Specializations of multigradings and the arithmetical rank of lattice ideals

Details Specializations of multigradings and the arithmetical rank of lattice ideals Specializations of multigradings and the arithmetical rank of lattice ideals

2008-11-24

arxiv.org/abs/0811.3830v2

Mathematics

Commutative Algebra

14 pages

Scientific paper

In this article we study specializations of multigradings and apply them to the problem of the computation of the arithmetical rank of a lattice ideal $I_{L_{\mathcal{G}}} \subset K[x_{1},...,x_{n}]$. The arithmetical rank of $I_{L_{\mathcal{G}}}$ equals the $\mathcal{F}$-homogeneous arithmetical rank of $I_{L_{\mathcal{G}}}$, for an appropriate specialization $\mathcal{F}$ of $\mathcal{G}$. To the lattice ideal $I_{L_{\mathcal{G}}}$ and every specialization $\mathcal{F}$ of $\mathcal{G}$ we associate a simplicial complex. We prove that combinatorial invariants of the simplicial complex provide lower bounds for the $\mathcal{F}$-homogeneous arithmetical rank of $I_{L_{\mathcal{G}}}$.

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