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

TL;DR

This paper generalizes a theorem on joint reductions and Hilbert coefficients for complete ideals in dimension 3, using local cohomology of multi-Rees algebras to extend results from two to three dimensions.

Contribution

It provides a new formula for the third local cohomology module of extended Rees algebras in three dimensions, generalizing previous results on joint reductions and Hilbert coefficients.

Findings

Generalization of Rees's theorem to dimension 3

Formula for third local cohomology of multi-Rees algebras

Vanishing results for the second normal Hilbert coefficient

Abstract

In this paper, we obtain a generalization, in dimension $3$, of a theorem of David Rees about joint reductions of the bigraded filtration $\{ \overline{I^rJ^s}\}$ 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 $\{\overline{I^rJ^sK^t}\}$ with support in the ideal $(x_1t_1,x_2t_2,x_3t_3)$ where $(x_1,x_2,x_3)$ is a good joint reduction of $\{\overline{I^rJ^sK^t}\}.$