The Stacks project

Lemma 70.4.3. Let $S$ be a scheme. Let $I$ be a directed set. Let $(X_ i, f_{i'i}) \to (Y_ i, g_{i'i})$ be a morphism of inverse systems over $I$ of algebraic spaces over $S$. Assume

  1. the morphisms $f_{i'i} : X_{i'} \to X_ i$ are affine,

  2. the morphisms $g_{i'i} : Y_{i'} \to Y_ i$ are affine,

  3. the morphisms $X_ i \to Y_ i$ are closed immersions.

Then $\mathop{\mathrm{lim}}\nolimits X_ i \to \mathop{\mathrm{lim}}\nolimits Y_ i$ is a closed immersion.

Proof. Observe that $\mathop{\mathrm{lim}}\nolimits X_ i$ and $\mathop{\mathrm{lim}}\nolimits Y_ i$ exist by Lemma 70.4.1. Pick $0 \in I$ and choose an affine scheme $V_0$ and an étale morphism $V_0 \to Y_0$. Then the morphisms $V_ i = Y_ i \times _{Y_0} V_0 \to U_ i = X_ i \times _{Y_0} V_0$ are closed immersions of affine schemes. Hence the morphism $V = Y \times _{Y_0} V_0 \to U = X \times _{Y_0} V_0$ is a closed immersion because $V = \mathop{\mathrm{lim}}\nolimits V_ i$, $U = \mathop{\mathrm{lim}}\nolimits U_ i$ and because a limit of closed immersions of affine schemes is a closed immersion: a filtered colimit of surjective ring maps is surjective. Since the étale morphisms $V \to Y$ form an étale covering of $Y$ as we vary our choice of $V_0 \to Y_0$ we see that the lemma is true. $\square$

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