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

 

Catenarity in module-finite algebras


Authors: Shiro Goto and Kenji Nishida
Journal: Proc. Amer. Math. Soc. 127 (1999), 3495-3502
MSC (1991): Primary 13E05, 16A18; Secondary 13H10, 16A33
Published electronically: May 13, 1999
MathSciNet review: 1618674
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Abstract: The main theorem says that any module-finite (but not necessarily commutative) algebra $\Lambda$ over a commutative Noetherian universally catenary ring $R$ is catenary. Hence the ring $\Lambda$ is catenary if $R$ is Cohen-Macaulay. When $R$ is local and $\Lambda$ is a Cohen-Macaulay $R$-module, we have that $\Lambda$ is a catenary ring, $\dim\Lambda=\dim\Lambda/Q+\mathrm{ht}_\Lambda Q$ for any $Q\in\operatorname{Spec}\Lambda$, and the equality $n=\mathrm{ht}_\Lambda Q- \mathrm{ht}_\Lambda P$ holds true for any pair $P\subseteq Q$ of prime ideals in $\Lambda$ and for any saturated chain $P=P_0\subset P_1\subset \cdots\subset P_n=Q$ of prime ideals between $P$ and $Q$.


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Additional Information

Shiro Goto
Affiliation: Department of Mathematics, School of Science and Technology, Meiji University, Kawasaki 214-71, Japan
Email: goto@math.meiji.ac.jp

Kenji Nishida
Affiliation: Department of Mathematics, Faculty of Science, Shinsyu University, Matsumoto, 390-0802 Japan
Email: kenisida@math.shinsyu-u.ac.jp

DOI: http://dx.doi.org/10.1090/S0002-9939-99-04962-X
PII: S 0002-9939(99)04962-X
Received by editor(s): October 27, 1997
Received by editor(s) in revised form: February 24, 1998
Published electronically: May 13, 1999
Additional Notes: The first author was supported by the Grant-in-Aid for Scientific Researches (C)
Communicated by: Ken Goodearl
Article copyright: © Copyright 1999 American Mathematical Society