An extension of Rees theorem and two interpretations of a vector in the joint reduction lattice

TL;DR

This paper extends Rees' theorem to characterize joint reduction numbers for pairs of m-primary ideals in a Cohen-Macaulay local ring of dimension two, using homological and cohomological methods.

Contribution

It generalizes Rees' theorem to ordinary powers of ideals and introduces the joint reduction lattice with cohomological interpretations.

Findings

Characterization of joint reduction number zero via vanishing modules

Introduction of the joint reduction lattice for pairs of ideals

Extension of results to bigraded Hilbert coefficients and local cohomology

Abstract

In \cite{rees} Rees gave a characterization for the normal joint reduction number zero of two $\m$-primary ideals in an analytically unramified Cohen-Macaulay local ring of dimension two. Rees' result is a generalization of Zariski's product theorem for complete ideals in a regular local ring of dimension two. The aim of this paper is to extend Rees' theorem for the ordinary powers of $\m$-primary ideals $I$ and $J$ in a Cohen-Macaulay local ring of dimension two. Following Rees' approach, we define the modified Koszul homology modules $M^1_{r,s}(a^k,b^k)$ for a joint reduction $(a,b)$ of $I$ and $J$. Under the additional assumption that the associated graded rings of $I$ and $J$ have positive depth, we obtain a characterization of the joint reduction number zero of $I$ and $J$ in terms of the vanishing of the module $M^1_{0,0}(a,b)$, as well as in terms of the Hilbert coefficients and the bigraded Hilbert coefficients. More generally, we introduce the joint reduction lattice and study the vanishing of $M^1_{r,s}(a,b)$ for any $r, s \geq 0$. This gives a characterization for a vector $(r,s)$ to be in the joint reduction lattice of $I$ and $J$. We also give a cohomological interpretation of these theorems by investigating the local cohomology modules of the bigraded extended Rees algebra. This gives another characterization for a vector $(r,s)$ to be in the joint reduction lattice and also extends a recent result of Masuti and Verma in \cite{masuti-verma} for ordinary powers of ideals.