The Stacks project

Lemma 69.23.3. In Situation 69.23.1. Let $X \to B$ be a quasi-separated and finite type morphism of algebraic spaces. Given $i \in I$ and a diagram

\[ \vcenter { \xymatrix{ X \ar[r] \ar[d] & W \ar[d] \\ B \ar[r] & B_ i } } \]

as in (69.23.2.1) for $i' \geq i$ let $X_{i'}$ be the scheme theoretic image of $X \to B_{i'} \times _{B_ i} W$. Then $X = \mathop{\mathrm{lim}}\nolimits _{i' \geq i} X_{i'}$.

Proof. Since $X$ is quasi-compact and quasi-separated formation of the scheme theoretic image of $X \to B_{i'} \times _{B_ i} W$ commutes with ├ętale localization (Morphisms of Spaces, Lemma 66.16.3). Hence we may and do assume $W$ is affine and maps into an affine $U_ i$ ├ętale over $B_ i$. Then

\[ B_{i'} \times _{B_ i} W = B_{i'} \times _{B_ i} U_ i \times _{U_ i} W = U_{i'} \times _{U_ i} W \]

where $U_{i'} = B_{i'} \times _{B_ i} U_ i$ is affine as the transition morphisms are affine. Thus the lemma follows from the case of schemes which is Limits, Lemma 32.22.3. $\square$


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