Resolutions of Monomial Ideals Via Quasi-Trees

Abstract

We examine ways in which simplicial complexes can be used for describing, classifying, and studying multigraded free resolutions of monomial ideals. By using homgenizations of frames and dehomogenizations of resolutions we can, under appropriate circumstances, describe the structure of a resolution of a monomial ideal by a simiplicial complex. We discuss the successes and failures of this approach. We finish by applying the tools we have presented to quasi-trees, providing a new proof to a theorem of Herzog, Hibi, and Zheng which classifies monomial ideals with minimal projective dimension.