The Stacks project

Lemma 37.14.2. Let $\mathcal{I} \to (\mathit{Sch}/S)_{fppf}$, $i \mapsto X_ i$ be a diagram of schemes. Let $(W, X_ i \to W)$ be a cocone for the diagram in the category of schemes (Categories, Remark 4.14.5). If there exists a fpqc covering $\{ W_ a \to W\} _{a \in A}$ of schemes such that

  1. for all $a \in A$ we have $W_ a = \mathop{\mathrm{colim}}\nolimits X_ i \times _ W W_ a$ in the category of schemes, and

  2. for all $a, b \in A$ we have $W_ a \times _ W W_ b = \mathop{\mathrm{colim}}\nolimits X_ i \times _ W W_ a \times _ W W_ b$ in the category of schemes,

then $W = \mathop{\mathrm{colim}}\nolimits X_ i$ in the category of schemes.

Proof. Namely, for a scheme $T$ a morphism $W \to T$ is the same thing as collection of morphism $W_ a \to T$, $a \in A$ which agree on the overlaps $W_ a \times _ W W_ b$, see Descent, Lemma 35.13.7. $\square$

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