An Extension of Rees' Theorem and Two Interpretations of a Vector in the Joint Reduction Lattice

In D. Rees, Hilbert functions and pseudo-rational local rings of dimension two, J. London Math. Soc. (2) 24 (1981) 467–479, Rees gave a characterization for the normal joint reduction number zero of two Formula: see text-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 Formula: see text-primary ideals Formula: see text and Formula: see text in a Cohen–Macaulay local ring of dimension two. Following Rees’ approach, we define the modified Koszul homology modules Formula: see text for a joint reduction Formula: see text of Formula: see text and Formula: see text. Under the additional assumption that the associated graded rings of Formula: see text and Formula: see text have positive depth, we obtain a characterization of the joint reduction number zero of Formula: see text and Formula: see text in terms of the vanishing of the module Formula: see text, 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 Formula: see text for any Formula: see text. This gives a characterization for a vector Formula: see text to be in the joint reduction lattice of Formula: see text and Formula: see text. 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 Formula: see text to be in the joint reduction lattice and also extends a recent result of Masuti and Verma in Local cohomology of bigraded Rees algebras and normal Hilbert coefficients, J. Pure Appl. Algebra 218(5) (2014) 904–918 for ordinary powers of ideals.