
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"

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 0AS4. Beware of the difference between the letter 'O' and the digit '0'.