Castelnuovo-Mumford regularity and projective dimension of a squarefree monomial ideal
Let \(S = K[x_1, x_2,\dots, x_n]\) be the polynomial ring in \(n\) variables over a field \(K\), and \(I\) be an ideal of \(S\). The (Castelnuovo-Mumford) regularity of an ideal \(I\), denoted by \(\mathrm{reg}(I)\), is defined to be the minimal number \(r\) such that the \(i\)-th syzygy module of \(I\) is generated by elements of degree \(\leq i+r\) for all \(i\geq 0\). \textit{Lê Tuân Hoa} and \textit{Ngô Viêt Trung} [Math. Z. 229, No. 3, 519--537 (1998; Zbl 0931.13015)] proved that the regularity conjecture [\textit{D. Eisenbud} and \textit{S. Goto}, J. Algebra 88, 89--133 (1984; Zbl 0531.13015)] is true if \(I\) is a squarefree monomial ideal, that is: \[ \mathrm{reg}(I)\leq \deg(S/I) -\mathrm{codim}(S/I) + 1. \] Let \(I\) be a squarefree monomial ideal minimally generated by the monomials \(u_1, u_2,\dots, u_m\). The authors of this paper shows that the regularity of \(I\) can be bounded by \(\mathrm{bigsize}_S(I) + 1 \), where the bigsize of a monomial ideal was used firstly by \textit{D. Popescu} [Commun. Algebra 41, No. 11, 4351--4362 (2013; Zbl 1327.13044)] to consider the Stanley Conjecture. Also, as a corollary, they show that projective dimension of a squarefree monomial ideal \(I\) is bounded by the number \(w\), where \(w\) is the smallest number \(t\) with the property that for all integers \(1\leq i_1 < i_2 <\dots < i_t\leq m\) such that \[ \mathrm{lcm}(u_{i_{1}}, u_{i_{2}},\dots , u_{i_{t}}) =\mathrm{lcm}(u_1, u_2,\dots , u_m). \]
scientific article; zbMATH DE number 6909655
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