Arithmetical rank of the cyclic and bicyclic graphs. (English) Zbl 1242.13004
The aim of this article is to study the arithmetical rank of a monomial ideal which is the edge ideal of a cycle or a bicycle graph, a bycicle graph is the union of two cycles with only one common vertex. More precisely, the edge ideal \(I\) of a cycle is generated by the monomials \(x_1x_2, x_2x_3,\dots,x_nx_1\) on the polynomial ring in \(n\) variables \(S\). Recall that the arithmetical rank of \(I\) (ara\((I)\)) is the minimal number of generators of \(I\) up to radical, and there is not a systematic way to compute this invariant. The main result of the paper under review is that ara\((I)\) coincides with the projective dimension of the ring \(S/I\), which was previously computed by Jacques in his thesis work.
Reviewer: Marcel Morales (Saint-Martin-d’Hères)
MSC:
13A15 | Ideals and multiplicative ideal theory in commutative rings |
13F55 | Commutative rings defined by monomial ideals; Stanley-Reisner face rings; simplicial complexes |
13D02 | Syzygies, resolutions, complexes and commutative rings |
05C38 | Paths and cycles |
Keywords:
arithmetical rank; projective dimension; edge ideals; set-theoretic complete intersection idealsReferences:
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