1. INTRODUCTION 3 face incidences, whereas algebraic discrete Morse theory still can be applied to its cellular chain complex. Often algebraic discrete Morse theory allows statements on minimal resolutions, Betti-numbers, regularity, etc., without explicitly constructing the minimal free resolution. For example, in [31] the first author constructs a sequence of acyclic matchings on the Taylor resolution of any monomial ideal a in the polynomial ring S such that the resulting Morse complex is a minimal free resolution. Clearly, we cannot explicitly construct this resolution, but we retrieve information about its structure, which in special cases allows the calculation of the Betti-numbers of k viewed as an S/a-module. In this volume we present applications of algebraic discrete Morse theory to free resolutions of R-modules, where R is a quotient a the commutative or non- commutative polynomial ring in a finite number of variables. We proceed as follows: In Chapter 2 we describe how algebraic discrete Morse theory works and provide the basic constructions. We do not provide proofs in that chapter. Even though our version of algebraic discrete Morse theory is almost identical with the one developed by Sk¨ oldberg [44] we do give proofs in Appendix B since our theory is formulated in a slightly more general framework and also contains a description of the differential of the resulting complex. The remaining chapters of the manuscript are devoted to applications of algebraic discrete Morse theory which comprise the central results of our work. In Chapter 3 we consider resolutions of the field k over a quotient A = S/a of the commutative polynomial ring S = k[x1, . . . , xn] in n variables by an ideal a. We construct a free resolution of k as an A-module which can be seen as a generalization of the Anick resolution to the commutative case. We show that our resolution is minimal if a admits a quadratic Gr¨ obner basis. Similarly, we give an explicit description of the minimal free resolution of k, if there is a term order for which the the initial ideal of a is a complete intersection. Chapter 4 considers the same situation in the non-commutative case. We ap- ply algebraic discrete Morse theory in order to obtain the Anick resolution of the residue field k over A = k x1, . . . , xn /a from the normalized Bar resolution, where k x1, . . . , xn is the polynomial ring in n non-commuting indeterminates, and a is a two-sided ideal. This result has also been obtained by Sk¨oldberg [44]. In addition we give a description of the differential. Using this description we get, in addition to [44], conditions on Gr¨ obner bases, which imply minimality of the resolution. For example we prove the minimality of the resolution when a is monomial or the Gr¨ obner basis consists of homogeneous polynomials which all have the same degree. The first case provides an alternative proof of a result by Anick [1]. In the two cases, it follows that the Poincar´ e-Betti series is rational. This is well known when the ideal is monomial [4]. In particular, it follows that the Hilbert series of A/a is rational if a admits a quadratic Gr¨ obner basis. Also, this fact is well known (see [41] or [3]). In the case when A/a k[Λ] is the (commutative) semigroup ring of an affine semigroup Λ and a admits a quadratic Gr¨ obner basis, then we derive a conjecture by Sturmfels [47] from our construction. Sturmfels had described for this type of ideals a complex of free A-modules which he conjectured to define a minimal free resolution of k as an A-module. Now it is easily seen that in this
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