Algebraic Discrete Morse Theory and Applications to Commutative Algebra
In dieser Doktorarbeit verallgemeinern wir die Diskrete Morse Theorie von Forman auf eine algebraische Version, die wir Algebraische Diskrete Morse Theorie nennen. Ziel der Theorie ist es zu einem gegebenem algebraischem Kettenkomplex freier RModuln einen Homotopieäquivalenten Kettenkomplex...
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Format:  Dissertation 
Language:  English 
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PhilippsUniversität Marburg
2005
Reine und Angewandte Mathematik 
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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 Rmodules. We call this generalization Algebraic Discrete Morse Theory. The idea is to construct from a given chain complex of free Rmodules 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 Amodules, 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 PoincareBetti 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 PoincareBetti 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 PoincareBetti 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 pBorelfixed ideals over the polynomial ring. Here we construct an algorithm, which produces a minimal (even cellular) free resolution for a large class of pBorelfixed ideals. In particular we can give formulas for the multigraded PoincareBetti series and for the regularity of this class of pBorel fixed ideals. Our formulas generalize known results. Finally we study two related problems in algebraic combinatorics. The first problem concerns the homology of nilpotent Liealgebras and the second studied problem is the NeggersStanleyConjecture about realrootness and unimodality of a special class of polynomials. For both problems we present some results and new ideas for the general approach.