The Stacks project

Lemma 10.37.17. Let $(R_ i, \varphi _{ii'})$ be a directed system (Categories, Definition 10.8.1) of rings. If each $R_ i$ is a normal ring so is $R = \mathop{\mathrm{colim}}\nolimits _ i R_ i$.

Proof. Let $\mathfrak p \subset R$ be a prime ideal. Set $\mathfrak p_ i = R_ i \cap \mathfrak p$ (usual abuse of notation). Then we see that $R_{\mathfrak p} = \mathop{\mathrm{colim}}\nolimits _ i (R_ i)_{\mathfrak p_ i}$. Since each $(R_ i)_{\mathfrak p_ i}$ is a normal domain we reduce to proving the statement of the lemma for normal domains. If $a, b \in R$ and $a/b$ satisfies a monic polynomial $P(T) \in R[T]$, then we can find a (sufficiently large) $i \in I$ such that $a, b$ come from objects $a_ i, b_ i$ over $R_ i$, $P$ comes from a monic polynomial $P_ i\in R_ i[T]$ and $P_ i(a_ i/b_ i)=0$. Since $R_ i$ is normal we see $a_ i/b_ i \in R_ i$ and hence also $a/b \in R$. $\square$

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