Arithmetical rank of Gorenstein squarefree monomial ideals of height three
Let \(S\) be a polynomial ring over a field \(K\) and \(I\) a monomial ideal of \(S\). We say that the elements \(u_1,\ldots,u_r\in S\) generate \(I\) up to radical if \(\sqrt{(u_1,\ldots,u_r)}=\sqrt{I}\). The minimal number \(r\) with this property is called the arithmetical rank of \(I\), denoted by \(\mathrm{ara} I\). By Krull's principal ideal theorem, we have \(\mathrm{height} I\leq \mathrm{ara} I\). When the equality holds, we say that \(I\) is a set-theoretic complete intersection. It is known by \textit{G. Lyubeznik} [Complete intersections, Lect. 1st Sess. C.I.M.E., Acireale/Italy 1983, Lect. Notes Math. 1092, 214--220 (1984; Zbl 0578.13012)] that if \(I\) is a squarefree monomial ideal, then \(\mathrm{pd}_S S/I\leq \mathrm{ara} I\), where \(\mathrm{pd}_S S/I\) is the projective dimension of \(S/I\). Thus for a squarefree monomial ideal \(I\) we have that \(\mathrm{height} I\leq \mathrm{pd}_S S/I\leq \mathrm{ara} I\). It is natural to ask when \(\mathrm{ara} I = \mathrm{pd}_S S/I\) for a squarefree monomial ideal \(I\). In particular, which squarefree monomial ideal \(I\) is a set-theoretic complete intersection? Many authors investigated this question. \textit{K. Kimura} [J. Commut. Algebra 3, No. 1, 31--46 (2011; Zbl 1237.13043)] showed that a Cohen-Macaulay squarefree monomial ideal of height 2 is a set-theoretic complete intersection. However \textit{Z. Yan} [J. Pure Appl. Algebra 146, No. 3, 305--318 (2000; Zbl 1005.14007)] found a Cohen-Macaulay squarefree monomial ideal \(I\) of height 3 (when \(\mathrm{char} K\neq 2\)) which is not set-theoretic complete intersection. In the paper under review, the authors prove that a Gorenstein squarefree monomial ideal of height 3 is a set-theoretic complete intersection.
scientific article; zbMATH DE number 6370241
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