Cellular resolutions of cointerval ideals

Minimal cellular resolutions of the edge ideals of cointerval hypergraphs are constructed. This class of d-uniform hypergraphs coincides with the complements of interval graphs (for the case d?=?2), and strictly contains the class of ‘strongly stable’ hypergraphs corresponding to pure shifted simplicial complexes. The polyhedral complexes supporting the resolutions are described as certain spaces of directed graph homomorphisms, and are realized as subcomplexes of mixed subdivisions of the Minkowski sums of simplices. Resolutions of more general hypergraphs are obtained by considering decompositions into cointerval hypergraphs.     

Bernstein-Sato ideals and hyperplane arrangements

We study the relation between zero loci of Bernstein-Sato ideals and roots of b-functions and obtain a criterion to guarantee that roots of b-functions of a reducible polynomial are determined by the zero locus of the associated Bernstein-Sato ideal. Applying the criterion together with a result of Maisonobe we prove that the set of roots of the b-function of a free hyperplane arrangement is determined by its intersection lattice.We also study the zero loci of Bernstein-Sato ideals and the associated relative characteristic cycles for arbitrary central hyperplane arrangements. We prove the multivariable $n/d$ conjecture of Budur for complete factorizations of arbitrary hyperplane arrangements, which in turn proves the strong monodromy conjecture for the associated multivariable topological zeta functions.     

Hyperplane arrangements and linear strands in resolutions

The cohomology ring of the complement of a central complex hyperplane arrangement is the well-studied Orlik-Solomon algebra. The homotopy group of the complement is interesting, complicated, and few results are known about it. We study the ranks for the lower central series of such a homotopy group via the linear strand of the minimal free resolution of the field over the Orlik-Solomon algebra.

     

Multiplier ideals of hyperplane arrangements

In this note we compute multiplier ideals of hyperplane arrangements. This is done using the interpretation of multiplier ideals in terms of spaces of arcs by Ein, Lazarsfeld, and Mustata (2004).

     

Divisors on graphs,binomial and monomial ideals,and cellular resolutions

We study various binomial and monomial ideals arising in the theory of divisors, orientations, and matroids on graphs. We use ideas from potential theory on graphs and from the theory of Delaunay decompositions for lattices to describe their minimal polyhedral cellular free resolutions. We show that the resolutions of all these ideals are closely related and that their \({\mathbb {Z}}\)-graded Betti tables coincide. As corollaries, we give conceptual proofs of conjectures and questions posed by Postnikov and Shapiro, by Manjunath and Sturmfels, and by Perkinson, Perlman, and Wilmes. Various other results related to the theory of chip-firing games on graphs also follow from our general techniques and results.     

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.     

Powers of edge ideals with linear resolutions

We show that if G is a gap-free and diamond-free graph, then I(G)^s has a linear minimal free resolution for every s≥2.     

The structure of DGA resolutions of monomial ideals

Let $I?\mathbb{k}[x_1,\dots ,x_n]$ be a squarefree monomial ideal in a polynomial ring. In this paper we study multiplications on the minimal free resolution $\mathbb{F}$ of $\mathbb{k}[x_1,\dots ,x_n]/I$. In particular, we characterize the possible vectors of total Betti numbers for such ideals which admit a differential graded algebra (DGA) structure on $\mathbb{F}$. We also show that under these assumptions the maximal shifts of the graded Betti numbers are subadditive.On the other hand, we present an example of a strongly generic monomial ideal which does not admit a DGA structure on its minimal free resolution. In particular, this demonstrates that the Hull resolution and the Lyubeznik resolution do not admit DGA structures in general.Finally, we show that it is enough to modify the last map of $\mathbb{F}$ to ensure that it admits the structure of a DG algebra.     

The colorful Helly theorem and colorful resolutions of ideals

We demonstrate that the topological Helly theorem and the algebraic Auslander-Buchsbaum theorem may be viewed as different versions of the same phenomenon. Using this correspondence we show how the colorful Helly theorem of I. Barany and its generalizations by G. Kalai and R. Meshulam translate to the algebraic side. Our main results are algebraic generalizations of these translations, which in particular give a syzygetic version of Helly’s theorem.     

Lower central series and free resolutions of hyperplane arrangements

If is the complement of a hyperplane arrangement, and is the cohomology ring of over a field of characteristic , then the ranks, , of the lower central series quotients of can be computed from the Betti numbers, , of the linear strand in a minimal free resolution of over . We use the Cartan-Eilenberg change of rings spectral sequence to relate these numbers to the graded Betti numbers, , of a minimal resolution of over the exterior algebra .

From this analysis, we recover a formula of Falk for , and obtain a new formula for . The exact sequence of low-degree terms in the spectral sequence allows us to answer a question of Falk on graphic arrangements, and also shows that for these arrangements, the algebra is Koszul if and only if the arrangement is supersolvable.

We also give combinatorial lower bounds on the Betti numbers, , of the linear strand of the free resolution of over ; if the lower bound is attained for , then it is attained for all . For such arrangements, we compute the entire linear strand of the resolution, and we prove that all components of the first resonance variety of are local. For graphic arrangements (which do not attain the lower bound, unless they have no braid subarrangements), we show that is determined by the number of triangles and subgraphs in the graph.

     

Cellular resolutions of ideals defined by nondegenerate simplicial homomorphisms

In this paper we introduce the class of ordered homomorphism ideals and prove that these ideals admit minimal cellular resolutions constructed as homomorphism complexes. Motivated by work of Dochtermann and Engström, we introduce the class of cointerval simplicial complexes and investigate their combinatorial and topological properties. As a concrete illustration of these structural results, we introduce and study nonnesting monomial ideals, an interesting family of combinatorially defined ideals.