study a family of ideals of regularity 4 that play an important role in our main result and whose graded Betti numbers can be completely described through closed combinatorial formulas.

/pdf/regularity-3-in-edge-ideals-associated-to-bipartite-graphs-3kl0xsxy4m.pdf

Regularity 3 in edge ideals associated to bipartite graphs

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.

Dependence of Betti Numbers on Characteristic

We consider semigroup S of a rank 1 valuation ? centered on a local ring R. We show that the Hilbert polynomial of R gives a bound on the growth of the valuation semigroup S. This allows us to give a very simple example of a well ordered subsemigroup of Q+, which is not a value semigroup of a local domain. We also consider the rates of growth which are possible for S. We show that quite exotic behavior can occur. In the final section, we consider the general question of characterizing rank 1 value semigroups.

https://faculty.missouri.edu/~cutkoskys/valuesemigroup1.7.pdf

Growth of Rank 1 Valuation Semigroups

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, Stanley--Reisner ideals of vertex-decomposable complexes and ideals with componentwise linear resolutions. 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. We also prove that the Betti table of a monomial ideal over the field of rational numbers can be obtained from the Betti table over any field by a sequence of consecutive cancellations.

We consider the question of which semigroups can occur as the semigroup $S_R(\nu)$ of positive values of a rank 1 valuation dominating a Noetherian local ring $R$. We give a number of bounds of polynomial type on the growth of $\phi(n)=S_R(\nu)\cap (0,n)$ for $n\in\NN$, starting with the upper bound of $P_R(n)$, where $P_R(n)$ is the Hilbert function of $R$. This bound is generalized to an extremely general bound for arbitrary rank valuations in the paper "Semigroups of valuations on local rings, II", by Cutkosky and Teissier, arXiv:0805.3788. This bound is already enough to give simple examples of rank 1 well ordered semigroups which are not the value semigroup $S_R(\nu)$ of a valuation dominating a Noetherian local ring. 
In the case of rank 1, it is possible to give more precise estimates of $\phi(n)$, which we prove in this paper. We also give examples showing that many different rates of growth are possible for $\phi(n)$ on a regular local ring of dimension 2, such as $n({\alpha}$ for any rational $\alpha$ with $1\le\alpha\le 2$, and $n{log}(n)$.

Growth of rank 1 valuation semigroups

Suppose that (R,mR) is a Noetherian local domain, with quotient field K, and ν is a valuation of K which dominates R. Let Γν be the value group of ν. We consider the problem of determining the valuation semigroup S(R) = {ν(f) | f ∈ R \mR}. We give some general results and examples, and give a complete discription in the case of regular local rings of dimension 2. This generalizes the classification of valuation semigroups of regular local rings of dimension two with algebraically closed residue fields obtained by Spivakovsky in S [46]. This article presents recent results of the author, in joint work with Bernard Teissier, Kia Dalili, Olga Kashcheyeva and Vinh An Pham. It is a write up of a lecture given at the second international valuation theory conference.

/pdf/valuation-semigroups-of-noethierian-local-domains-22lbq2gm1r.pdf

Kia Dalili

Papers

Dependence of Betti Numbers on Characteristic

Growth of Rank 1 Valuation Semigroups

Dependence of Betti Numbers on Characteristic

Growth of rank 1 valuation semigroups

Valuation semigroups of noethierian local domains