Algebraic Discrete Morse Theory and Applications to Commutative Algebra

In this PhD thesis we generalize Forman's Discrete Morse Theory to an algebraic version in order to calculate the homology of arbitrary algebraic chain complexes of free R-modules. We call this generalization Algebraic Discrete Morse Theory. The idea is to construct from a given chain complex of free R-modules an homotopic equivalent chain complex with fewer copies of R. In order to do so we associate to a given complex a directed graph, which represents the complex completely. In this graph we look for so called acyclic matchings, which are as large as possible and then construct smaller graphs, which can be interpreted as a chain complex. We prove that this chain complex has same the homology as the original one. The main part of this thesis consists of applications of our theory to commutative algebra. Using Algebraic Discrete Morse Theory, we construct minimal multigraded free resolutions for several classes of A-modules, where A is a polynomial ring (not necessarily commutative), divided by an arbitrary ideal. If one knows the minimal resolution one can study the multigraded Poincare-Betti series of the module. We use Algebraic Discrete Morse Theory in order to formulate a conjecture about the vectorspace structure of the minimal multigraded free resolution of the residue class field A/ over a monomial ring A=k[x1,...,xn]/I. This conjecture implies an explicit form of the multigraded Poincare-Betti series in this situation, which is a precise formulation of a vague conjecture about the series made by Charalambous and Reeves. We prove our conjecture for several classes of algebras A. Knowing the Poincare-Betti series we can find new combinatorial criteria for a monomial ring to be Golod. For example we prove that - in case our conjecture is true - Golodness is equivalent to the fact that the first Massey operation on the Koszul homology vanishes. Compared to the original definition of Golodness, this is a rather easy condition. We develop further purely combinatorial conditions for Golodness of k[x1,...,xn]/I, which depend only on the minimal monomial generating system of the ideal I. The next field of applications studied by us is the construction of minimal free resolutions of p-Borel-fixed ideals over the polynomial ring. Here we construct an algorithm, which produces a minimal (even cellular) free resolution for a large class of p-Borel-fixed ideals. In particular we can give formulas for the multigraded Poincare-Betti series and for the regularity of this class of p-Borel fixed ideals. Our formulas generalize known results. Finally we study two related problems in algebraic combinatorics. The first problem concerns the homology of nilpotent Lie-algebras and the second studied problem is the Neggers-Stanley-Conjecture about real-rootness and unimodality of a special class of polynomials. For both problems we present some results and new ideas for the general approach.