The Stacks project

87.27 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 87.23 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 87.23. 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

87.27.0.1
\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 87.23.9.

Lemma 87.27.1. 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 (87.27.0.1) 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 87.27.2. 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 (87.27.0.1) 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 87.25.5. 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 87.25.2.

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 86.10.5). 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 86.8.4). By Lemma 87.25.1 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 87.25.2 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 87.25.7 starting with $X \amalg X$ over $X$.

Proof. We already know the functor is fully faithful, see Lemma 87.27.2. 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 87.25.1. Thus we see that $W$ is in the essential image by Lemma 87.25.7.

Step 2: $W$ is separated. Choose $\{ W_ i \to W\} $ as in Formal Spaces, Definition 86.7.1. 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 87.25.6 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 79.10.1. 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 86.7.1. 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 87.27.4. 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 (87.27.0.1)

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

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

Proof. The functor is fully faithful by Lemma 87.27.2. 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 87.27.1. 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 87.27.3 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 73.21.3) $(X'_ i, a_{ij})$ relative to the family $\{ X_ i \to X\} $. By Bootstrap, Lemma 79.11.3, 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. $\square$


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