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The Stacks project

Lemma 70.5.1. Let S be a scheme. Let X = \mathop{\mathrm{lim}}\nolimits _{i \in I} X_ i be the limit of a directed inverse system of algebraic spaces over S with affine transition morphisms (Lemma 70.4.1). If each X_ i is decent (for example quasi-separated or locally separated) then |X| = \mathop{\mathrm{lim}}\nolimits _ i |X_ i| as sets.

Proof. There is a canonical map |X| \to \mathop{\mathrm{lim}}\nolimits |X_ i|. Choose 0 \in I. If W_0 \subset X_0 is an open subspace, then we have f_0^{-1}W_0 = \mathop{\mathrm{lim}}\nolimits _{i \geq 0} f_{i0}^{-1}W_0, see Lemma 70.4.1. Hence, if we can prove the lemma for inverse systems where X_0 is quasi-compact, then the lemma follows in general. Thus we may and do assume X_0 is quasi-compact.

Choose an affine scheme U_0 and a surjective étale morphism U_0 \to X_0. Set U_ i = X_ i \times _{X_0} U_0 and U = X \times _{X_0} U_0. Set R_ i = U_ i \times _{X_ i} U_ i and R = U \times _ X U. Recall that U = \mathop{\mathrm{lim}}\nolimits U_ i and R = \mathop{\mathrm{lim}}\nolimits R_ i, see proof of Lemma 70.4.1. Recall that |X| = |U|/|R| and |X_ i| = |U_ i|/|R_ i|. By Limits, Lemma 32.4.6 we have |U| = \mathop{\mathrm{lim}}\nolimits |U_ i| and |R| = \mathop{\mathrm{lim}}\nolimits |R_ i|.

Surjectivity of |X| \to \mathop{\mathrm{lim}}\nolimits |X_ i|. Let (x_ i) \in \mathop{\mathrm{lim}}\nolimits |X_ i|. Denote S_ i \subset |U_ i| the inverse image of x_ i. This is a finite nonempty set by the definition of decent spaces (Decent Spaces, Definition 68.6.1). Hence \mathop{\mathrm{lim}}\nolimits S_ i is nonempty, see Categories, Lemma 4.21.7. Let (u_ i) \in \mathop{\mathrm{lim}}\nolimits S_ i \subset \mathop{\mathrm{lim}}\nolimits |U_ i|. By the above this determines a point u \in |U| which maps to an x \in |X| mapping to the given element (x_ i) of \mathop{\mathrm{lim}}\nolimits |X_ i|.

Injectivity of |X| \to \mathop{\mathrm{lim}}\nolimits |X_ i|. Suppose that x, x' \in |X| map to the same point of \mathop{\mathrm{lim}}\nolimits |X_ i|. Choose lifts u, u' \in |U| and denote u_ i, u'_ i \in |U_ i| the images. For each i let T_ i \subset |R_ i| be the set of points mapping to (u_ i, u'_ i) \in |U_ i| \times |U_ i|. This is a finite set by the definition of decent spaces (Decent Spaces, Definition 68.6.1). Moreover T_ i is nonempty as we've assumed that x and x' map to the same point of X_ i. Hence \mathop{\mathrm{lim}}\nolimits T_ i is nonempty, see Categories, Lemma 4.21.7. As before let r \in |R| = \mathop{\mathrm{lim}}\nolimits |R_ i| be a point corresponding to an element of \mathop{\mathrm{lim}}\nolimits T_ i. Then r maps to (u, u') in |U| \times |U| by construction and we see that x = x' in |X| as desired.

Parenthetical statement: A quasi-separated algebraic space is decent, see Decent Spaces, Section 68.6 (the key observation to this is Properties of Spaces, Lemma 66.6.7). A locally separated algebraic space is decent by Decent Spaces, Lemma 68.15.2. \square


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