Betti numbers and regularity of projective monomial curves
In this thesis we describe how the balancing of the $\operatorname{Tor}$ functor can be used to compute the minimal free resolution of a graded module $M$ over the polynomial ring $B=\mathbb{K}[X_0,\dots,X_m]$ ($\mathbb{K}$ a field $X_i$'s indeterminates). Using a correspondence due to R. Stan...
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| Format: | Others |
| Language: | en en |
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2008
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| Online Access: | http://hdl.handle.net/1974/1474 |
| Summary: | In this thesis we describe how the balancing of the $\operatorname{Tor}$ functor can be used to compute the minimal free resolution of a graded module $M$ over the polynomial ring $B=\mathbb{K}[X_0,\dots,X_m]$ ($\mathbb{K}$ a field $X_i$'s indeterminates). Using a correspondence due to R. Stanley and M. Hochster, we explicitly show how this approach can be used in the case when $M=\mathbb{K}[S]$,
the semigroup ring of a subsemigroup $S\subseteq \mathbb{N}^l$ (containing $0$) over $\mathbb{K}$ and when $M$ is a monomial ideal of $B$.
We also study the class of affine semigroup rings for which $\mathbb{K}[S]\cong B/\mathfrak{p}$ is the homogeneous coordinate ring of a monomial curve in $\mathbb{P}^n_{\mathbb{K}}$. We use easily computable combinatorial and arithmetic properties of $S$ to define a notion which we call stabilization. We provide a direct proof showing how stabilization gives a bound on the $\mathbb{N}$-graded degree of minimal generators of $\mathfrak{p}$ and also show that it is related to the regularity of $\mathfrak{p}$. Moreover, we partition the above mentioned class into three cases and show that this partitioning is reflected in how the regularity is attained. An interesting consequence is that the regularity of $\mathfrak{p}$ can be effectively computed by elementary means. === Thesis (Master, Mathematics & Statistics) -- Queen's University, 2008-09-24 09:49:35.462 |
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