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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 horizontal 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


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