This paper gives the additivity and reduction formulas for mixed multiplicities of multi-graded modules M and mixed multiplicities of arbitrary ideals, and establishes the recursion formulas for the sum of all the mixed multiplicities of M. As an application of these formulas we get the recursion formulas for the multiplicity of multi-graded Rees modules.
Let (A, m) be a noetherian local ring with maximal ideal m and infinite residue field k = A/m. Let J be an m-primary ideal, I 1 , . . . , I s ideals of A, and M a finitely generated A-module. In this paper, we interpret mixed multiplicities of (I 1 , . . . , I s , J) with respect to M as multiplicities of joint reductions of them. This generalizes the Rees's theorem on mixed multiplicity [12, Theorem 2.4]. As an application we show that mixed multiplicities are also multiplicities of Rees's superficial sequences.
Abstract. Let A be a Noetherian local ring with the maximal ideal m and an m-primary ideal J . Let F = {I n } n≥0 be a good filtration of ideals in A. Denote by F J (F ) = n≥0 (I n /JI n )t n the fiber cone of F with respect to J. In this paper we characterize the multiplicity and the Cohen-Macaulayness of F J (F ) in terms of minimal reductions of F .
Let S = n 1 ,...,ns ≥0 S (n 1 ,...,ns ) be a finitely generated standard multi-graded algebra over a Noetherian local ring A. This paper investigates the positivity of mixed multiplicities of S and characterizes them in terms of Hilbert-Samuel multiplicities. As an application, we get some results on the mixed multiplicities of ideals that covers the main results in [Vi] and [TV].
This paper establishes mixed multiplicity formulas concerning the relationship between mixed multiplicities of modules and mixed multiplicities of rings via rank of modules.
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