Lemma 87.38.6. Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $Z \subset X$ be a closed subspace and set $T = |Z|$. The canonical morphism $c : X^\wedge _ Z \to X_{/T}$ is an isomorphism if $Z \to X$ is of finite presentation, but not in general.
Proof. Both constructions commute with étale localization, hence it suffices to prove this when $X$ is affine. Say $X = \mathop{\mathrm{Spec}}(A)$ and $Z$ corresponds to the ideal $I \subset A$. If $I$ is finitely generated then both $X^\wedge _ Z$ and $X_{/T}$ are equal to $\text{Spf}(A^\wedge )$ where $A^\wedge $ is the $I$-adic completion of $A$, see Lemmas 87.14.6 and 87.38.4. If $A = \mathbf{Z}[x_1, x_2, \ldots ]$ and $I = (x_1, x_2, \ldots )$ then $\mathop{\mathrm{Spec}}(A/(x_ n^ n, n \geq 1)) \to X$ factors through $X_{/T}$ but not through $X^\wedge _ Z$ and hence $c$ is not an isomorphism. $\square$
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