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Let V be a vector space of dimension ℓ ≥ 1 over the field K. An arrangement A = { H _1, …, H _n} is a set of n ≥ 0 hyperplanes in V . In dimension 1, we consider n points in the real line ℝ or in the complex line ℂ. We shall see later that these seemingly innocent examples lead to interesting problems. In dimension 2, the Selberg arrangement of five lines is shown below. We shall use this arrangement to illustrate definitions and results in Section 1.11.

Let V be a vector space of dimension ℓ. Let A be an arrangement of n hyperplanes in V . Let L = L (A) be the set of nonempty intersections of elements of A. An element X ∈ L is called an edge of A.

Much of the algebraic combinatorics described in Chapter 1 was originally developed with topological applications in mind. We give a brief description of some of the main features of these applications.

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].

In this chapter we define the concept of a cellular resolution, which lays the ground for the application of discrete Morse theory in Commutative Algebra. Since the concept of a cellular resolution uses the concept of multigraded free resolutions and CW-complexes as an ingredient we define these concepts first. Indeed we have already given definitions of resolutions and simplicial complexes in Chapter 1 in a more informal way. In this chapter will be more rigorous and also add more details and examples. More precisely, Sections 2.1 - 2.4 give the definitions of free resolutions, CW-complexes, simplicial complexes, cellular free resolutions, monomial modules and co-Artinian monomial modules.

This chapter contains more facts about cellular resolutions and in particular many examples of cellular resolutions. The set of these examples is chosen with some personal bias from a big set of examples of cellular resolutions that have emerged over the last years. We try to be a bit more complete by covering in the exercises some of the examples that are left out.

This chapter contains a presentation of discrete Morse theory as developed by Robin Forman (see e.g. [20], [21]). This theory allows to combinatorially construct from a given (regular, finite) CW-complex a second CW-complex that is homotopy equivalent to the first but has fewer cells. As the upshot of this chapter we then show that one can use this theory in order to construct minimal free resolutions (see also [3]). Discrete Morse theory has found many more applications in Geometric Combinatorics and other fields of mathematics, we will not be able to speak about them. We refer the reader for example to [29] where most applications of discrete Morse theory to complexes of graphs are reviewed. There are even promising attempts to find real world applications of discrete Morse theory (see [31]) to image analysis.