Lemma 37.34.5. Let $S$ be a scheme. Let $s \in S$. Then we have

where the colimit is over the filtered category which is opposite to the category of elementary étale neighbourhoods $(U, u)$ of $(S, s)$.

Lemma 37.34.5. Let $S$ be a scheme. Let $s \in S$. Then we have

\[ \mathcal{O}_{S, s}^ h = \mathop{\mathrm{colim}}\nolimits _{(U, u)} \mathcal{O}(U) \]

where the colimit is over the filtered category which is opposite to the category of elementary étale neighbourhoods $(U, u)$ of $(S, s)$.

**Proof.**
Let $\mathop{\mathrm{Spec}}(A) \subset S$ be an affine neighbourhood of $s$. Let $\mathfrak p \subset A$ be the prime ideal corresponding to $s$. With these choices we have canonical isomorphisms $\mathcal{O}_{S, s} = A_{\mathfrak p}$ and $\kappa (s) = \kappa (\mathfrak p)$. A cofinal system of elementary étale neighbourhoods is given by those elementary étale neighbourhoods $(U, u)$ such that $U$ is affine and $U \to S$ factors through $\mathop{\mathrm{Spec}}(A)$. In other words, we see that the right hand side is equal to $\mathop{\mathrm{colim}}\nolimits _{(B, \mathfrak q)} B$ where the colimit is over étale $A$-algebras $B$ endowed with a prime $\mathfrak q$ lying over $\mathfrak p$ with $\kappa (\mathfrak p) = \kappa (\mathfrak q)$. Thus the lemma follows from Algebra, Lemma 10.155.7.
$\square$

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