The Stacks project

Lemma 70.19.3. Let $S$ be a scheme. Let $X$ be a decent algebraic space over $S$. Let $x \in |X|$ be a closed point such that $U = X \setminus \{ x\} \to X$ is quasi-compact. With $V = \mathop{\mathrm{Spec}}(\mathcal{O}_{X, x}^ h) \setminus \{ \mathfrak m_ x^ h\} $ the base change functor

\[ \left\{ \begin{matrix} f : Y \to X\text{ of finite presentation} \\ f^{-1}(U) \to U\text{ is an isomorphism} \end{matrix} \right\} \longrightarrow \left\{ \begin{matrix} g : Y \to \mathop{\mathrm{Spec}}(\mathcal{O}_{X, x}^ h)\text{ of finite presentation} \\ g^{-1}(V) \to V\text{ is an isomorphism} \end{matrix} \right\} \]

is an equivalence of categories.

Proof. Let $a : (W, w) \to (X, x)$ be an elementary étale neighbourhood of $x$ with $W$ affine as in Decent Spaces, Lemma 68.11.4. Since $x$ is a closed point of $X$ and $w$ is the unique point of $W$ lying over $x$, we see that $w$ is a closed point of $W$. Since $a$ is étale and identifies residue fields at $x$ and $w$, it follows that $a$ induces an isomorphism $a^{-1}x \to x$ (as closed subspaces of $X$ and $W$). Thus we may apply Lemma 70.19.1 and 70.19.2 to reduce the problem to the case where $X$ is an affine scheme.

Assume $X$ is an affine scheme. Recall that $\mathcal{O}_{X, x}^ h$ is the colimit of $\Gamma (U, \mathcal{O}_ U)$ over affine elementary étale neighbourhoods $(U, u) \to (X, x)$. Recall that the category of these neighbourhoods is cofiltered, see Decent Spaces, Lemma 68.11.6 or More on Morphisms, Lemma 37.35.4. Then $\mathop{\mathrm{Spec}}(\mathcal{O}_{X, x}^ h) = \mathop{\mathrm{lim}}\nolimits U$ and $V = \mathop{\mathrm{lim}}\nolimits U \setminus \{ u\} $ (Lemma 70.4.1) where the limits are taken over the same category. Thus by Lemma 70.7.1 The category on the right is the colimit of the categories for the pairs $(U, u)$. And by the material in the first paragraph, each of these categories is equivalent to the category for the pair $(X, x)$. This finishes the proof. $\square$

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