A note on joint reductions and mixed multiplicities

Details A note on joint reductions and mixed multiplicities A note on joint reductions and mixed multiplicities

2011-10-28

arxiv.org/abs/1110.6239v1

Mathematics

Commutative Algebra

Scientific paper

Let $(A, \frak m)$ be a noetherian local ring with maximal ideal $\frak{m}$ and infinite residue field $k = A/\frak{m}.$ Let $J$ be an $\frak m$-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 \cite[Theorem 2.4]{Re}. As an application we show that mixed multiplicities are also multiplicities of Rees's superficial sequences.

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