Extended Rees algebras and mixed multiplicities (Q1109087)

Let \((R,m)\) be a local ring of positive dimension \(d\). Let \(I\) be an \(m\)-primary ideal and \(J\) an ideal of positive height with analytic spread \(a\). It is proved that the coefficient of \(r^{d-1-i}s i\) in the polynomial \(\ell (I\) rJ \(s/I^{r+1}J s)\) for large \(r\) and \(s\) is nonzero if and only if \(i=0,1,...,a-1\). A formula is developed for the multiplicity of the ideal \((t^{-1},I,Jt)\) in the extended Rees algebra \(R[Jt,t^{-1}]\) localized at \((t^{-1},m,Jt)\). A criterion is given for the Cohen-Macaulayness of \(R[It,t^{-1}]\). Using these results, the class of parameter ideals \(I\) in \(R\) is completely determined for \(R[It,t^{-1}]\) to be Cohen-Macaulay with minimal multiplicity at its maximal homogeneous ideal \((t^{-1},m,It)\).