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:  Doctoral Thesis 
Language:  English 
Published: 
PhilippsUniversität Marburg
2005

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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/