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

In this paper, we obtain a generalization, in dimension $3$, of a theorem of David Rees about joint reductions of the bigraded filtration ${ \overline{I^rJ^s}}$ofcomplete${\mathfrak m}$-primary ideals and vanishing of the second normal Hilbert coefficient ${\bar{e}}_2(IJ)$ where $R$ is a two-dimensional Cohen-Macaulay analytically unramified local ring with maximal ideal $\mathfrak{m}\mathrm{.}$ 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 $\{\overline{I^r{J}^s{K}^t}\}\mathrm{.}$