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Catenarity in module-finite algebras

Authors:	Shiro Goto Kenji Nishida

Institution:	Department of Mathematics, School of Science and Technology, Meiji University, Kawasaki 214-71, Japan ; Department of Mathematics, Faculty of Science, Shinsyu University, Matsumoto, 390-0802 Japan

Abstract:	The main theorem says that any module-finite (but not necessarily commutative) algebra $\Lambda$ over a commutative Noetherian universally catenary ring is catenary. Hence the ring $\Lambda$ is catenary if is Cohen-Macaulay. When is local and $\Lambda$ is a Cohen-Macaulay -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 and .

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