Monomial and toric ideals associated to Ferrers graphs
Alberto Corso,Uwe Nagel +1 more
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In this article, it was shown that a Ferrers ideal has a 2-linear minimal free resolution, i.e., it defines a small subscheme of a bipartite graph.Abstract:
Each partition A = (λ 1 , λ 2 ,..., An) determines a so-called Ferrers tableau or, equivalently, a Ferrers bipartite graph. Its edge ideal, dubbed a Ferrers ideal, is a squarefree monomial ideal that is generated by quadrics. We show that such an ideal has a 2-linear minimal free resolution; i.e. it defines a small subscheme. In fact, we prove that this property characterizes Ferrers graphs among bipartite graphs. Furthermore, using a method of Bayer and Sturmfels, we provide an explicit description of the maps in its minimal free resolution. This is obtained by associating a suitable polyhedral cell complex to the ideal/graph. Along the way, we also determine the irredundant primary decomposition of any Ferrers ideal. We conclude our analysis by studying several features of toric rings of Ferrers graphs. In particular we recover/establish formulae for the Hilbert series, the Castelnuovo-Mumford regularity, and the multiplicity of these rings. While most of the previous works in this highly investigated area of research involve path counting arguments, we offer here a new and self-contained approach based on results from Gorenstein liaison theory.read more
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Edge Ideals: Algebraic and Combinatorial Properties
Susan Morey,Rafael H. Villarreal +1 more
TL;DR: A survey of algebraic and combinatorial properties of R/I(C) and C, respectively, can be found in this paper, where the authors give a criterion to estimate the regularity of the ideal of vertex covers of C. They also examine the associated primes of powers of edge ideals and show that these sets form an ascending chain.
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Monomial ideals, edge ideals of hypergraphs, and their graded Betti numbers
Huy Tài Hà,Adam Van Tuyl +1 more
TL;DR: In this paper, the problem of computing the graded Betti numbers of a simple graph H = G in terms of its sub-hypergraphs is studied, where H is defined as a triangulated hypergraph.
Journal ArticleDOI
Algebraic properties of edge ideals via combinatorial topology
TL;DR: Some basic notions from combinatorial topology are applied to establish various algebraic properties of edge ideals of graphs and more general Stanley-Reisner rings.
Journal ArticleDOI
Minimal generators of toric ideals of graphs
TL;DR: This work characterize in graph theoretical terms the primitive, the minimal, the indispensable and the fundamental binomials of the toric ideal I"G.
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Resolutions of square-free monomial ideals via facet ideals: a survey
Huy Tài Hà,Adam Van Tuyl +1 more
TL;DR: A survey of recent results on the minimal graded free resolution of a monomial ideal can be found in this article, where the point-of-view that the generators of monomial ideals correspond to the maximal faces (the facets) of a simplicial complex is discussed.
References
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Cohen-Macaulay rings
Winfried Bruns,H. Jürgen Herzog +1 more
TL;DR: In this article, the authors present a self-contained introduction to the homological and combinatorial aspects of the theory of Cohen-Macaulay rings, Gorenstein rings, local cohomology, and canonical modules.
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Combinatorial Commutative Algebra
Ezra Miller,Bernd Sturmfels +1 more
TL;DR: In this paper, the authors present a set of monomial ideals for three-dimensional staircases and cellular resolutions, including two-dimensional lattice ideals, and a threedimensional staircase with cellular resolutions.