A note on joint reductions and mixed multiplicities

Viet, Duong; Dinh, Le; Thanh, Truong

doi:10.1090/S0002-9939-2014-11916-2

A note on joint reductions and mixed multiplicities
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by Duong Quoc Viet, Le Van Dinh and Truong Thi Hong Thanh

Proc. Amer. Math. Soc. 142 (2014), 1861-1873

DOI: https://doi.org/10.1090/S0002-9939-2014-11916-2

Published electronically: February 28, 2014

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Abstract:

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, $I_1,\ldots ,I_s$ ideals of $A$, and $M$ a finitely generated $A$-module. In this paper, we interpret mixed multiplicities of $(I_1,\ldots , I_s,J)$ with respect to $M$ as multiplicities of joint reductions of them. This generalizes Rees’s theorem on mixed multiplicity. As an application we show that mixed multiplicities are also multiplicities of Rees superficial sequences.

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