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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

A note on joint reductions and mixed multiplicities


Authors: Duong Quoc Viet, Le Van Dinh and Truong Thi Hong Thanh
Journal: Proc. Amer. Math. Soc. 142 (2014), 1861-1873
MSC (2010): Primary 13H15; Secondary 14C17, 13D40, 13C15
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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Additional Information

Duong Quoc Viet
Affiliation: Department of Mathematics, Hanoi National University of Education, 136 Xuan Thuy Street, Hanoi, Vietnam
Email: duongquocviet@fmail.vnn.vn

Le Van Dinh
Affiliation: Department of Mathematics, Hanoi National University of Education, 136 Xuan Thuy Street, Hanoi, Vietnam
Email: dinhlevands@gmail.com

Truong Thi Hong Thanh
Affiliation: Department of Mathematics, Hanoi National University of Education, 136 Xuan Thuy Street, Hanoi, Vietnam
Email: thanhtth@hnue.edu.vn

DOI: http://dx.doi.org/10.1090/S0002-9939-2014-11916-2
PII: S 0002-9939(2014)11916-2
Keywords: Mixed multiplicity, multiplicity, joint reduction, (FC)-sequence
Received by editor(s): February 29, 2012
Received by editor(s) in revised form: June 30, 2012
Published electronically: February 28, 2014
Additional Notes: This research was partially supported by a grant from NAFOSTED
Communicated by: Irena Peeva
Article copyright: © Copyright 2014 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.