Regularity of canonical and deficiency modules for monomial ideals (Q627401)

Let \(R=k[x_1,\dots,x_n]\) be a standard graded polynomial ring over a field \(k\) and let \(\mathfrak m=(x_1,\dots,x_n)\) be the homogeneous maximal ideal of \(R\). We consider a monomial ideal \(I\subset R\). In considering the vanishing degrees of graded local cohomology modules \(H^i_{\mathfrak m}(R/I)\), it is sometimes useful to consider the deficiency modules \(\text{Ext}^i_R(R/I,\omega_R)\) for \(i>n-\dim R/I\) or the canonical module \(\text{Ext}^{n-\dim R/I}_R(R/I,\omega_R)\), where \(\omega_R=R(-n)\) is the canonical module of \(R\), since they are linked by the local duality theorem of Grothendieck. The aim of the paper under review is to prove the following result \[ \text{reg}\text{Ext}^i_R(R/I,\omega_R)\leq\dim\text{Ext}^i_R(R/I,\omega_R)\qquad\text{for all }0\leq i\leq n. \] If \(I\subset R\) is a square-free case, this can be proved by the theory of square-free modules [\textit{K.~Yanagawa}, J. Algebra 225, No. 2, 630--645 (2000; Zbl 0981.13011)], but for the general case the authors use a multigraded filtration for \(\text{Ext}^i_R(R/I,\omega_R)\) called the Stanley filtration [\textit{D.~Maclagan} and \textit{G.~G.~Smoth}, J. Algebr. Geom. 14, No. 1, 137--164 (2005; Zbl 1070.14006)].