The stalk of the structure sheaf of a scheme in the etale topology is the strict henselization.

Lemma 59.33.1. Let $S$ be a scheme. Let $\overline{s}$ be a geometric point of $S$ lying over $s \in S$. Let $\kappa = \kappa (s)$ and let $\kappa \subset \kappa ^{sep} \subset \kappa (\overline{s})$ denote the separable algebraic closure of $\kappa$ in $\kappa (\overline{s})$. Then there is a canonical identification

$(\mathcal{O}_{S, s})^{sh} \cong (\mathcal{O}_ S)_{\overline{s}}$

where the left hand side is the strict henselization of the local ring $\mathcal{O}_{S, s}$ as described in Theorem 59.32.8 and right hand side is the stalk of the structure sheaf $\mathcal{O}_ S$ on $S_{\acute{e}tale}$ at the geometric point $\overline{s}$.

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 $\kappa (s) = \kappa (\mathfrak p)$. Thus we have $\kappa (\mathfrak p) \subset \kappa ^{sep} \subset \kappa (\overline{s})$. Recall that

$(\mathcal{O}_ S)_{\overline{s}} = \mathop{\mathrm{colim}}\nolimits _{(U, \overline{u})} \mathcal{O}(U)$

where the limit is over the étale neighbourhoods of $(S, \overline{s})$. A cofinal system is given by those étale neighbourhoods $(U, \overline{u})$ such that $U$ is affine and $U \to S$ factors through $\mathop{\mathrm{Spec}}(A)$. In other words, we see that

$(\mathcal{O}_ S)_{\overline{s}} = \mathop{\mathrm{colim}}\nolimits _{(B, \mathfrak q, \phi )} B$

where the colimit is over étale $A$-algebras $B$ endowed with a prime $\mathfrak q$ lying over $\mathfrak p$ and a $\kappa (\mathfrak p)$-algebra map $\phi : \kappa (\mathfrak q) \to \kappa (\overline{s})$. Note that since $\kappa (\mathfrak q)$ is finite separable over $\kappa (\mathfrak p)$ the image of $\phi$ is contained in $\kappa ^{sep}$. Via these translations the result of the lemma is equivalent to the result of Algebra, Lemma 10.155.11. $\square$

Comment #1229 by David Corwin on

Suggested slogan: The stalk in the etale topology is the strict henselization

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