Introduction

Abstract

These lecture notes present topics from Algebraic Combinatorics that lie on the borderline to Algebraic Topology and Commutative Algebra. In particular, we will present and review combinatorial and geometric methods for studying minimal free resolutions of ideals in polynomial rings. Before we give the exposition of the topics we will spend time in order to outline the basic mathematical theory behind. Thus these notes will also include definitions and examples for CW-complexes and free resolutions. All this basic material is geared towards the applications given in the later sections and is therefore not presented in utmost generality. A comprehensive exposition of the interaction between Combinatorics and Commutative Algebra and the history of this interaction can be found in the books by Miller and Sturmfels [35] and Stanley [57].