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

Abstract

In this paper, we obtain a generalization, in dimension $3$, of a theorem of David Rees about joint reductions of the bigraded filtration $\{\overline{I^r{J}^s}\}$ of complete $\mathfrak{m}$-primary ideals and vanishing of the second normal Hilbert coefficient ${\overline{e}}_2(IJ)$ where $R$ is a two-dimensional Cohen-Macaulay analytically unramified local ring with maximal ideal $\mathfrak{m}.$ This generalization is obtained as a consequence of a formula for the third local cohomology module of the extended Rees algebras of the ${\mathbb{Z}}^3$-graded filtration $\{\overline{I^r{J}^s{K}^t}\}$ with support in the ideal $(x_1{t}_1,x_2{t}_2,x_3{t}_3)$ where $(x_1,x_2,x_3)$ is a good joint reduction of $\{\overline{I^r{J}^s{K}^t}\}.$

Dedicated to Ngo Viet Trung on his 60th Birthday