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Tag 0AS4

Chapter 10: Commutative Algebra > Section 10.49: Valuation rings

Lemma 10.49.5. Let $I$ be a directed set. Let $(A_i, \varphi_{ij})$ be a system of valuation rings over $I$. Then $A = \mathop{\rm colim}\nolimits A_i$ is a valuation ring.

Proof. It is clear that $A$ is a domain. Let $a, b \in A$. Lemma 10.49.4 tells us we have to show that either $a | b$ or $b | a$ in $A$. Choose $i$ so large that there exist $a_i, b_i \in A_i$ mapping to $a, b$. Then Lemma 10.49.3 applied to $a_i, b_i$ in $A_i$ implies the result for $a, b$ in $A$. $\square$

    The code snippet corresponding to this tag is a part of the file algebra.tex and is located in lines 11211–11216 (see updates for more information).

    \begin{lemma}
    \label{lemma-colimit-valuation-rings}
    Let $I$ be a directed set. Let $(A_i, \varphi_{ij})$
    be a system of valuation rings over $I$.
    Then $A = \colim A_i$ is a valuation ring.
    \end{lemma}
    
    \begin{proof}
    It is clear that $A$ is a domain. Let $a, b \in A$.
    Lemma \ref{lemma-x-or-x-inverse-valuation-ring} tells us we have
    to show that either $a | b$ or $b | a$ in $A$. Choose $i$
    so large that there exist $a_i, b_i \in A_i$ mapping to $a, b$.
    Then Lemma \ref{lemma-valuation-ring-x-or-x-inverse}
    applied to $a_i, b_i$ in $A_i$ implies the result for $a, b$ in $A$.
    \end{proof}

    Comments (2)

    Comment #2477 by Dario WeiƟmann on April 7, 2017 a 12:10 pm UTC

    (Why) Is the condition $\varphi_{ij}$ local necessary?

    Comment #2510 by Johan (site) on April 14, 2017 a 12:33 am UTC

    OK, I guess not. Thanks! Change is here.

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