Valuation rings are stable under filtered direct limits

Lemma 10.49.5. Let $I$ be a directed set. Let $(A_ i, \varphi _{ij})$ be a system of valuation rings over $I$. Then $A = \mathop{\mathrm{colim}}\nolimits A_ i$ 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 $A$. $\square$

Comment #2477 by Dario Weißmann on

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

Comment #3796 by slogan_bot on

Suggested slogan: "Valuation rings are stable under filtered direct limits"

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