The Stacks project

Lemma 73.11.1. Let $S$ be a scheme. The property $\mathcal{P}(f) =$“$f$ is quasi-compact” is fpqc local on the base on algebraic spaces over $S$.

Proof. We will use Lemma 73.10.4 to prove this. Assumptions (1) and (2) of that lemma follow from Morphisms of Spaces, Lemma 66.8.8. Let $Z' \to Z$ be a surjective flat morphism of affine schemes over $S$. Let $f : X \to Z$ be a morphism of algebraic spaces, and assume that the base change $f' : Z' \times _ Z X \to Z'$ is quasi-compact. We have to show that $f$ is quasi-compact. To see this, using Morphisms of Spaces, Lemma 66.8.8 again, it is enough to show that for every affine scheme $Y$ and morphism $Y \to Z$ the fibre product $Y \times _ Z X$ is quasi-compact. Here is a picture:
\begin{equation} \label{spaces-descent-equation-cube} \vcenter { \xymatrix{ Y \times _ Z Z' \times _ Z X \ar[dd] \ar[rr] \ar[rd] & & Z' \times _ Z X \ar '[d][dd]^{f'} \ar[rd] \\ & Y \times _ Z X \ar[dd] \ar[rr] & & X \ar[dd]^ f \\ Y \times _ Z Z' \ar '[r][rr] \ar[rd] & & Z' \ar[rd] \\ & Y \ar[rr] & & Z } } \end{equation}

Note that all squares are cartesian and the bottom square consists of affine schemes. The assumption that $f'$ is quasi-compact combined with the fact that $Y \times _ Z Z'$ is affine implies that $Y \times _ Z Z' \times _ Z X$ is quasi-compact. Since

\[ Y \times _ Z Z' \times _ Z X \longrightarrow Y \times _ Z X \]

is surjective as a base change of $Z' \to Z$ we conclude that $Y \times _ Z X$ is quasi-compact, see Morphisms of Spaces, Lemma 66.8.6. This finishes the proof. $\square$

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