Splittable ideals and the resolutions of monomial ideals

By Huy Tài Hà, Adam Van Tuyl. 2005

100 citations recorded by the source corpus.

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.

Subfields: Algebra and Number Theory, Computational Theory and Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/4e2451e92b18fcdcab0aca1bde55e50d7cf4751c

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Algebra and Number Theory

In Computational Theory and Mathematics

In Geometry and Topology

Citations in the corpus

Bounds on the regularity and projective dimension of ideals associated to graphs (2011, background): “It should be noted that the projective dimension and regularity of Stanley-Reisner ideals, or more specifically edge ideals, have attracted a lot of attention; see [3, 10, 11, 12, 16, 21, 23] and references given therein.”
Regularity of Squarefree Monomial Ideals (2013, background): “The following result was originally stated and proved by Wegner [50] using topological language, and re-stated in terms of monomial ideals by Fröberg [22] (see also [17, 26]).”
A Beginner’s Guide to Edge and Cover Ideals (2013, methodology): “Hà and myself [34] introduced what we called edge splittings, i.”
A CLASS OF HYPERGRAPHS THAT GENERALIZES CHORDAL GRAPHS (2008, background): “If H is a simple hypergraph, the complex (H ) = {F ⊆ [n]; E ⊆ F, ∀E ∈ E (H )}. is called the independence complex of H .”
Powers of edge ideals of regularity three bipartite graphs (2014, background): “Francisco, Ha and Van Tuyl [4] showed that if any power of an edge ideal has linear minimal free resolution, then the complement of the corresponding graph has no induced four cycles, which is equivalent to having a linear presentation due to [6].”
Regularity of powers of forests and cycles (2014, methodology): “[18, 13, 21]) imply that if the complement graph of G contains no induced cycles of length ≥ 4 then b = 0 and s0 = 1.”
The regularity of powers of edge ideals (2014, background): “Bounds on the regularity of edge ideals have been studied by a number of researchers (see [1], [2], [3], [4], [5], [6], [7],[8] [9], [11],[12],[13]).”