Multigraded Rees algebras and mixed multiplicities

AbstractLet I1, I2,…, Ig be ideals of positive height in a local ring (R, m). Let I0 be m-primary. Set S = R[I1t1, I2t2,…, Igtg], where t1,…, tg are indeterminates and M = (m, I1t1, I2t2,…, Igtg). A formula is developed for the multiplicity of the ideal (I0, I1t1, I2t2,…, Igtg)SM in terms of mixed multiplicities of I 0, I1,…, Ig. This formula is used to prove that if R is Cohen-Macaulay of dimension 2 and I1 = ma1,…, Ig = mag for positive integers a1, a2,…, ag, then SM is Cohen-Macaulay with minimal multiplicity if and only if R has minimal multiplicity