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.
70.5 Descending properties
This section is the analogue of Limits, Section 32.4.
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
Lemma 70.5.2. With same notation and assumptions as in Lemma 70.5.1 we have |X| = \mathop{\mathrm{lim}}\nolimits _ i |X_ i| as topological spaces.
Proof. We will use the criterion of Topology, Lemma 5.14.3. We have seen that |X| = \mathop{\mathrm{lim}}\nolimits _ i |X_ i| as sets in Lemma 70.5.1. The maps f_ i : X \to X_ i are morphisms of algebraic spaces hence determine continuous maps |X| \to |X_ i|. Thus f_ i^{-1}(U_ i) is open for each open U_ i \subset |X_ i|. Finally, let x \in |X| and let x \in V \subset |X| be an open neighbourhood. We have to find an i and an open neighbourhood W_ i \subset |X_ i| of the image x with f_ i^{-1}(W_ i) \subset V. Choose 0 \in I. Choose a scheme U_0 and a surjective étale morphism U_0 \to X_0. Set U = X \times _{X_0} U_0 and U_ i = X_ i \times _{X_0} U_0 for i \geq 0. Then U = \mathop{\mathrm{lim}}\nolimits _{i \geq 0} U_ i in the category of schemes by Lemma 70.4.1. Choose u \in U mapping to x. By the result for schemes (Limits, Lemma 32.4.2) we can find an i \geq 0 and an open neighbourhood E_ i \subset U_ i of the image of u whose inverse image in U is contained in the inverse image of V in U. Then we can set W_ i \subset |X_ i| equal to the image of E_ i. This works because |U_ i| \to |X_ i| is open. \square
Lemma 70.5.3. 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 quasi-compact and nonempty, then |X| is nonempty.
Proof. Choose 0 \in I. 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. Then each U_ i is a nonempty affine scheme. Hence U = \mathop{\mathrm{lim}}\nolimits U_ i is nonempty (Limits, Lemma 32.4.3) and thus X is nonempty. \square
Lemma 70.5.4. 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). Let x \in |X| with images x_ i \in |X_ i|. If each X_ i is decent, then \overline{\{ x\} } = \mathop{\mathrm{lim}}\nolimits _ i \overline{\{ x_ i\} } as sets and as algebraic spaces if endowed with reduced induced scheme structure.
Proof. Set Z = \overline{\{ x\} } \subset |X| and Z_ i = \overline{\{ x_ i\} } \subset |X_ i|. Since |X| \to |X_ i| is continuous we see that Z maps into Z_ i for each i. Hence we obtain an injective map Z \to \mathop{\mathrm{lim}}\nolimits Z_ i because |X| = \mathop{\mathrm{lim}}\nolimits |X_ i| as sets (Lemma 70.5.1). Suppose that x' \in |X| is not in Z. Then there is an open subset U \subset |X| with x' \in U and x \not\in U. Since |X| = \mathop{\mathrm{lim}}\nolimits |X_ i| as topological spaces (Lemma 70.5.2) we can write U = \bigcup _{j \in J} f_ j^{-1}(U_ j) for some subset J \subset I and opens U_ j \subset |X_ j|, see Topology, Lemma 5.14.2. Then we see that for some j \in J we have f_ j(x') \in U_ j and f_ j(x) \not\in U_ j. In other words, we see that f_ j(x') \not\in Z_ j. Thus Z = \mathop{\mathrm{lim}}\nolimits Z_ i as sets.
Next, endow Z and Z_ i with their reduced induced scheme structures, see Properties of Spaces, Definition 66.12.5. The transition morphisms X_{i'} \to X_ i induce affine morphisms Z_{i'} \to Z_ i and the projections X \to X_ i induce compatible morphisms Z \to Z_ i. Hence we obtain morphisms Z \to \mathop{\mathrm{lim}}\nolimits Z_ i \to X of algebraic spaces. By Lemma 70.4.3 we see that \mathop{\mathrm{lim}}\nolimits Z_ i \to X is a closed immersion. By Lemma 70.4.4 the algebraic space \mathop{\mathrm{lim}}\nolimits Z_ i is reduced. By the above Z \to \mathop{\mathrm{lim}}\nolimits Z_ i is bijective on points. By uniqueness of the reduced induced closed subscheme structure we find that this morphism is an isomorphism of algebraic spaces. \square
Situation 70.5.5. 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). We assume that X_ i is quasi-compact and quasi-separated for all i \in I. We also choose an element 0 \in I.
Lemma 70.5.6. Notation and assumptions as in Situation 70.5.5. Suppose that \mathcal{F}_0 is a quasi-coherent sheaf on X_0. Set \mathcal{F}_ i = f_{0i}^*\mathcal{F}_0 for i \geq 0 and set \mathcal{F} = f_0^*\mathcal{F}_0. Then
Proof. Choose a surjective étale morphism U_0 \to X_0 where U_0 is an affine scheme (Properties of Spaces, Lemma 66.6.3). Set U_ i = X_ i \times _{X_0} U_0. Set R_0 = U_0 \times _{X_0} U_0 and R_ i = R_0 \times _{X_0} X_ i. In the proof of Lemma 70.4.1 we have seen that there exists a presentation X = U/R with U = \mathop{\mathrm{lim}}\nolimits U_ i and R = \mathop{\mathrm{lim}}\nolimits R_ i. Note that U_ i and U are affine and that R_ i and R are quasi-compact and separated (as X_ i is quasi-separated). Hence Limits, Lemma 32.4.7 implies that
The lemma follows as \Gamma (X, \mathcal{F}) = \mathop{\mathrm{Ker}}(\mathcal{F}(U) \to \mathcal{F}(R)) and similarly \Gamma (X_ i, \mathcal{F}_ i) = \mathop{\mathrm{Ker}}(\mathcal{F}_ i(U_ i) \to \mathcal{F}_ i(R_ i)) \square
Lemma 70.5.7. Notation and assumptions as in Situation 70.5.5. For any quasi-compact open subspace U \subset X there exists an i and a quasi-compact open U_ i \subset X_ i whose inverse image in X is U.
Proof. Follows formally from the construction of limits in Lemma 70.4.1 and the corresponding result for schemes: Limits, Lemma 32.4.11. \square
The following lemma will be superseded by the stronger Lemma 70.6.10.
Lemma 70.5.8. Notation and assumptions as in Situation 70.5.5. Let f_0 : Y_0 \to Z_0 be a morphism of algebraic spaces over X_0. Assume (a) Y_0 \to X_0 and Z_0 \to X_0 are representable, (b) Y_0, Z_0 quasi-compact and quasi-separated, (c) f_0 locally of finite presentation, and (d) Y_0 \times _{X_0} X \to Z_0 \times _{X_0} X an isomorphism. Then there exists an i \geq 0 such that Y_0 \times _{X_0} X_ i \to Z_0 \times _{X_0} X_ i is an isomorphism.
Proof. Choose an affine scheme U_0 and a surjective étale morphism U_0 \to X_0. Set U_ i = U_0 \times _{X_0} X_ i and U = U_0 \times _{X_0} X. Apply Limits, Lemma 32.8.11 to see that Y_0 \times _{X_0} U_ i \to Z_0 \times _{X_0} U_ i is an isomorphism of schemes for some i \geq 0 (details omitted). As U_ i \to X_ i is surjective étale, it follows that Y_0 \times _{X_0} X_ i \to Z_0 \times _{X_0} X_ i is an isomorphism (details omitted). \square
Lemma 70.5.9. Notation and assumptions as in Situation 70.5.5. If X is separated, then X_ i is separated for some i \in I.
Proof. Choose an affine scheme U_0 and a surjective étale morphism U_0 \to X_0. For i \geq 0 set U_ i = U_0 \times _{X_0} X_ i and set U = U_0 \times _{X_0} X. Note that U_ i and U are affine schemes which come equipped with surjective étale morphisms U_ i \to X_ i and U \to X. Set R_ i = U_ i \times _{X_ i} U_ i and R = U \times _ X U with projections s_ i, t_ i : R_ i \to U_ i and s, t : R \to U. Note that R_ i and R are quasi-compact separated schemes (as the algebraic spaces X_ i and X are quasi-separated). The maps s_ i : R_ i \to U_ i and s : R \to U are of finite type. By definition X_ i is separated if and only if (t_ i, s_ i) : R_ i \to U_ i \times U_ i is a closed immersion, and since X is separated by assumption, the morphism (t, s) : R \to U \times U is a closed immersion. Since R \to U is of finite type, there exists an i such that the morphism R \to U_ i \times U is a closed immersion (Limits, Lemma 32.4.16). Fix such an i \in I. Apply Limits, Lemma 32.8.5 to the system of morphisms R_{i'} \to U_ i \times U_{i'} for i' \geq i (this is permissible as indeed R_{i'} = R_ i \times _{U_ i \times U_ i} U_ i \times U_{i'}) to see that R_{i'} \to U_ i \times U_{i'} is a closed immersion for i' sufficiently large. This implies immediately that R_{i'} \to U_{i'} \times U_{i'} is a closed immersion finishing the proof of the lemma. \square
Lemma 70.5.10. Notation and assumptions as in Situation 70.5.5. If X is affine, then there exists an i such that X_ i is affine.
Proof. Choose 0 \in I. Choose an affine scheme U_0 and a surjective étale morphism U_0 \to X_0. Set U = U_0 \times _{X_0} X and U_ i = U_0 \times _{X_0} X_ i for i \geq 0. Since the transition morphisms are affine, the algebraic spaces U_ i and U are affine. Thus U \to X is an étale morphism of affine schemes. Hence we can write X = \mathop{\mathrm{Spec}}(A), U = \mathop{\mathrm{Spec}}(B) and
such that \Delta = \det (\partial g_\lambda /\partial x_\mu ) is invertible in B, see Algebra, Lemma 10.143.2. Set A_ i = \mathcal{O}_{X_ i}(X_ i). We have A = \mathop{\mathrm{colim}}\nolimits A_ i by Lemma 70.5.6. After increasing 0 we may assume we have g_{1, i}, \ldots , g_{n, i} \in A_ i[x_1, \ldots , x_ n] mapping to g_1, \ldots , g_ n. Set
for all i \geq 0. Increasing 0 if necessary we may assume that \Delta _ i = \det (\partial g_{\lambda , i}/\partial x_\mu ) is invertible in B_ i for all i \geq 0. Thus A_ i \to B_ i is an étale ring map. After increasing 0 we may assume also that \mathop{\mathrm{Spec}}(B_ i) \to \mathop{\mathrm{Spec}}(A_ i) is surjective, see Limits, Lemma 32.8.15. Increasing 0 yet again we may choose elements h_{1, i}, \ldots , h_{n, i} \in \mathcal{O}_{U_ i}(U_ i) which map to the classes of x_1, \ldots , x_ n in B = \mathcal{O}_ U(U) and such that g_{\lambda , i}(h_{\nu , i}) = 0 in \mathcal{O}_{U_ i}(U_ i). Thus we obtain a commutative diagram
By construction B_ i = B_0 \otimes _{A_0} A_ i and B = B_0 \otimes _{A_0} A. Consider the morphism
This is a morphism of quasi-compact and quasi-separated algebraic spaces representable, separated and étale over X_0. The base change of f_0 to X is an isomorphism by our choices. Hence Lemma 70.5.8 guarantees that there exists an i such that the base change of f_0 to X_ i is an isomorphism, in other words the diagram (70.5.10.1) is cartesian. Thus Descent, Lemma 35.39.1 applied to the fppf covering \{ \mathop{\mathrm{Spec}}(B_ i) \to \mathop{\mathrm{Spec}}(A_ i)\} combined with Descent, Lemma 35.37.1 give that X_ i \to \mathop{\mathrm{Spec}}(A_ i) is representable by a scheme affine over \mathop{\mathrm{Spec}}(A_ i) as desired. (Of course it then also follows that X_ i = \mathop{\mathrm{Spec}}(A_ i) but we don't need this.) \square
Lemma 70.5.11. Notation and assumptions as in Situation 70.5.5. If X is a scheme, then there exists an i such that X_ i is a scheme.
Proof. Choose a finite affine open covering X = \bigcup W_ j. By Lemma 70.5.7 we can find an i \in I and open subspaces W_{j, i} \subset X_ i whose base change to X is W_ j \to X. By Lemma 70.5.10 we may assume that each W_{j, i} is an affine scheme. This means that X_ i is a scheme (see for example Properties of Spaces, Section 66.13). \square
Lemma 70.5.12. Let S be a scheme. Let B be an algebraic space over S. Let X = \mathop{\mathrm{lim}}\nolimits X_ i be a directed limit of algebraic spaces over B with affine transition morphisms. Let Y \to X be a morphism of algebraic spaces over B.
If Y \to X is a closed immersion, X_ i quasi-compact, and Y \to B locally of finite type, then Y \to X_ i is a closed immersion for i large enough.
If Y \to X is an immersion, X_ i quasi-separated, Y \to B locally of finite type, and Y quasi-compact, then Y \to X_ i is an immersion for i large enough.
If Y \to X is an isomorphism, X_ i quasi-compact, X_ i \to B locally of finite type, the transition morphisms X_{i'} \to X_ i are closed immersions, and Y \to B is locally of finite presentation, then Y \to X_ i is an isomorphism for i large enough.
If Y \to X is a monomorphism, X_ i quasi-separated, Y \to B locally of finite type, and Y quasi-compact, then Y \to X_ i is a monomorphism for i large enough.
Proof. Proof of (1). Choose 0 \in I. As X_0 is quasi-compact, we can choose an affine scheme W and an étale morphism W \to B such that the image of |X_0| \to |B| is contained in |W| \to |B|. Choose an affine scheme U_0 and an étale morphism U_0 \to X_0 \times _ B W such that U_0 \to X_0 is surjective. (This is possible by our choice of W and the fact that X_0 is quasi-compact; details omitted.) Let V \to Y, resp. U \to X, resp. U_ i \to X_ i be the base change of U_0 \to X_0 (for i \geq 0). It suffices to prove that V \to U_ i is a closed immersion for i sufficiently large. Thus we reduce to proving the result for V \to U = \mathop{\mathrm{lim}}\nolimits U_ i over W. This follows from the case of schemes, which is Limits, Lemma 32.4.16.
Proof of (2). Choose 0 \in I. Choose a quasi-compact open subspace X'_0 \subset X_0 such that Y \to X_0 factors through X'_0. After replacing X_ i by the inverse image of X'_0 for i \geq 0 we may assume all X_ i' are quasi-compact and quasi-separated. Let U \subset X be a quasi-compact open such that Y \to X factors through a closed immersion Y \to U (U exists as Y is quasi-compact). By Lemma 70.5.7 we may assume that U = \mathop{\mathrm{lim}}\nolimits U_ i with U_ i \subset X_ i quasi-compact open. By part (1) we see that Y \to U_ i is a closed immersion for some i. Thus (2) holds.
Proof of (3). Choose 0 \in I. 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, U = X \times _{X_0} U_0 = Y \times _{X_0} U_0. Then U = \mathop{\mathrm{lim}}\nolimits U_ i is a limit of affine schemes, the transition maps of the system are closed immersions, and U \to U_0 is of finite presentation (because U \to B is locally of finite presentation and U_0 \to B is locally of finite type and Morphisms of Spaces, Lemma 67.28.9). Thus we've reduced to the following algebra fact: If A = \mathop{\mathrm{lim}}\nolimits A_ i is a directed colimit of R-algebras with surjective transition maps and A of finite presentation over A_0, then A = A_ i for some i. Namely, write A = A_0/(f_1, \ldots , f_ n). Pick i such that f_1, \ldots , f_ n map to zero under the surjective map A_0 \to A_ i.
Proof of (4). Set Z_ i = Y \times _{X_ i} Y. As the transition morphisms X_{i'} \to X_ i are affine hence separated, the transition morphisms Z_{i'} \to Z_ i are closed immersions, see Morphisms of Spaces, Lemma 67.4.5. We have \mathop{\mathrm{lim}}\nolimits Z_ i = Y \times _ X Y = Y as Y \to X is a monomorphism. Choose 0 \in I. Since Y \to X_0 is locally of finite type (Morphisms of Spaces, Lemma 67.23.6) the morphism Y \to Z_0 is locally of finite presentation (Morphisms of Spaces, Lemma 67.28.10). The morphisms Z_ i \to Z_0 are locally of finite type (they are closed immersions). Finally, Z_ i = Y \times _{X_ i} Y is quasi-compact as X_ i is quasi-separated and Y is quasi-compact. Thus part (3) applies to Y = \mathop{\mathrm{lim}}\nolimits _{i \geq 0} Z_ i over Z_0 and we conclude Y = Z_ i for some i. This proves (4) and the lemma. \square
Lemma 70.5.13. Let S be a scheme. Let Y be an algebraic space over S. Let X = \mathop{\mathrm{lim}}\nolimits X_ i be a directed limit of algebraic spaces over Y with affine transition morphisms. Assume
Y is quasi-separated,
X_ i is quasi-compact and quasi-separated,
the morphism X \to Y is separated.
Then X_ i \to Y is separated for all i large enough.
Proof. Let 0 \in I. Choose an affine scheme W and an étale morphism W \to Y such that the image of |W| \to |Y| contains the image of |X_0| \to |Y|. This is possible as X_0 is quasi-compact. It suffices to check that W \times _ Y X_ i \to W is separated for some i \geq 0 because the diagonal of W \times _ Y X_ i over W is the base change of X_ i \to X_ i \times _ Y X_ i by the surjective étale morphism (X_ i \times _ Y X_ i) \times _ Y W \to X_ i \times _ Y X_ i. Since Y is quasi-separated the algebraic spaces W \times _ Y X_ i are quasi-compact (as well as quasi-separated). Thus we may base change to W and assume Y is an affine scheme. When Y is an affine scheme, we have to show that X_ i is a separated algebraic space for i large enough and we are given that X is a separated algebraic space. Thus this case follows from Lemma 70.5.9. \square
Lemma 70.5.14. Let S be a scheme. Let Y be an algebraic space over S. Let X = \mathop{\mathrm{lim}}\nolimits X_ i be a directed limit of algebraic spaces over Y with affine transition morphisms. Assume
Y quasi-compact and quasi-separated,
X_ i quasi-compact and quasi-separated,
X \to Y affine.
Then X_ i \to Y is affine for i large enough.
Proof. Choose an affine scheme W and a surjective étale morphism W \to Y. Then X \times _ Y W is affine and it suffices to check that X_ i \times _ Y W is affine for some i (Morphisms of Spaces, Lemma 67.20.3). This follows from Lemma 70.5.10. \square
Lemma 70.5.15. Let S be a scheme. Let Y be an algebraic space over S. Let X = \mathop{\mathrm{lim}}\nolimits X_ i be a directed limit of algebraic spaces over Y with affine transition morphisms. Assume
Y quasi-compact and quasi-separated,
X_ i quasi-compact and quasi-separated,
the transition morphisms X_{i'} \to X_ i are finite,
X_ i \to Y locally of finite type
X \to Y integral.
Then X_ i \to Y is finite for i large enough.
Proof. Choose an affine scheme W and a surjective étale morphism W \to Y. Then X \times _ Y W is finite over W and it suffices to check that X_ i \times _ Y W is finite over W for some i (Morphisms of Spaces, Lemma 67.45.3). By Lemma 70.5.11 this reduces us to the case of schemes. In the case of schemes it follows from Limits, Lemma 32.4.19. \square
Lemma 70.5.16. Let S be a scheme. Let Y be an algebraic space over S. Let X = \mathop{\mathrm{lim}}\nolimits X_ i be a directed limit of algebraic spaces over Y with affine transition morphisms. Assume
Y quasi-compact and quasi-separated,
X_ i quasi-compact and quasi-separated,
the transition morphisms X_{i'} \to X_ i are closed immersions,
X_ i \to Y locally of finite type
X \to Y is a closed immersion.
Then X_ i \to Y is a closed immersion for i large enough.
Proof. Choose an affine scheme W and a surjective étale morphism W \to Y. Then X \times _ Y W is a closed subspace of W and it suffices to check that X_ i \times _ Y W is a closed subspace W for some i (Morphisms of Spaces, Lemma 67.12.1). By Lemma 70.5.11 this reduces us to the case of schemes. In the case of schemes it follows from Limits, Lemma 32.4.20. \square
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