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 (88.27.0.1) 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\} $. $\square$


Comments (0)


Post a comment

Your email address will not be published. Required fields are marked.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.




In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 0GDR. Beware of the difference between the letter 'O' and the digit '0'.