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Chapter 10: Commutative Algebra > Section 10.104: Catenary rings

Lemma 10.104.5. Let $R$ be a ring. The following are equivalent

  1. $R$ is catenary,
  2. $R_\mathfrak p$ is catenary for all prime ideals $\mathfrak p$,
  3. $R_\mathfrak m$ is catenary for all maximal ideals $\mathfrak m$.

Assume $R$ is Noetherian. The following are equivalent

  1. $R$ is universally catenary,
  2. $R_\mathfrak p$ is universally catenary for all prime ideals $\mathfrak p$,
  3. $R_\mathfrak m$ is universally catenary for all maximal ideals $\mathfrak m$.

Proof. The implication (1) $\Rightarrow$ (2) follows from Lemma 10.104.4 in both cases. The implication (2) $\Rightarrow$ (3) is immediate in both cases. Assume $R_\mathfrak m$ is catenary for all maximal ideals $\mathfrak m$ of $R$. If $\mathfrak p \subset \mathfrak q$ are primes in $R$, then choose a maximal ideal $\mathfrak q \subset \mathfrak m$. Chains of primes ideals between $\mathfrak p$ and $\mathfrak q$ are in 1-to-1 correspondence with chains of prime ideals between $\mathfrak pR_\mathfrak m$ and $\mathfrak qR_\mathfrak m$ hence we see $R$ is catenary. Assume $R$ is Noetherian and $R_\mathfrak m$ is universally catenary for all maximal ideals $\mathfrak m$ of $R$. Let $R \to S$ be a finite type ring map. Let $\mathfrak q$ be a prime ideal of $S$ lying over the prime $\mathfrak p \subset R$. Choose a maximal ideal $\mathfrak p \subset \mathfrak m$ in $R$. Then $R_\mathfrak p$ is a localization of $R_\mathfrak m$ hence universally catenary by Lemma 10.104.4. Then $S_\mathfrak p$ is catenary as a finite type ring over $R_\mathfrak p$. Hence $S_\mathfrak q$ is catenary as a localization. Thus $S$ is catenary by the first case treated above. $\square$

    The code snippet corresponding to this tag is a part of the file algebra.tex and is located in lines 24147–24163 (see updates for more information).

    \begin{lemma}
    \label{lemma-catenary-check-local}
    Let $R$ be a ring. The following are equivalent
    \begin{enumerate}
    \item $R$ is catenary,
    \item $R_\mathfrak p$ is catenary for all prime ideals $\mathfrak p$,
    \item $R_\mathfrak m$ is catenary for all maximal ideals $\mathfrak m$.
    \end{enumerate}
    Assume $R$ is Noetherian. The following are equivalent
    \begin{enumerate}
    \item $R$ is universally catenary,
    \item $R_\mathfrak p$ is universally catenary for all prime ideals
    $\mathfrak p$,
    \item $R_\mathfrak m$ is universally catenary for all maximal ideals
    $\mathfrak m$.
    \end{enumerate}
    \end{lemma}
    
    \begin{proof}
    The implication (1) $\Rightarrow$ (2) follows from
    Lemma \ref{lemma-localization-catenary} in both cases.
    The implication (2) $\Rightarrow$ (3) is immediate in both cases.
    Assume $R_\mathfrak m$ is catenary for all maximal ideals
    $\mathfrak m$ of $R$. If $\mathfrak p \subset \mathfrak q$ are primes
    in $R$, then choose a maximal ideal $\mathfrak q \subset \mathfrak m$.
    Chains of primes ideals between $\mathfrak p$ and $\mathfrak q$ are
    in 1-to-1 correspondence with chains of prime ideals between
    $\mathfrak pR_\mathfrak m$ and $\mathfrak qR_\mathfrak m$ hence we
    see $R$ is catenary. Assume $R$ is Noetherian and $R_\mathfrak m$ is
    universally catenary for all maximal ideals $\mathfrak m$ of $R$.
    Let $R \to S$ be a finite type ring map. Let $\mathfrak q$ be a prime
    ideal of $S$ lying over the prime $\mathfrak p \subset R$.
    Choose a maximal ideal $\mathfrak p \subset \mathfrak m$ in $R$.
    Then $R_\mathfrak p$ is a localization of $R_\mathfrak m$ hence
    universally catenary by Lemma \ref{lemma-localization-catenary}.
    Then $S_\mathfrak p$ is catenary as a finite type ring over $R_\mathfrak p$.
    Hence $S_\mathfrak q$ is catenary as a localization. Thus $S$ is catenary
    by the first case treated above.
    \end{proof}

    Comments (2)

    Comment #2374 by Matthew Emerton on February 11, 2017 a 8:13 pm UTC

    Very minor: there is a typo on line 2 of the proof: $R\mathfrak m$ should be $R_{\mathfrak m}$.

    Comment #2430 by Johan (site) on February 17, 2017 a 2:58 pm UTC

    THanks, fixed here.

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