Minimal Free Resolutions and Monomial Ideals of Projective Dimension <=1

Abstract

Let R= k[x1,x2,...,xn] be the polynomial ring in n variables and I anideal in R. We first define the notions of minimal free resolutions of algebras R/I and multigraded minimal resolutions of monomial ideals I. We then discuss the following established result in [6]: projdim (I)≤1⇐⇒a graph tree supports the minimal free resolution of R/I.