Adam Van Tuyl

Sequentially Cohen-Macaulay edge ideals

Let G be a simple undirected graph on n vertices, and let I(G) C R = k[x 1 ,...,x n ] denote its associated edge ideal. We show that all chordal graphs G are sequentially Cohen-Macaulay; our proof depends upon showing that the Alexander dual of I(G) is componentwise linear. Our result complements Faridis theorem that the facet ideal of a simplicial tree is sequentially Cohen-Macaulay and implies Herzog, Hibi, and Zhengs theorem that a chordal graph is Cohen-Macaulay if and only if its edge ideal is unmixed. We also characterize the sequentially Cohen-Macaulay cycles and produce some examples of nonchordal sequentially Cohen-Macaulay graphs.

Sequentially Cohen–Macaulay bipartite graphs: vertex decomposability and regularity

Let G be a bipartite graph with edge ideal I(G) whose quotient ring R/I(G) is sequentially Cohen–Macaulay. We prove: (1) the independence complex of G must be vertex decomposable, and (2) the Castelnuovo–Mumford regularity of R/I(G) can be determined from the invariants of G.

Splittings of monomial ideals

We provide some new conditions under which the graded Betti numbers of a monomial ideal can be computed in terms of the graded Betti numbers of smaller ideals, thus complementing Eliahou and Kervaires splitting approach. As applications, we show that edge ideals of graphs are splittable, and we provide an iterative method for computing the Betti numbers of the cover ideals of Cohen-Macaulay bipartite graphs. Finally, we consider the frequency with which one can find particular splittings of monomial ideals and raise questions about ideals whose resolutions are characteristic-dependent.

Algebraic Properties of the Path Ideal of a Tree

The path ideal (of length t ≥ 2) of a directed graph Γ is the monomial ideal, denoted I t (Γ), whose generators correspond to the directed paths of length t in Γ. We study some of the algebraic properties of I t (Γ) when Γ is a tree. We first show that I t (Γ) is the facet ideal of a simplicial tree. As a consequence, the quotient ring R/I t (Γ) is always sequentially Cohen–Macaulay, and the Betti numbers of R/I t (Γ) do not depend upon the characteristic of the field. We study the case of the line graph in greater detail at the end of the article. We give an exact formula for the projective dimension of these ideals, and in some cases, we compute their arithmetical rank.

Some families of componentwise linear monomial ideals

Let R=k[x_1,...,x_n] be a polynomial ring over a field k. Let J={j_1,...,j_t} be a subset of [n]={1,...,n}, and let m_J denote the ideal (x_{j_1},...,x_{j_t}) of R. Given subsets J_1,...,J_s of [n] and positive integers a_1,...,a_s, we study ideals of the form I=m_{J_1}^{a_1} \cap ... \cap m_{J_s}^{a_s}. These ideals arise naturally, for example, in the study of fat points, tetrahedral curves, and Alexander duality of squarefree monomial ideals. Our main focus is determining when ideals of this form are componentwise linear. Using polymatroidality, we prove that I is always componentwise linear when s = 4, we give examples to show that I may or may not be componentwise linear. We apply these results to ideals of small sets of general fat points in multiprojective space, and we extend work of Fatabbi, Lorenzini, Valla, and the first author by computing the graded Betti numbers in the s=2 case. Since componentwise linear ideals satisfy the Multiplicity Conjecture of Herzog, Huneke, and Srinivasan when char(k)=0, our work also yields new cases in which this conjecture holds.

On the Linear Strand of an Edge Ideal

Let I(G) be the edge ideal associated to a simple graph G. We study the graded Betti numbers that appear in the linear strand of the minimal free resolution of I(G).

The border of the Hilbert function of a set of points in Pn1×⋯×Pnk

Abstract We describe the eventual behaviour of the Hilbert function of a set of distinct points in P n 1 ×⋯× P n k . As a consequence of this result, we show that the Hilbert function of a set of points in P n 1 ×⋯× P n k can be determined by computing the Hilbert function at only a finite number of values. Our result extends the result that the Hilbert function of a set of points in P n stabilizes at the cardinality of the set of points. Motivated by our result, we introduce the notion of the border of the Hilbert function of a set of points. By using the Gale–Ryser Theorem, a classical result about (0,1)-matrices, we characterize all the possible borders for the Hilbert function of a set of distinct points in P 1 × P 1 .

The Hilbert functions of ACM sets of points in Pn1×⋯×Pnk

Abstract If X is a set of points in P n 1 ×⋯× P n k , then the associated coordinate ring R/I X is an N k -graded ring. The Hilbert function of X , defined by H X ( i ):= dim k (R/I X ) i for all i ∈ N k , is studied. Since the ring R/I X may or may not be Cohen–Macaulay, we consider only those X that are ACM. Generalizing the case of k =1 to all k , we show that a function is the Hilbert function of an ACM set of points if and only if its first difference function is the Hilbert function of a multi-graded Artinian quotient. We also give a new characterization of ACM sets of points in P 1 × P 1 , and show how the graded Betti numbers (and hence, Hilbert function) of ACM sets of points in this space can be obtained solely through combinatorial means.

ACM sets of points in multiprojective space

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