Hilbert's Syzygy Theorem for monomial ideals

Abstract

We give a new proof of Hilbert's Syzygy Theorem for monomial ideals. In addition, we prove the following. If S=k[x1,…,xn]S=k[x1,…,xn] is a polynomial ring over a field, MM is a squarefree monomial ideal in SS, and each minimal generator of MM has degree larger than ii, then pd(S/M)≤n−i\pd(S/M)≤n−i.