The Stacks project

80.11 Formal glueing of algebraic spaces

In Situation 80.10.6 we consider the category $\textit{Spaces}(X \to Y, Z)$ of commutative diagrams of algebraic spaces over $S$ of the form

\[ \xymatrix{ U' \ar[d] & V' \ar[l] \ar[d] \ar[r] & Y' \ar[d] \\ U & V \ar[l] \ar[r] & Y } \]

where both squares are cartesian. There is a canonical functor

80.11.0.1
\begin{equation} \label{spaces-pushouts-equation-formal-glueing-spaces} \textit{Spaces}/X \longrightarrow \textit{Spaces}(Y \to X, Z) \end{equation}

which maps $X' \to X$ to the morphisms $U \times _ X X' \leftarrow V \times _ X X' \rightarrow Y \times _ X X'$.

Lemma 80.11.1. In Situation 80.10.6 the functor (80.11.0.1) restricts to an equivalence

  1. from the category of algebraic spaces affine over $X$ to the full subcategory of $\textit{Spaces}(Y \to X, Z)$ consisting of $(U' \leftarrow V' \rightarrow Y')$ with $U' \to U$, $V' \to V$, and $Y' \to Y$ affine,

  2. from the category of closed immersions $X' \to X$ to the full subcategory of $\textit{Spaces}(Y \to X, Z)$ consisting of $(U' \leftarrow V' \rightarrow Y')$ with $U' \to U$, $V' \to V$, and $Y' \to Y$ closed immersions, and

  3. same statement as in (2) for finite morphisms.

Proof. The category of algebraic spaces affine over $X$ is equivalent to the category of quasi-coherent sheaves $\mathcal{A}$ of $\mathcal{O}_ X$-algebras. The full subcategory of $\textit{Spaces}(Y \to X, Z)$ consisting of $(U' \leftarrow V' \rightarrow Y')$ with $U' \to U$, $V' \to V$, and $Y' \to Y$ affine is equivalent to the category of algebra objects of $\mathit{QCoh}(Y \to X, Z)$. In both cases this follows from Morphisms of Spaces, Lemma 66.20.7 with quasi-inverse given by the relative spectrum construction (Morphisms of Spaces, Definition 66.20.8) which commutes with arbitrary base change. Thus part (1) of the lemma follows from Proposition 80.10.9.

Fully faithfulness in part (2) follows from part (1). For essential surjectivity, we reduce by part (1) to proving that $X' \to X$ is a closed immersion if and only if both $U \times _ X X' \to U$ and $Y \times _ X X' \to Y$ are closed immersions. By Lemma 80.10.11 $\{ U \to X, Y \to X\} $ can be refined by an fpqc covering. Hence the result follows from Descent on Spaces, Lemma 73.11.17.

For (3) use the argument proving (2) and Descent on Spaces, Lemma 73.11.23. $\square$

Proof. By a formal argument with base change, this reduces to the following question: A morphism $a : X' \to X$ of algebraic spaces such that $U \times _ X X' \to U$ and $Y \times _ X X' \to Y$ are isomorphisms, is an isomorphism. The family $\{ U \to X, Y \to X\} $ can be refined by an fpqc covering by Lemma 80.10.11. Hence the result follows from Descent on Spaces, Lemma 73.11.15. $\square$

Lemma 80.11.3. In Situation 80.10.6 the functor (80.11.0.1) is fully faithful on algebraic spaces separated over $X$. More precisely, it induces a bijection

\[ \mathop{\mathrm{Mor}}\nolimits _ X(X'_1, X'_2) \longrightarrow \mathop{\mathrm{Mor}}\nolimits _{\textit{Spaces}(Y \to X, Z)}(F(X'_1), F(X'_2)) \]

whenever $X'_2 \to X$ is separated.

Proof. Since $X'_2 \to X$ is separated, the graph $i : X'_1 \to X'_1 \times _ X X'_2$ of a morphism $X'_1 \to X'_2$ over $X$ is a closed immersion, see Morphisms of Spaces, Lemma 66.4.6. Moreover a closed immersion $i : T \to X'_1 \times _ X X'_2$ is the graph of a morphism if and only if $\text{pr}_1 \circ i$ is an isomorphism. The same is true for

  1. the graph of a morphism $U \times _ X X'_1 \to U \times _ X X'_2$ over $U$,

  2. the graph of a morphism $V \times _ X X'_1 \to V \times _ X X'_2$ over $V$, and

  3. the graph of a morphism $Y \times _ X X'_1 \to Y \times _ X X'_2$ over $Y$.

Moreover, if morphisms as in (1), (2), (3) fit together to form a morphism in the category $\textit{Spaces}(Y \to X, Z)$, then these graphs fit together to give an object of $\textit{Spaces}(Y \times _ X (X'_1 \times _ X X'_2) \to X'_1 \times _ X X'_2, Z \times _ X (X'_1 \times _ X X'_2))$ whose triple of morphisms are closed immersions. The proof is finished by applying Lemmas 80.11.1 and 80.11.2. $\square$


Comments (1)

Comment #7780 by Laurent Moret-Bailly on

First line of section: it should be , not .


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