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Tag 00NK

Chapter 10: Commutative Algebra > Section 10.104: Catenary rings

Lemma 10.104.6. Any quotient of a (universally) catenary ring is (universally) catenary.

Proof. Let $A$ be a ring and let $I \subset A$ be an ideal. The description of $\mathop{\rm Spec}(A/I)$ in Lemma 10.16.7 shows that if $A$ is catenary, then so is $A/I$. If $A/I \to B$ is of finite type, then $A \to B$ is of finite type. Hence if $A$ is universally catenary, then $B$ is catenary. Combined with Lemma 10.30.1 this proves the lemma. $\square$

    The code snippet corresponding to this tag is a part of the file algebra.tex and is located in lines 24262–24265 (see updates for more information).

    \begin{lemma}
    \label{lemma-quotient-catenary}
    Any quotient of a (universally) catenary ring is (universally) catenary.
    \end{lemma}
    
    \begin{proof}
    Let $A$ be a ring and let $I \subset A$ be an ideal.
    The description of $\Spec(A/I)$ in Lemma \ref{lemma-spec-closed}
    shows that if $A$ is catenary, then so is $A/I$. If $A/I \to B$
    is of finite type, then $A \to B$ is of finite type. Hence if $A$
    is universally catenary, then $B$ is catenary. Combined with
    Lemma \ref{lemma-Noetherian-permanence} this proves the lemma.
    \end{proof}

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