Minimal Reductions and Cores of Edge Ideals

Details Minimal Reductions and Cores of Edge Ideals Minimal Reductions and Cores of Edge Ideals

2010-12-28

arxiv.org/abs/1012.5746v1

Mathematics

Commutative Algebra

28 pages, submitted for publication

Scientific paper

We study minimal reductions of edge ideals of graphs containing a unique even cycle, and determine restrictions on the coefficients of the generators of these minimal reductions. We focus our attention on two special subclasses of edge ideals; the first is edge ideals of even cycles and the second is edge ideals of even cycles with an arbitrary number of whiskers. We prove that $\rm{core}(I)=\mathfrak{m} I$, where $I$ is the edge ideal in the corresponding localized polynomial ring and $\mathfrak{m}$ is the maximal ideal of this ring. Moreover, we show that the core is obtained as a finite intersection of homogeneous minimal reductions in the case of even cycles. The formula for the core does not hold in general for the edge ideal of any graph and we provide a counterexample. In particular, we show in this example that the core is not obtained as a finite intersection of general minimal reductions.

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