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The Stacks project

Lemma 70.4.1. Let S be a scheme. Let I be a directed set. Let (X_ i, f_{ii'}) be an inverse system over I in the category of algebraic spaces over S. If the morphisms f_{ii'} : X_ i \to X_{i'} are affine, then the limit X = \mathop{\mathrm{lim}}\nolimits _ i X_ i (as an fppf sheaf) is an algebraic space. Moreover,

  1. each of the morphisms f_ i : X \to X_ i is affine,

  2. for any i \in I and any morphism of algebraic spaces T \to X_ i we have

    X \times _{X_ i} T = \mathop{\mathrm{lim}}\nolimits _{i' \geq i} X_{i'} \times _{X_ i} T.

    as algebraic spaces over S.

Proof. Part (2) is a formal consequence of the existence of the limit X = \mathop{\mathrm{lim}}\nolimits X_ i as an algebraic space over S. Choose an element 0 \in I (this is possible as a directed set is nonempty). Choose a scheme U_0 and a surjective étale morphism U_0 \to X_0. Set R_0 = U_0 \times _{X_0} U_0 so that X_0 = U_0/R_0. For i \geq 0 set U_ i = X_ i \times _{X_0} U_0 and R_ i = X_ i \times _{X_0} R_0 = U_ i \times _{X_ i} U_ i. By Limits, Lemma 32.2.2 we see that U = \mathop{\mathrm{lim}}\nolimits _{i \geq 0} U_ i and R = \mathop{\mathrm{lim}}\nolimits _{i \geq 0} R_ i are schemes. Moreover, the two morphisms s, t : R \to U are the base change of the two projections R_0 \to U_0 by the morphism U \to U_0, in particular étale. The morphism R \to U \times _ S U defines an equivalence relation as directed a limit of equivalence relations is an equivalence relation. Hence the morphism R \to U \times _ S U is an étale equivalence relation. We claim that the natural map

70.4.1.1
\begin{equation} \label{spaces-limits-equation-isomorphism-sheaves} U/R \longrightarrow \mathop{\mathrm{lim}}\nolimits X_ i \end{equation}

is an isomorphism of fppf sheaves on the category of schemes over S. The claim implies X = \mathop{\mathrm{lim}}\nolimits X_ i is an algebraic space by Spaces, Theorem 65.10.5.

Let Z be a scheme and let a : Z \to \mathop{\mathrm{lim}}\nolimits X_ i be a morphism. Then a = (a_ i) where a_ i : Z \to X_ i. Set W_0 = Z \times _{a_0, X_0} U_0. Note that W_0 = Z \times _{a_ i, X_ i} U_ i for all i \geq 0 by our choice of U_ i \to X_ i above. Hence we obtain a morphism W_0 \to \mathop{\mathrm{lim}}\nolimits _{i \geq 0} U_ i = U. Since W_0 \to Z is surjective and étale, we conclude that (70.4.1.1) is a surjective map of sheaves. Finally, suppose that Z is a scheme and that a, b : Z \to U/R are two morphisms which are equalized by (70.4.1.1). We have to show that a = b. After replacing Z by the members of an fppf covering we may assume there exist morphisms a', b' : Z \to U which give rise to a and b. The condition that a, b are equalized by (70.4.1.1) means that for each i \geq 0 the compositions a_ i', b_ i' : Z \to U \to U_ i are equal as morphisms into U_ i/R_ i = X_ i. Hence (a_ i', b_ i') : Z \to U_ i \times _ S U_ i factors through R_ i, say by some morphism c_ i : Z \to R_ i. Since R = \mathop{\mathrm{lim}}\nolimits _{i \geq 0} R_ i we see that c = \mathop{\mathrm{lim}}\nolimits c_ i : Z \to R is a morphism which shows that a, b are equal as morphisms of Z into U/R.

Part (1) follows as we have seen above that U_ i \times _{X_ i} X = U and U \to U_ i is affine by construction. \square


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