Lemma 81.3.2 (0GFQ)—The Stacks project

Lemma 81.3.2. Let $S$ be a scheme. Let $B$ be an algebraic space over $S$ . Let $\mathcal{I} \to (\mathit{Sch}/S)_{fppf}$ , $i \mapsto X_ i$ be a diagram of algebraic spaces over $B$ . Let $(X, X_ i \to X)$ be a cocone for the diagram in the category of algebraic spaces over $B$ (Categories, Remark 4.14.5). If there exists a fpqc covering $\{ U_ a \to X\} _{a \in A}$ such that

for all $a \in A$ we have $U_ a = \mathop{\mathrm{colim}}\nolimits X_ i \times _ X U_ a$ in the category of algebraic spaces over $B$ , and
for all $a, b \in A$ we have $U_ a \times _ X U_ b = \mathop{\mathrm{colim}}\nolimits X_ i \times _ X U_ a \times _ X U_ b$ in the category of algebraic spaces over $B$ ,

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

Proof. Namely, for an algebraic space $Y$ over $B$ a morphism $X \to Y$ over $B$ is the same thing as a collection of morphism $U_ a \to Y$ which agree on the overlaps $U_ a \times _ X U_ b$ for all $a, b \in A$ , see Descent on Spaces, Lemma 74.7.2. $\square$

Comments (2)

Comment #7782 by Laurent Moret-Bailly on September 22, 2022 at 12:05

I think condition (2) can be weakened: one only needs that for all $a,b\in A$ the family $(X_i\times_X U_a \times_X U_b \to U_a \times_X U_b)_{i\in\mathcal{I}}$ is epimorphic in the category of algebraic spaces.

Comment #8023 by Stacks Project on January 18, 2023 at 15:21

Yes, I agree. But will this ever be useful? Going to leave as is for now.

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