Algebraic Discrete Morse Theory and Applications to Commutative Algebra

Download or read book Algebraic Discrete Morse Theory and Applications to Commutative Algebra written by and published by . This book was released on 2005 with total page pages. Available in PDF, EPUB and Kindle. Book excerpt: 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/x1 ..., xn 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 pure.