Multiplicity and Hilbert-Kunz Multiplicity of Monoid Rings

Abstract

In this paper, we will give a method to compute the multiplicity and the Hilbert-Kunz multiplicity of monoid rings. The multiplicity and the Hilbert-Kunz multiplicity are fundamental invariants of rings. For example, the multiplicity (resp. the Hilbert-Kunz multiplicity) of a regular local ring equals to one. Monoid rings are defined by lattice ideals, which are binomial ideals I in a polynomial ring R over a field such that any monomial is a non zero divisor on R/I. Affine semigrouprings are monoidrings. Hencewe want to extendthe thoery of affine semigroup rings to that of monoid rings. 1. Main Result. Let N> 0b e an integer andZ the ring of integers. For α ∈ Z N , we denote the i-th entry of α by αi .W e say α> 0i fα � 0a ndαi ≥ 0 for each i .A nd α>α � if α − α � > 0. Let R = k[X1, ··· ,X N ] be a polynomial ring over a field k .F or α> 0, we simply write X α in place of N=1 X αi i . For a positive submodule V of Z N of rank r, we define an ideal I( V )of R ,w hich is generatedby all binomials X α −X β with α−β ∈ V (we say that V is positiveif it is contained in the kernel of a map Z N → Z which is defined by positive integers). Put d = N − r .T hen R/I (V ) is naturally a Z d -graded ring, which is called a monoid ring. Further, there is a