Hilbert's Syzygy Theorem for monomial ideals

TL;DR

This paper presents a new proof of Hilbert's Syzygy Theorem specifically for monomial ideals and establishes an upper bound on the projective dimension based on generator degrees.

Contribution

It provides a novel proof for monomial ideals and introduces a bound on projective dimension related to generator degrees in polynomial rings.

Findings

New proof of Hilbert's Syzygy Theorem for monomial ideals

Bound on projective dimension based on minimal generator degrees

Applicable to squarefree monomial ideals in polynomial rings

Abstract

We give a new proof of Hilbert's Syzygy Theorem for monomial ideals. In addition, we prove the following. If S=k[x_1,...,x_n] is a polynomial ring over a field, M is a squarefree monomial ideal in S, and each minimal generator of M has degree larger than i, then the projective dimension of S/M is at most n-i.