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=(I ₁,…,I _d ) 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 (x _{i j} ) 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.