The Stacks project

Lemma 80.3.1. Let $S$ be a scheme. Let $\mathcal{I} \to (\mathit{Sch}/S)_{fppf}$, $i \mapsto X_ i$ be a diagram of schemes over $S$ as above. Assume that

  1. $X = \mathop{\mathrm{colim}}\nolimits X_ i$ exists in the category of schemes,

  2. $\coprod X_ i \to X$ is surjective,

  3. if $U \to X$ is étale and $U_ i = X_ i \times _ X U$, then $U = \mathop{\mathrm{colim}}\nolimits U_ i$ in the category of schemes, and

  4. every object $(U_ i \to X_ i)$ of $\mathop{\mathrm{lim}}\nolimits X_{i, {\acute{e}tale}}$ with $U_ i \to X_ i$ separated is in the essential image of the functor $X_{\acute{e}tale}\to \mathop{\mathrm{lim}}\nolimits X_{i, {\acute{e}tale}}$.

Then $X = \mathop{\mathrm{colim}}\nolimits X_ i$ in the category of algebraic spaces over $S$ also.

Proof. Let $Z$ be an algebraic space over $S$. Suppose that $f_ i : X_ i \to Z$ is a family of morphisms such that for each $i \to j$ the composition $X_ i \to X_ j \to Z$ is equal to $f_ i$. We have to construct a morphism of algebraic spaces $f : X \to Z$ such that we can recover $f_ i$ as the composition $X_ i \to X \to Z$. Let $W \to Z$ be a surjective étale morphism of a scheme to $Z$. We may assume that $W$ is a disjoint union of affines and in particular we may assume that $W \to Z$ is separated. For each $i$ set $U_ i = W \times _{Z, f_ i} X_ i$ and denote $h_ i : U_ i \to W$ the projection. Then $U_ i \to X_ i$ forms an object of $\mathop{\mathrm{lim}}\nolimits X_{i, {\acute{e}tale}}$ with $U_ i \to X_ i$ separated. By assumption (4) we can find an étale morphism $U \to X$ and (functorial) isomorphisms $U_ i = X_ i \times _ X U$. By assumption (3) there exists a morphism $h : U \to W$ such that the compositions $U_ i \to U \to W$ are $h_ i$. Let $g : U \to Z$ be the composition of $h$ with the map $W \to Z$. To finish the proof we have to show that $g : U \to Z$ descends to a morphism $X \to Z$. To do this, consider the morphism $(h, h) : U \times _ X U \to W \times _ S W$. Composing with $U_ i \times _{X_ i} U_ i \to U \times _ X U$ we obtain $(h_ i, h_ i)$ which factors through $W \times _ Z W$. Since $U \times _ X U$ is the colimit of the schemes $U_ i \times _{X_ i} U_ i$ by (3) we see that $(h, h)$ factors through $W \times _ Z W$. Hence the two compositions $U \times _ X U \to U \to W \to Z$ are equal. Because each $U_ i \to X_ i$ is surjective and assumption (2) we see that $U \to X$ is surjective. As $Z$ is a sheaf for the étale topology, we conclude that $g : U \to Z$ descends to $f : X \to Z$ as desired. $\square$

Comments (2)

Comment #7008 by Laurent Moret-Bailly on

Typo in (4): "the essential image of..."

There are also:

  • 2 comment(s) on Section 80.3: Colimits of algebraic spaces

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