Splittable ideals and the resolutions of monomial ideals

Abstract

We provide a new combinatorial approach to study the minimal free resolutions of edge ideals, that is, quadratic square-free monomial ideals. With this method we can recover most of the known results on resolutions of edge ideals with fuller generality, and at the same time, obtain new results. Past investigations on the resolutions of edge ideals usually reduced the problem to computing the dimensions of reduced homology or Koszul homology groups. Our approach circumvents the highly nontrivial problem of computing the dimensions of these groups and turns the problem into combinatorial questions about the associated simple graph. We also show that our technique extends successfully to the study of graded Betti numbers of arbitrary square-free monomial ideals viewed as facet ideals of simplicial complexes.

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

“…Subsequently, many people, including [1, 12, 13, 15-18, 27-30, 32], have been working on a program to build a dictionary between the algebraic properties of I(G) and combinatorial structure of G. Of particular relevance to this paper, the minimal graded resolution of I(G) was investigated in [5, 7, 9, 19-22, 26, 33] (see also [23] for a survey). In this paper, we extend some of these results to the hypergraph case, most notably, the results of [22], thereby extending our understanding of quadratic squarefree monomial ideals to arbitrary squarefree monomial ideals. At the same time, we also derive new results which, even when restricted to graphs, give new and interesting corollaries.…”

“…The starting point of this paper is to determine how the splitting technique used in [22] to study the resolution of edge ideals of graphs can be extended to hypergraphs. Recall that Eliahou and Kervaire [8] call a monomial ideal I splittable if I = J + K for two monomial ideas J and K such that the minimal generators of J, K, and J ∩ K satisfy a technical condition (see Definition 2.3 for the precise statement).…”

Splittings of monomial ideals

“…(Isolated vertices do not affect the Betti numbers of ( ), and if ∖ { } consists only of isolated vertices, the Betti numbers of ( ) are easy to compute since is a complete bipartite graph plus possibly some isolated vertices.) Using Corollary 2.7, we recover [9, Theorem 4.2], which was instrumental in [9] in unifying a number of previous works on resolutions of edge ideals.…”

“…Following [9], if is a vertex of that is not isolated and such that ∖ { } is not a graph of isolated vertices, we call a splitting vertex of . (Isolated vertices do not affect the Betti numbers of ( ), and if ∖ { } consists only of isolated vertices, the Betti numbers of ( ) are easy to compute since is a complete bipartite graph plus possibly some isolated vertices.)…”

A class of hypergraphs that generalizes chordal graphs

“…After that, hypergraph algebras have been widely studied. See for instance [5], [7], [10], [11], [12], [13], [14], [16], [19]. In [10], the authors use certain connectedness properties to determine a class of hypergraphs such that the hypergraph algebras have linear resolutions.…”