Local cohomology of bigraded Rees algebras and normal Hilbert coefficients

Abstract

Let (R,m) be an analytically unramified Cohen–Macaulay local ring of dimension 2 with infinite residue field and I¯ be the integral closure of an ideal I in R. Necessary and sufficient conditions are given for Ir+1Js+1¯=aIrJs+1¯+bIr+1Js¯ to hold for all r⩾r0 and s⩾s0 in terms of vanishing of [H(at1,bt2)2(R′¯(I,J))](r0,s0), where a∈I,b∈J is a good joint reduction of the filtration {IrJs¯}. This is used to derive a theorem due to Rees on normal joint reduction number zero. The vanishing of e¯2(IJ) is shown to be equivalent to Cohen–Macaulayness of R¯(I,J).