Local Cohomology of Multi-Rees Algebras with Applications to joint Reductions and Complete Ideals

Abstract

Let \((R,\mathfrak {m})\) be a Cohen-Macaulay local ring of dimension d and I=(I1,…,Id) be \(\mathfrak {m}-\)primary ideals in R. We prove that \({\lambda }_{R}([H^{d}_{(x_{ii}t_{i}:1\leq i \leq d)}({\mathcal {R}}^{\prime }({\mathcal {F}})]_{\mathbf {n}})\)\(< \infty \), for all \(\mathbf {n} \in \mathbb {N}^{d}\), where \({\mathcal {F}}=\{{\mathcal {F}}(\mathbf {n}):\mathbf {n}\in \mathbb {Z}^{d}\}\) is an I−admissible filtration and (xij) is a strict complete reduction of \({\mathcal {F}}\) and \({\mathcal {R}}^{\prime }({\mathcal {F}})\) is the extended multi-Rees algebra of \({\mathcal {F}}.\) As a consequence, we prove that the normal joint reduction number of I,J,K is zero in an analytically unramified Cohen-Macaulay local ring of dimension 3 if and only if \(\overline {e}_{3}(IJK)-[\overline {e}_{3}(IJ)+\overline {e}_{3}(IK)\)\(+\overline {e}_{3}(JK)]+\overline {e}_{3}(I)+ \overline {e}_{3}(J)+\overline {e}_{3}(K)=0\). This generalizes a theorem of Rees on joint reduction number zero in dimension 2. We apply this theorem to generalize a theorem of M. A. Vitulli in dimension 3.