Arithmetical rank of a squarefree monomial ideal whose Alexander dual is of deviation two
Let \(S\) be a standard graded polynomial ring over a field. Let \(I\) be a homogeneous ideal of \(S\). The arithmetical rank of \(I\), denoted \(\text{ara} I\), is the minimal number \(r\) such that \(I\) can be generated by \(r\) forms up to radical. \textit{G. Lyubeznik} [Lect. Notes Math. 1092, 214--220 (1984; Zbl 0578.13012)] showed that for a monomial ideal \(I\), the arithmetical rank is bounded below by the projective dimension of \(S/I\). There is no known criteria for when the equality holds. It is expected that the equality holds when the arithmetic degree of \(I\), namely the number of prime components of \(I\), is small. The main result of the paper (Theorem 1.1) is that the equality holds when \(I\) has a linear resolution and the difference between the arithmetic degree and the initial degree is (at most) \(2\). Using Alexander's duality, the main tool is the classification of Cohen-Macaulay squarefree monomial ideals of deviation \(2\) via hypergraphs given by Kimura, Terai, and Yoshida [\textit{K. Kimura} et al., Contemp. Math. 502, 73--112 (2009; Zbl 1194.13025)].
scientific article; zbMATH DE number 6496838
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