Introduction to Binomial Ideals

Abstract

In this chapter we introduce the main topic of this book: binomials and binomial ideals. Special attention is given to toric ideals. These are binomial ideals arising from an integer matrix which represents the exponent vectors of the monomial generators of a toric ring. It will be shown that the toric ideal IA attached to the matrix A is graded if and only if A is a configuration matrix. Furthermore, it will be shown that an arbitrary binomial ideal is a toric ideal if and only if it is a prime ideal. Then we study the Grobner basis of a binomial ideal and show that its reduced Grobner basis consists of binomials. We introduce Graver bases and show that the reduced Grobner basis of a binomial ideal is contained in its Graver basis. Naturally attached to a lattice \(L\subset {\mathbb Z}^n\) (i.e. a subgroup of the abelian group \({\mathbb Z}^n\)) there is a binomial ideal IL, called the lattice ideal of L. It will be shown that the saturation of any binomial ideal is a lattice ideal, and that the lattice ideals are exactly those which are saturated. The ideal generated by the binomials corresponding to the basis vectors of a basis of the lattice L is called a lattice basis ideal. Its saturation is the lattice ideal IL. The chapter closes with an introduction to Lawrence ideals and to squarefree divisor complexes.