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
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