set-theoretic complete intersection ideals
Arithmetical rank of the cyclic and bicyclic graphs
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.
scientific article; zbMATH DE number 6037686