Reductions and Mixed Multiplicities of Ideals#

Abstract

The notion of (FC)-sequences was built in Viet [Viet, D. Q. (2000). Mixed multiplicities of arbitrary ideals in local rings. Comm. Algebra 28(8):3803–3821] containing important information for multiplicities and reductions of ideals [see Viet, D. Q. 2000; Viet, D. Q. (2003a). On some properties of (FC)-sequences of ideals in local rings. Proc. Am. Math. Soc. 131:45–53; Viet, D. Q. (2003b). Sequences determining mixed multiplicities and reductions of ideals. Comm. Algebra 31(10):5047–5069; Viet, D. Q. (2003c). On the mixed multiplicity and the multiplicity of blow-up rings of equimultiple. J. Pure Appl. Algebra 183:313–327]. In this paper, we will construct generalized joint reductions generated by maximal weak-(FC)-sequences of a set of arbitrary ideals. We showed that the mixed multiplicity of ideals and the mixed multiplicity of their reductions are the same. The paper also determines the vanishing and non-vanishing of mixed multiplicities and as an application of above results we compute mixed multiplicities and the multiplicity of multi-graded Rees algebras in some particular cases.