Local Cohomology of Bigraded Rees Algebras and Normal Hilbert Coefficients

TL;DR

This paper investigates the conditions under which certain equalities hold for the integral closures of products of ideals in a Cohen-Macaulay local ring, linking local cohomology vanishing to properties of the Rees algebra.

Contribution

It provides necessary and sufficient conditions for integral closure equalities using local cohomology vanishing, extending Rees's theorem on normal joint reduction number zero.

Findings

Vanishing of specific local cohomology modules characterizes integral closure equalities.

Normal joint reduction number zero is characterized via local cohomology conditions.

Cohen-Macaulayness of the Rees algebra is equivalent to the vanishing of a certain Hilbert coefficient.

Abstract

Let $(R,\m)$ be an analytically unramified Cohen-Macaulay local ring of dimension 2 with infinite residue field and $\ov{I}$ be the integral closure of an ideal $I$ in $R$. Necessary and sufficient conditions are given for $\ov{I^{r+1}J^{s+1}}=a\ov{I^rJ^{s+1}}+b\ov{I^{r+1}J^s}$ to hold ${for all}r \geq r_0{and}s \geq s_0$ in terms of vanishing of $[H^2_{(at_1,bt_2)}(\ov{\mathcal{R}^\prime}(I,J))]_{(r_0,s_0)}$, where $a \in I,b \in J$ is a good joint reduction of the filtration $\{\ov{I^rJ^s}\}.$ This is used to derive a theorem due to Rees on normal joint reduction number zero. The vanishing of $\ov{e}_2(IJ)$ is shown to be equivalent to Cohen-Macaulayness of $\ov{\mathcal{R}}(I,J)$.