Polytopale Konstruktionen in der Algebra

Die zentralen Objekte dieser Arbeit sind einerseits Ideale in einem Polynomring mehrerer Veränderlicher und andererseits simpliziale Komplexe. Klassische Invarianten der Ideale sind die minimalen freien Auflösungen, deren Betti-Zahlen, ihre Hilbert-Reihe und die Krull-Dimension, die der simplizi...

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Bibliographic Details
Main Author: Soll, Daniel
Contributors: Welker, Volkmar (Prof. Dr.) (Thesis advisor)
Format: Dissertation
Published: Philipps-Universität Marburg 2006
Reine und Angewandte Mathematik
Online Access:PDF Full Text
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Table of Contents: For $n\geq 3$, let $\Omega_n$ be the set of line segments between the vertices of a convex $n$-gon. For $j\geq 2$, a $j$-crossing is a set of $j$ line segments pairwise intersecting in the relative interior of the $n$-gon. For $k\geq 1$, let $\Delta_{n,k}$ be the simplicial complex of (type-A) generalized triangulations, i.e. the simplicial complex of subsets of $\Omega_n$ not containing any $(k+1)$-crossing. The complex $\Delta_{n,k}$ has been the central object of numerous papers. Here we continue this work by considering the complex of type-B generalized triangulations. For this we identify line-segments in $\Omega_{2n}$ which can be transformed into each other by a $180^\circ$-rotation of the $2n$-gon. Let $\F_n$ be the set $\Omega_{2n}$ after identification, then the complex $\D_{n,k}$ of type-B generalized triangulations is the simplicial complex of subsets of $\F_n$ not containing any $(k+1)$-crossing in the above sense. For $k = 1$, we have that $\D_{n,1}$ is the simplicial complex of type-B triangulations of the $2n$-gon as defined in \cite{Si} and decomposes into a join of an $(n-1)$-simplex and the boundary of the $n$-dimensional cyclohedron. We demonstrate that $\D_{n,k}$ is a pure, $k(n-k)-1+kn$ dimensional complex that decomposes into a $kn-1$-simplex and a $k(n-k)-1$ dimensional homology sphere. For $k=n-2$ we show that this homology-sphere is in fact the boundary of a cyclic polytope. We provide a lower and an upper bound for the number of maximal faces of $\D_{n,k}$. On the algebraical side we give a term-order on the monomials in the variables $X_{ij}, 1\leq i,j\leq n$, such that the corresponding initial ideal of the determinantal ideal generated by the $(k+1)$ times $(k+1)$ minors of the generic $n \times n$ matrix contains the Stanley-Reisner ideal of $\D_{n,k}$. We show that the minors form a Gr\"obner-Basis whenever $k\in\{1,n-2,n-1\}$ thereby proving the equality of both ideals and the unimodality of the $h$-vector of the determinantal ideal in these cases. We conjecture this result to be true for all values of $k