The Stacks project

Theorem 85.10.9. Let $S$ be a scheme. Let $X$ be a locally Noetherian algebraic space over $S$. Let $T \subset |X|$ be a closed subset. The functor $F_{X, T}$ (

\[ \left\{ \begin{matrix} \text{algebraic spaces }Y\text{ locally of finite} \\ \text{type over }X\text{ such that }Y \to X \\ \text{is an isomorphism over }X \setminus T \end{matrix} \right\} \longrightarrow \left\{ \begin{matrix} \text{formal algebraic spaces }W\text{ endowed} \\ \text{with a rig-étale morphism }W \to X_{/T} \end{matrix} \right\} \]

given by formal completion is an equivalence.

Proof. The theorem is essentially a formal consequence of Lemma 85.10.8. We give the details but we encourage the reader to think it through for themselves. Let $g : U \to X$ be a surjective étale morphism with $U = \coprod U_ i$ and each $U_ i$ affine. Denote $F_{U, T}$ the functor for $U$ and the inverse image of $T$ in $|U|$.

Since $U = \coprod U_ i$ both the category $\mathcal{C}_{U, T}$ and the category $\mathcal{C}_{U_{/T}}$ decompose as a product of categories, one for each $i$. Since the functors $F_{U_ i, T}$ are equivalences for all $i$ by the lemma we find that the same is true for $F_{U, T}$.

Since $F_{U, T}$ is faithful, it follows that $F_{X, T}$ is faithful too. Namely, if $a, b : Y \to Y'$ are morphisms in $\mathcal{C}_{X, T}$ such that $a_{/T} = b_{/T}$, then we find on pulling back that the base changes $a_ U, b_ U : U \times _ X Y \to U \times _ X Y'$ are equal. Since $U \times _ X Y \to Y$ is surjective étale, this implies that $a = b$.

At this point we know that $F_{X, T}$ is faithful for every situation as in the theorem. Let $R = U \times _ X U$ where $U$ is as above. Let $t, s : R \to U$ be the projections. Since $X$ is Noetherian, so is $R$. Thus the functor $F_{R, T}$ (defined in the obvious manner) is faithful. Let $Y \to X$ and $Y' \to X$ be objects of $\mathcal{C}_{X, T}$. Let $a' : Y_{/T} \to Y'_{/T}$ be a morphism in the category $\mathcal{C}_{X_{/T}}$. Taking the base change to $U$ we obtain a morphism $a'_ U : (U \times _ X Y)_{/T} \to (U \times _ X Y')_{/T}$ in the category $\mathcal{C}_{U_{/T}}$. Since the functor $F_{U, T}$ is fully faithful we obtain a morphism $a_ U : U \times _ X Y \to U \times _ X Y'$ with $F_{U, T}(a_ U) = a'_ U$. Since $s^*(a'_ U) = t^*(a'_ U)$ and since $F_{R, T}$ is faithful, we find that $s^*(a_ U) = t^*(a_ U)$. Since

\[ \xymatrix{ R \times _ X Y \ar@<1ex>[r] \ar@<-1ex>[r] & U \times _ X Y \ar[r] & Y } \]

is an equalizer diagram of sheaves, we find that $a_ U$ descends to a morphism $a : Y \to Y'$. We omit the proof that $F_{X, T}(a) = a'$.

At this point we know that $F_{X, T}$ is faithful for every situation as in the theorem. To finish the proof we show that $F_{X, T}$ is essentially surjective. Let $W \to X_{/T}$ be an object of $\mathcal{C}_{X_{/T}}$. Then $U \times _ X W$ is an object of $\mathcal{C}_{U_{/T}}$. By the affine case we find an object $V \to U$ of $\mathcal{C}_{U, T}$ and an isomorphism $\alpha : F_{U, T}(V) \to U \times _ X W$ in $\mathcal{C}_{U_{/T}}$. By fully faithfulness of $F_{R, T}$ we find a unique morphism $h : s^*V \to t^*V$ in the category $\mathcal{C}_{R, T}$ such that $F_{R, T}(h)$ corresponds, via the isomorphism $\alpha $, to the canonical descent datum on $U \times _ X W$ in the category $\mathcal{C}_{R_{/T}}$. Using faithfulness of our functor on $R \times _{s, U, t} R$ we see that $h$ satisfies the cocycle condition. We conclude, for example by the much more general Bootstrap, Lemma 77.11.3, that there exists an object $Y \to X$ of $\mathcal{C}_{X, T}$ and an isomorphism $\beta : U \times _ X Y \to V$ such that the descent datum $h$ corresponds, via $\beta $, to the canonical descent datum on $U \times _ X Y$. We omit the verification that $F_{X, T}(Y)$ is isomorphic to $W$; hint: in the category of formal algebraic spaces there is descent for morphisms along étale coverings. $\square$

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