The Stacks project

Lemma 88.27.2. In the situation above. Let $f : X' \to X$ be a morphism of algebraic spaces which is locally of finite type and an isomorphism over $U$. Let $g : Y \to X$ be a morphism with $Y$ locally Noetherian. Then completion defines a bijection

\[ \mathop{\mathrm{Mor}}\nolimits _ X(Y, X') \longrightarrow \mathop{\mathrm{Mor}}\nolimits _{X_{/T}}(Y_{/T}, X'_{/T}) \]

In particular, the functor ( is fully faithful.

Proof. Let $a, b : Y \to X'$ be morphisms over $X$ such that $a_{/T} = b_{/T}$. Then we see that $a$ and $b$ agree over the open subspace $g^{-1}U$ and after completion along $g^{-1}T$. Hence $a = b$ by Lemma 88.25.5. In other words, the completion map is always injective.

Let $\alpha : Y_{/T} \to X'_{/T}$ be a morphism of formal algebraic spaces over $X_{/T}$. We have to prove there exists a morphism $a : Y \to X'$ over $X$ such that $\alpha = a_{/T}$. The proof proceeds by a standard but cumbersome reduction to the affine case and then applying Lemma 88.25.2.

Let $\{ h_ i : Y_ i \to Y\} $ be an étale covering of algebraic spaces. If we can find for each $i$ a morphism $a_ i : Y_ i \to X'$ over $X$ whose completion $(a_ i)_{/T} : (Y_ i)_{/T} \to X'_{/T}$ is equal to $\alpha \circ (h_ i)_{/T}$, then we get a morphism $a : Y \to X'$ with $\alpha = a_{/T}$. Namely, we first observe that $(a_ i)_{/T} \circ \text{pr}_1 = (a_ j)_{/T} \circ \text{pr}_2$ as morphisms $(Y_ i \times _ Y Y_ j)_{/T} \to X'_{/T}$ by the agreement with $\alpha $ (this uses that completion ${}_{/T}$ commutes with fibre products). By the injectivity already proven this shows that $a_ i \circ \text{pr}_1 = a_ j \circ \text{pr}_2$ as morphisms $Y_ i \times _ Y Y_ j \to X'$. Since $X'$ is an fppf sheaf this means that the collection of morphisms $a_ i$ descends to a morphism $a : Y \to X'$. We have $\alpha = a_{/T}$ because $\{ (a_ i)_{/T} : (Y_ i)_{/T} \to X'_{/T}\} $ is an étale covering.

By the result of the previous paragraph, to prove existence, we may assume that $Y$ is affine and that $g : Y \to X$ factors as $g_1 : Y \to X_1$ and an étale morphism $X_1 \to X$ with $X_1$ affine. Then we can consider $T_1 \subset |X_1|$ the inverse image of $T$ and we can set $X'_1 = X' \times _ X X_1$ with projection $f_1 : X'_1 \to X_1$ and

\[ \alpha _1 = (\alpha , (g_1)_{/T_1}) : Y_{/T_1} = Y_{/T} \longrightarrow X'_{/T} \times _{X_{/T}} (X_1)_{/T_1} = (X'_1)_{/T_1} \]

We conclude that it suffices to prove the existence for $\alpha _1$ over $X_1$, in other words, we may replace $X, T, X', Y, f, g, \alpha $ by $X_1, T_1, X'_1, Y, g_1, \alpha _1$. This reduces us to the case described in the next paragraph.

Assume $Y$ and $X$ are affine. Recall that $(Y_{/T})_{red}$ is an affine scheme (isomorphic to the reduced induced scheme structure on $g^{-1}T \subset Y$, see Formal Spaces, Lemma 87.14.5). Hence $\alpha _{red} : (Y_{/T})_{red} \to (X'_{/T})_{red}$ has quasi-compact image $E$ in $f^{-1}T$ (this is the underlying topological space of $(X'_{/T})_{red}$ by the same lemma as above). Thus we can find an affine scheme $V$ and an étale morpism $h : V \to X'$ such that the image of $h$ contains $E$. Choose a solid cartesian diagram

\[ \xymatrix{ Y'_{/T} \ar@{..>}[rd] \ar@{..>}[r] & W \ar[d] \ar[r] & V_{/T} \ar[d]^{h_{/T}} \\ & Y_{/T} \ar[r]^\alpha & X'_{/T} } \]

By construction, the morphism $W \to Y_{/T}$ is representable by algebraic spaces, étale, and surjective (surjectivity can be seen by looking at the reductions, see Formal Spaces, Lemma 87.12.4). By Lemma 88.25.1 we can write $W = Y'_{/T}$ for $Y' \to Y$ étale and $Y'$ affine. This gives the dotted arrows in the diagram. Since $W \to Y_{/T}$ is surjective, we see that the image of $Y' \to Y$ contains $g^{-1}T$. Hence $\{ Y' \to Y, Y \setminus g^{-1}T \to Y\} $ is an étale covering. As $f$ is an isomorphism over $U$ we have a (unique) morphism $Y \setminus g^{-1}T \to X'$ over $X$ agreeing with $\alpha $ on completions (as the completion of $Y \setminus g^{-1}T$ is empty). Thus it suffices to prove the existence for $Y'$ which reduces us to the case studied in the next paragraph.

By the result of the previous paragraph, we may assume that $Y$ is affine and that $\alpha $ factors as $Y_{/T} \to V_{/T} \to X'_{/T}$ where $V$ is an affine scheme étale over $X'$. We may still replace $Y$ by the members of an affine étale covering. By Lemma 88.25.2 we may find an étale morphism $b : Y' \to Y$ of affine schemes which induces an isomorphism $b_{/T} : Y'_{/T} \to Y_{/T}$ and a morphism $c : Y' \to V$ such that $c_{/T} \circ b_{/T}^{-1}$ is the given morphism $Y_{/T} \to V_{/T}$. Setting $a' : Y' \to X'$ equal to the composition of $c$ and $V \to X'$ we find that $a'_{/T} = \alpha \circ b_{/T}$, in other words, we have existence for $Y'$ and $\alpha \circ b_{/T}$. Then we are done by replacing considering once more the étale covering $\{ Y' \to Y, Y \setminus g^{-1}T \to Y\} $. $\square$

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