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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フォーマット: | Dissertation |
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Philipps-Universität Marburg
2006
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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