Integral closures of monomial ideals and Fulkersonian hypergraphs

Proceedings of the Edinburgh Mathematical Society, 2016

We investigate symbolic and regular powers of monomial ideals. For a square-free monomial ideal I ⊆ 𝕜[x0, … , xn] we show that for all positive integers m, t and r, where e is the big-height of I and . This captures two conjectures (r = 1 and r = e): one of Harbourne and Huneke, and one of Bocci et al. We also introduce the symbolic polyhedron of a monomial ideal and use this to explore symbolic powers of non-square-free monomial ideals.

Commutative Algebra, 2012

We survey research relating algebraic properties of powers of squarefree monomial ideals to combinatorial structures. In particular, we describe how to detect important properties of (hyper)graphs by solving ideal membership problems and computing associated primes. This work leads to algebraic characterizations of perfect graphs independent of the Strong Perfect Graph Theorem. In addition, we discuss the equivalence between the Conforti-Cornuéjols conjecture from linear programming and the question of when symbolic and ordinary powers of squarefree monomial ideals coincide.

2013

In a 2008 paper, the first author and Van Tuyl proved that the regularity of the edge ideal of a graph G is at most one greater than the matching number of G. In this note, we provide a generalization of this result to any square-free monomial ideal. We define a 2-collage in a simple hypergraph to be a collection of edges with the property that for any edge E of the hypergraph, there exists an edge F in the 2-collage such that |E \ F | ≤ 1. The Castelnuovo-Mumford regularity of the edge ideal of a simple hypergraph is bounded above by a multiple of the minimum size of a 2-collage. We also give a recursive formula to compute the regularity of a vertex-decomposable hypergraph. Finally, we show that regularity in the graph case is bounded by a certain statistic based on maximal packings of nondegenerate star subgraphs.

2021

We study the regularity of symbolic powers of square-free monomial ideals. We prove that if I = I ∆ is the Stanley-Reisner ideal of a simplicial complex ∆, then reg(I (n)) δ(n − 1) + b for all n 1, where δ = lim n→∞ reg(I (n))/n, and b = max{reg(I Γ) | Γ is a subcomplex of ∆ with F (Γ) ⊆ F (∆)}. This bound is sharp for any n. When I = I(G) is the edge ideal of a simple graph G, we obtain a general linear upper bound reg(I (n)) 2n + order-match(G) − 1, where order-match(G) is the ordered matching number of G.

Journal of Algebraic Combinatorics

We explore connections between the generalized multiplicities of squarefree monomial ideals and the combinatorial structure of the underlying hypergraphs using methods of commutative algebra and polyhedral geometry. For instance, we show that the j-multiplicity is multiplicative over the connected components of a hypergraph, and we explicitly relate the j-multiplicity of the edge ideal of a properly connected uniform hypergraph to the Hilbert-Samuel multiplicity of its special fiber ring. In addition, we provide general bounds for the generalized multiplicities of the edge ideals and compute these invariants for classes of uniform hypergraphs.

Mathematische Zeitschrift, 2021

Let I be a homogeneous ideal in a polynomial ring over a field. Let I (n) be the n-th symbolic power of I. Motivated by results about ordinary powers of I, we study the asymptotic behavior of the regularity function reg(I (n)) and the maximal generating degree function ω(I (n)), when I is a monomial ideal. It is known that both functions are eventually quasi-linear. We show that, in addition, the sequences {reg I (n) /n}n and {ω(I (n))/n}n converge to the same limit, which can be described combinatorially. We construct an example of an equidimensional, height two squarefree monomial ideal I for which ω(I (n)) and reg(I (n)) are not eventually linear functions. For the last goal, we introduce a new method for establishing the componentwise linearity of ideals. This method allows us to identify a new class of monomial ideals whose symbolic powers are componentwise linear.

Journal of Algebraic Combinatorics, 2007

We use the correspondence between hypergraphs and their associated edge ideals to study the minimal graded free resolution of squarefree monomial ideals. The theme of this paper is to understand how the combinatorial structure of a hypergraph H appears within the resolution of its edge ideal I(H). We discuss when recursive formulas to compute the graded Betti numbers of I(H) in terms of its subhypergraphs can be obtained; these results generalize our previous work (Hà, H.T., Van Tuyl, A. in J. Algebra 309:405-425, 2007) on the edge ideals of simple graphs. We introduce a class of hypergraphs, which we call properly-connected, that naturally generalizes simple graphs from the point of view that distances between intersecting edges are "well behaved." For such a hypergraph H (and thus, for any simple graph), we give a lower bound for the regularity of I(H) via combinatorial information describing H and an upper bound for the regularity when H = G is a simple graph. We also introduce triangulated hypergraphs that are properly-connected hypergraphs generalizing chordal graphs. When H is a triangulated hypergraph, we explicitly compute the regularity of I(H) and show that the graded Betti numbers of I(H) are independent of the ground field. As a consequence, many known results about the graded Betti numbers of forests can now be extended to chordal graphs. Keywords Hypergraphs • Chordal graphs • Monomial ideals • Graded resolutions • Regularity Dedicated to Anthony V. Geramita on the occasion of his 65th birthday.

Journal of Algebra, 2009

Two-dimensional squarefree monomial ideals can be seen as the Stanley-Reisner ideals of graphs. The main results of this paper are combinatorial characterizations for the Cohen-Macaulayness of ordinary and symbolic powers of such an ideal in terms of the associated graph.

Journal of Algebra, 2011

Let S = K[x 1 ,. .. , xn] be a polynomial ring over a field K. Let I(G) ⊆ S denote the edge ideal of a graph G. We show that the ℓth symbolic power I(G) (ℓ) is a Cohen-Macaulay ideal (i.e., S/I(G) (ℓ) is Cohen-Macaulay) for some integer ℓ ≥ 3 if and only if G is a disjoint union of finitely many complete graphs. When this is the case, all the symbolic powers I(G) (ℓ) are Cohen-Macaulay ideals. Similarly, we characterize graphs G for which S/I(G) (ℓ) has (FLC). As an application, we show that an edge ideal I(G) is complete intersection provided that S/I(G) ℓ is Cohen-Macaulay for some integer ℓ ≥ 3. This strengthens the main theorem in [5].

In dimension two, we study complete monomial ideals combinatorially, their Rees algebras and develop effective means to find their defining equations.