The Stacks project

88.27 Algebraization of rig-étale morphisms

In this section we prove a generalization of the result on dilatations from the paper of Artin [ArtinII].

The notation in this section will agree with the notation in Section 88.23 except our algebraic spaces and formal algebraic spaces will be locally Noetherian.

Thus, we first fix a base scheme $S$. All rings, topological rings, schemes, algebraic spaces, and formal algebraic spaces and morphisms between these will be over $S$. Next, we fix a locally Noetherian algebraic space $X$ and a closed subset $T \subset |X|$. We denote $U \subset X$ be the open subspace with $|U| = |X| \setminus T$. Picture

\[ U \to X \quad |X| = |U| \amalg T \]

Given a morphism of algebraic spaces $f : X' \to X$, we will use the notation $U' = f^{-1}U$, $T' = |f|^{-1}(T)$, and $f_{/T} : X'_{/T'} \to X_{/T}$ as in Section 88.23. We will sometimes write $X'_{/T}$ in stead of $X'_{/T'}$ and more generally for a morphism $a : X' \to X''$ of algebraic spaces over $X$ we will denote $a_{/T} : X'_{/T} \to X''_{/T}$ the induced morphism of formal algebraic spaces obtained by completing the morphism $a$ along the inverse images of $T$ in $X'$ and $X''$.

Given this setup we will consider the functor

88.27.0.1
\begin{equation} \label{restricted-equation-completion-functor} \left\{ \begin{matrix} \text{morphisms of algebraic spaces} \\ f : X' \to X\text{ which are locally} \\ \text{of finite type and such that} \\ U' \to U\text{ is an isomorphism} \end{matrix} \right\} \longrightarrow \left\{ \begin{matrix} \text{morphisms }g : W \to X_{/T} \\ \text{of formal algebraic spaces} \\ \text{with }W\text{ locally Noetherian} \\ \text{and }g\text{ rig-étale} \end{matrix} \right\} \end{equation}

sending $f : X' \to X$ to $f_{/T} : X'_{/T'} \to X_{/T}$. This makes sense because $f_{/T}$ is rig-étale by Lemma 88.23.9.

Lemma 88.27.1. In the situation above, let $X_1 \to X$ be a morphism of algebraic spaces with $X_1$ locally Noetherian. Denote $T_1 \subset |X_1|$ the inverse image of $T$ and $U_1 \subset X_1$ the inverse image of $U$. We denote

  1. $\mathcal{C}_{X, T}$ the category whose objects are morphisms of algebraic spaces $f : X' \to X$ which are locally of finite type and such that $U' = f^{-1}U \to U$ is an isomorphism,

  2. $\mathcal{C}_{X_1, T_1}$ the category whose objects are morphisms of algebraic spaces $f_1 : X_1' \to X_1$ which are locally of finite type and such that $f_1^{-1}U_1 \to U_1$ is an isomorphism,

  3. $\mathcal{C}_{X_{/T}}$ the category whose objects are morphisms $g : W \to X_{/T}$ of formal algebraic spaces with $W$ locally Noetherian and $g$ rig-étale,

  4. $\mathcal{C}_{X_{1, /T_1}}$ the category whose objects are morphisms $g_1 : W_1 \to X_{1, /T_1}$ of formal algebraic spaces with $W_1$ locally Noetherian and $g_1$ rig-étale.

Then the diagram

\[ \xymatrix{ \mathcal{C}_{X, T} \ar[d] \ar[r] & \mathcal{C}_{X_{/T}} \ar[d] \\ \mathcal{C}_{X_1, T_1} \ar[r] & \mathcal{C}_{X_{1, /T_1}} } \]

is commutative where the horizonal arrows are given by (88.27.0.1) and the vertical arrows by base change along $X_1 \to X$ and along $X_{1, /T_1} \to X_{/T}$.

Proof. This follows immediately from the fact that the completion functor $(h : Y \to X) \mapsto Y_{/T} = Y_{/|h|^{-1}T}$ on the category of algebraic spaces over $X$ commutes with fibre products. $\square$

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\} $. $\square$

Before we prove this lemma let us discuss an example. Suppose that $S = \mathop{\mathrm{Spec}}(k)$, $X = \mathbf{A}^1_ k$, and $T = \{ 0\} $. Then $X_{/T} = \text{Spf}(k[[x]])$. Let $W = \text{Spf}(k[[x]] \times k[[x]])$. Then the corresponding $f : X' \to X$ is the affine line with zero doubled mapping to the affine line (Schemes, Example 26.14.3). Moreover, this is the output of the construction in Lemma 88.25.7 starting with $X \amalg X$ over $X$.

Proof. We already know the functor is fully faithful, see Lemma 88.27.2. Essential surjectivity. Let $g : W \to X_{/T}$ be a morphism of formal algebraic spaces with $W$ locally Noetherian and $g$ rig-étale. We will prove $W$ is in the essential image in a number of steps.

Step 1: $W$ is an affine formal algebraic space. Then we can find $U \to X$ of finite type and étale over $X \setminus T$ such that $U_{/T}$ is isomorphic to $W$, see Lemma 88.25.1. Thus we see that $W$ is in the essential image by Lemma 88.25.7.

Step 2: $W$ is separated. Choose $\{ W_ i \to W\} $ as in Formal Spaces, Definition 87.11.1. By Step 1 the formal algebraic spaces $W_ i$ and $W_ i \times _ W W_ j$ are in the essential image. Say $W_ i = (X'_ i)_{/T}$ and $W_ i \times _ W W_ j = (X'_{ij})_{/T}$. By fully faithfulness we obtain morphisms $t_{ij} : X'_{ij} \to X'_ i$ and $s_{ij} : X'_{ij} \to X'_ j$ matching the projections $W_ i \times _ W W_ j \to W_ i$ and $W_ i \times _ W W_ j \to W_ j$. Consider the structure

\[ R = \coprod X'_{ij},\quad V = \coprod X'_ i,\quad s = \coprod s_{ij},\quad t = \coprod t_{ij} \]

(We can't use the letter $U$ as it has already been used.) Applying Lemma 88.25.6 we find that $(t, s) : R \to V \times _ X V$ defines an étale equivalence relation on $V$ over $X$. Thus we can take the quotient $X' = V/R$ and it is an algebraic space, see Bootstrap, Theorem 80.10.1. Since completion commutes with fibre products and taking quotient sheaves, we find that $X'_{/T} \cong W$ as formal algebraic spaces over $X_{/T}$.

Step 3: $W$ is general. Choose $\{ W_ i \to W\} $ as in Formal Spaces, Definition 87.11.1. The formal algebraic spaces $W_ i$ and $W_ i \times _ W W_ j$ are separated. Hence by Step 2 the formal algebraic spaces $W_ i$ and $W_ i \times _ W W_ j$ are in the essential image. Then we argue exactly as in the previous paragraph to see that $W$ is in the essential image as well. This concludes the proof. $\square$

Theorem 88.27.4. Let $S$ be a scheme. Let $X$ be a locally Noetherian algebraic space over $S$. Let $T \subset |X|$ be a closed subset. Let $U \subset X$ be the open subspace with $|U| = |X| \setminus T$. The completion functor (88.27.0.1)

\[ \left\{ \begin{matrix} \text{morphisms of algebraic spaces} \\ f : X' \to X\text{ which are locally} \\ \text{of finite type and such that} \\ f^{-1}U \to U\text{ is an isomorphism} \end{matrix} \right\} \longrightarrow \left\{ \begin{matrix} \text{morphisms }g : W \to X_{/T} \\ \text{of formal algebraic spaces} \\ \text{with }W\text{ locally Noetherian} \\ \text{and }g\text{ rig-étale} \end{matrix} \right\} \]

sending $f : X' \to X$ to $f_{/T} : X'_{/T'} \to X_{/T}$ is an equivalence.

Proof. The functor is fully faithful by Lemma 88.27.2. Let $g : W \to X_{/T}$ be a morphism of formal algebraic spaces with $W$ locally Noetherian and $g$ rig-étale. We will prove $W$ is in the essential image to finish the proof.

Choose an étale covering $\{ X_ i \to X\} $ with $X_ i$ affine for all $i$. Denote $U_ i \subset X_ i$ the inverse image of $U$ and denote $T_ i \subset X_ i$ the inverse image of $T$. Recall that $(X_ i)_{/T_ i} = (X_ i)_{/T} = (X_ i \times _ X X)_{/T}$ and $W_ i = X_ i \times _ X W = (X_ i)_{/T} \times _{X_{/T}} W$, see Lemma 88.27.1. Observe that we obtain isomorphisms

\[ \alpha _{ij} : W_ i \times _{X_{/T}} (X_ j)_{/T} \longrightarrow (X_ i)_{/T} \times _{X_{/T}} W_ j \]

satisfying a suitable cocycle condition. By Lemma 88.27.3 applied to $X_ i, T_ i, U_ i, W_ i \to (X_ i)_{/T}$ there exists a morphism $X'_ i \to X_ i$ of algebraic spaces which is locally of finite type and an isomorphism over $U_ i$ and an isomorphism $\beta _ i : (X'_ i)_{/T} \cong W_ i$ over $(X_ i)_{/T}$. By fully faithfullness we find an isomorphism

\[ a_{ij} : X'_ i \times _ X X_ j \longrightarrow X_ i \times _ X X'_ j \]

over $X_ i \times _ X X_ j$ such that $\alpha _{ij} = \beta _ j|_{X_ i \times _ X X_ j} \circ (a_{ij})_{/T} \circ \beta _ i^{-1}|_{X_ i \times _ X X_ j}$. By fully faithfulness again (this time over $X_ i \times _ X X_ j \times _ X X_ k$) we see that these morphisms $a_{ij}$ satisfy the same cocycle condition as satisfied by the $\alpha _{ij}$. In other words, we obtain a descent datum (as in Descent on Spaces, Definition 74.22.3) $(X'_ i, a_{ij})$ relative to the family $\{ X_ i \to X\} $. By Bootstrap, Lemma 80.11.3, this descent datum is effective. Thus we find a morphism $f : X' \to X$ of algebraic spaces and isomorphisms $h_ i : X' \times _ X X_ i \to X'_ i$ over $X_ i$ such that $a_{ij} = h_ j|_{X_ i \times _ X X_ j} \circ h_ i^{-1}|_{X_ i \times _ X X_ j}$. The reader can check that the ensuing isomorphisms

\[ (X' \times _ X X_ i)_{/T} \xrightarrow {\beta _ i \circ (h_ i)_{/T}} W_ i \]

over $X_ i$ glue to an isomorphism $X'_{/T} \to W$ over $X_{/T}$; some details omitted. $\square$


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 0AR1. Beware of the difference between the letter 'O' and the digit '0'.