Lemma 89.6.1. Let $(A, \mathfrak m, \kappa )$ be a local ring with finitely generated maximal ideal $\mathfrak m$. Let $X$ be a decent algebraic space over $A$. Let $Y = X \times _{\mathop{\mathrm{Spec}}(A)} \mathop{\mathrm{Spec}}(A^\wedge )$ where $A^\wedge $ is the $\mathfrak m$-adic completion of $A$. For a point $q \in |Y|$ with image $p \in |X|$ lying over the closed point of $\mathop{\mathrm{Spec}}(A)$ the map $\mathcal{O}_{X, p}^ h \to \mathcal{O}_{Y, q}^ h$ of henselian local rings induces an isomorphism on completions.
Proof. This follows immediately from the case of schemes by choosing an elementary étale neighbourhood $(U, u) \to (X, p)$ as in Decent Spaces, Lemma 68.11.4, setting $V = U \times _ X Y = U \times _{\mathop{\mathrm{Spec}}(A)} \mathop{\mathrm{Spec}}(A^\wedge )$ and $v = (u, q)$. The case of schemes is Resolution of Surfaces, Lemma 54.11.1. $\square$
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