Splittings of monomial ideals

By Christopher A. Francisco, Huy Tài Hà, Adam Van Tuyl. 2008

118 citations recorded by the source corpus.

We provide some new conditions under which the graded Betti numbers of a monomial ideal can be computed in terms of the graded Betti numbers of smaller ideals, thus complementing Eliahou and Kervaire's splitting approach. As applications, we show that edge ideals of graphs are splittable, and we provide an iterative method for computing the Betti numbers of the cover ideals of Cohen-Macaulay bipartite graphs. Finally, we consider the frequency with which one can find particular splittings of monomial ideals and raise questions about ideals whose resolutions are characteristic-dependent.

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

View Paper: https://www.semanticscholar.org/paper/e36c09bcb94d996de86197621476aa3b2b059966

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In Algebra and Number Theory

In Computational Theory and Mathematics

In Geometry and Topology

Citations in the corpus

On vertex decomposable simplicial complexes and their Alexander duals (2013, background): “Francisco, Ha and Van Tuyl in [10] defined the concept of x-partition for a monomial ideal, when x ∈ X .”
Sequentially Cohen–Macaulay bipartite graphs: vertex decomposability and regularity (2009, result): “As corollaries, we can recover a result of Zheng [25], who first linked the regularity to the invariant a(G), and a result of the author with Francisco and Hà [8].”
Matchings, coverings, and Castelnuovo-Mumford regularity (2010, background): “Several recent papers [13, 18, 28, 29, 31, 33, 36] have related the Castelnuovo-Mumford regularity of the edge ideal I(G) with various invariants of the graph G.”
A Beginner’s Guide to Edge and Cover Ideals (2013, background): “Further properties have been developed by He and Van Tuyl [35], Bouchat, Hà, and O’Keefe [3], and Alilooee and Faridi [1].”
Powers of sums and their homological invariants (2016, methodology): “Our method makes substantial use of Betti splittings, first introduced by Francisco, Hà and Van Tuyl [23] in their study of free resolutions of monomial ideals.”