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
the morphisms f_{i'i} : X_{i'} \to X_ i are affine,
the morphisms g_{i'i} : Y_{i'} \to Y_ i are affine,
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
Comments (0)