Extended Rees Algebras and Mixed Multiplicities

TLDR: In this article, the authors studied mixed multiplicities of ideals and used them to calculate multiplicity of certain homogeneous ideals of extended Rees algebras and to give a complete characterization of those parameter ideals whose extended rees algesbras are Cohen-Macaulay with minimal multiplicity at their maximal homogenous ideals.

Abstract: The aim of this paper is to study mixed multiplicities of ideals and use them to calculate multiplicities of certain homogeneous ideals of extended Rees algebras and to give a complete characterization of those parameter ideals whose extended Rees algebras are Cohen-Macaulay with minimal multiplicity at their maximal homogeneous ideals. Let (R, m) be a local ring of dimension d. Let I be an m-primary ideal. It is well known that the length of the artinian ring R/F, I(R/F), is a polynomial of degree d in r for all large r. The coefficient of nd/d ! in this polynomial, called the multiplicity of I, denoted by e(I), is a well-understood and useful invariant of/. Suppose J is another m-primary ideal. Then it is natural to consider l (R/FJ s) for positive integers r and s. It is proved in [B] that for large values of r and s, I(R/FJ ~) is given by a polynomial P(r, s) of total degree d in r and s. Moreover, the terms of total degree d in P(r, s) have the form