Stanley-Reisner rings and the radicals of lattice ideals

2003/10/20 by Anargyros Katsabekis, Katsabekis, Anargyros, Marcel Morales +3
Mathematics · #13F55 #14M25 #Algebraic Geometry (math.AG) #Commutative Algebra (math.AC) #FOS: Mathematics #math.AC #math.AG #msc:13F55 #msc:14M25

paper · pdf · doi:10.48550/arxiv.math/0310313

arxiv created 2003/10/20 · arxiv updated 2009/12/01

Abstract

In this article we associate to every lattice ideal I_L,ρ⊂ K[x₁,..., x_m] a cone σ and a graph G_σ with vertices the minimal generators of the Stanley-Reisner ideal of σ. To every polynomial F we assign a subgraph G_σ(F) of the graph G_σ. Every expression of the radical of I_L,ρ, as a radical of an ideal generated by some polynomials F₁,..., F_s gives a spanning subgraph of G_σ, the ∪_i=1^s G_σ(F_i). This result provides a lower bound for the minimal number of generators of I_L,ρ and therefore improves the generalized Krull's principal ideal theorem for lattice ideals. But mainly it provides lower bounds for the binomial arithmetical rank and the A-homogeneous arithmetical rank of a lattice ideal. Finally we show, by a family of examples, that the bounds given are sharp.

Related

• The *-core of the graded maximal ideal in a Stanley-Reisner ring
• Binomial generation of the radical of a lattice ideal
• The vertex ideal of a lattice
• Arithmetical ranks of Stanley-Reisner ideals of simplicial complexes with a cone
• Almost complete intersections and Stanley's conjecture