Linear resolutions of quadratic monomial ideals

We study the minimal free resolution of a quadratic monomial ideal in the case where the resolution is linear. First, we focus on the squarefree case, namely that of an edge ideal. We provide an explicit minimal free resolution under the assumption that the graph associated with the edge ideal satisfies specific combinatorial conditions. In addition, we construct a regular cellular structure on the resolution. Finally, we extend our results to non-squarefree ideals by means of polarization.     

Monomial ideals, edge ideals of hypergraphs, and their graded Betti numbers

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 sub-hypergraphs 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. Dedicated to Anthony V. Geramita on the occasion of his 65th birthday.     

The minimal free resolution of a Borel ideal

We consider the Eliahou–Kervaire minimal free resolution of a 0-Borel ideal and its applications.     

First Nonlinear Syzygies of Ideals Associated to Graphs

Consider an ideal I ? K[x ₁,…, x _n ], with K an arbitrary field, generated by monomials of degree two. Assuming that I does not have a linear resolution, we determine the step s of the minimal graded free resolution of I where nonlinear syzygies first appear, we show that at this step of the resolution nonlinear syzygies are concentrated in degree s + 3, and we compute the corresponding graded Betti number β _{s, s+3}. The multidegrees of these nonlinear syzygies are also determined and the corresponding multigraded Betti numbers are shown to be all equal to 1.     

An elementary approach to containment relations between symbolic and ordinary powers of certain monomial ideals

We provide an elementary explanation of a surprising result of Ein–Lazarsfeld–Smith and Hochster–Huneke on the containment between symbolic and ordinary powers of ideals for a certain class of simple monomial ideals.     

Stability of associated primes of integral closures of monomial ideals

Let I be a monomial ideal of a polynomial ring R=K[X₁,…,Xr] and d(I) the maximal degree of minimal generators of I. In this paper, we explicitly determine a number n₀ in terms of r and d(I) such that for all n?n₀. Furthermore, our n₀ is almost sharp.     

The Analytic Spread of Monomial Ideals

Abstract

We compute the analytic spread of a monomial ideal I of the ring ?[[x ₁,…,x _n ]] in terms of the Newton polyhedron of I.     

Specializations of Ferrers ideals

We introduce a specialization technique in order to study monomial ideals that are generated in degree two by using our earlier results about Ferrers ideals. It allows us to describe explicitly a cellular minimal free resolution of various ideals including any strongly stable and any squarefree strongly stable ideal whose minimal generators have degree two. In particular, this shows that threshold graphs can be obtained as specializations of Ferrers graphs, which explains their similar properties.     

The Integral Closure of Ideals in ℂ{x,y}

Abstract

We give some techniques to determine the ideal K _I generated by the monomials x ^{k ₁} y ^{k ₂} belonging to the integral closure ī of an ideal I ? ?{x, y}. We also give a sufficient condition for a weighted homogeneous ideal I ? ?{x, y} to satisfy the relation ī = I + K _I .     

Resolutions of 2 and 3 Dimensional Rings of Invariants for Cyclic Groups

Let G be the cyclic group of order n, and suppose F is a field containing a primitive nth root of unity. We consider the ring of invariants F[W] ^G of a three dimensional representation W of G where G ? SL(W). We describe minimal generators and relations for this ring and prove that the lead terms of the relations are quadratic. These minimal generators for the relations form a Gröbner basis with a surprisingly simple combinatorial structure. We describe the graded Betti numbers for a minimal free resolution of F[W] ^G . The case where W is any two dimensional representation of G is also handled.     

Squarefree Vertex Cover Algebras

In this paper we introduce squarefree vertex cover algebras and exhibit a duality for them. We study the question when these algebras are standard graded and when these algebras coincide with the ordinary vertex cover algebras. It is shown that this is the case for simplicial complexes corresponding to principal Borel sets. Moreover, the generators of these algebras are explicitly described.