Limits of Algebraic Spaces

Introduction

In this chapter we put material related to limits of algebraic spaces. A first topic is the characterization of algebraic spaces \(F\) locally of finite presentation over the base \(S\) as limit preserving functors. We continue with a study of limits of inverse systems over directed sets (Categories, Definition 00D3) with affine transition maps. We discuss absolute Noetherian approximation for quasi-compact and quasi-separated algebraic spaces following [CLO]. Another approach is due to David Rydh (see [rydh_approx]) whose results also cover absolute Noetherian approximation for certain algebraic stacks.

Conventions

The standing assumption is that all schemes are contained in a big fppf site \(\Sch_{fppf}\). And all rings \(A\) considered have the property that \(\Spec(A)\) is (isomorphic) to an object of this big site.

Let \(S\) be a scheme and let \(X\) be an algebraic space over \(S\). In this chapter and the following we will write \(X \times_S X\) for the product of \(X\) with itself (in the category of algebraic spaces over \(S\)), instead of \(X \times X\).

Morphisms of finite presentation

In this section we generalize Limits, Proposition 01ZC to morphisms of algebraic spaces. The motivation for the following definition comes from the proposition just cited.

Definition

Let \(S\) be a scheme.

1. A functor \(F : (\Sch/S)_{fppf}^{opp} \to \textit{Sets}\) is said to be limit preserving or locally of finite presentation if for every affine scheme \(T\) over \(S\) which is a limit \(T = \lim T_i\) of a directed inverse system of affine schemes \(T_i\) over \(S\), we have \[F(T) = \colim F(T_i).\] We sometimes say that \(F\) is locally of finite presentation over \(S\).

2. Let \(F, G : (\Sch/S)_{fppf}^{opp} \to \textit{Sets}\). A transformation of functors \(a : F \to G\) is limit preserving or locally of finite presentation if for every scheme \(T\) over \(S\) and every \(y \in G(T)\) the functor \[F_y : (\Sch/T)_{fppf}^{opp} \longrightarrow \textit{Sets}, \quad T'/T \longmapsto \{x \in F(T') \mid a(x) = y|_{T'}\}\] is locally of finite presentation over \(T\)¹. We sometimes say that \(F\) is relatively limit preserving over \(G\).

The functor \(F_y\) is in some sense the fiber of \(a : F \to G\) over \(y\), except that it is a presheaf on the big fppf site of \(T\). A formula for this functor is: [049K]\[\begin{equation} F_y = F|_{(\Sch/T)_{fppf}} {\times}_{G|_{(\Sch/T)_{fppf}}} * \end{equation}\] Here \(*\) is the final object in the category of (pre)sheaves on \((\Sch/T)_{fppf}\) (see Sites, Example 00W3) and the map \(* \to G|_{(\Sch/T)_{fppf}}\) is given by \(y\). Note that if \(j : (\Sch/T)_{fppf} \to (\Sch/S)_{fppf}\) is the localization functor, then the formula above becomes \(F_y = j^{-1}F \times_{j^{-1}G} *\) and \(j_!F_y\) is just the fiber product \(F \times_{G, y} T\). (See Sites, Section 00XZ, for information on localization, and especially Sites, Remark 0494 for information on \(j_!\) for presheaves.)

At this point we temporarily have two definitions of what it means for a morphism \(X \to Y\) of algebraic spaces over \(S\) to be locally of finite presentation. Namely, one by Morphisms of Spaces, Definition 03XP and one using that \(X \to Y\) is a transformation of functors so that Definition 049J applies (we will use the terminology “limit preserving” for this notion as much as possible). We will show in Proposition 04AK that these two definitions agree.

Lemma

Let \(S\) be a scheme. Let \(a : F \to G\) be a transformation of functors \((\Sch/S)_{fppf}^{opp} \to \textit{Sets}\). The following are equivalent

1. \(a : F \to G\) is limit preserving, and

2. for every affine scheme \(T\) over \(S\) which is a limit \(T = \lim T_i\) of a directed inverse system of affine schemes \(T_i\) over \(S\) the diagram of sets \[\xymatrix{ \colim_i F(T_i) \ar[r] \ar[d]_a & F(T) \ar[d]^a \\ \colim_i G(T_i) \ar[r] & G(T) }\] is a fibre product diagram.

Proof

Assume (1). Consider \(T = \lim_{i \in I} T_i\) as in (2). Let \((y, x_T)\) be an element of the fibre product \(\colim_i G(T_i) \times_{G(T)} F(T)\). Then \(y\) comes from \(y_i \in G(T_i)\) for some \(i\). Consider the functor \(F_{y_i}\) on \((\Sch/T_i)_{fppf}\) as in Definition 049J. We see that \(x_T \in F_{y_i}(T)\). Moreover \(T = \lim_{i' \geq i} T_{i'}\) is a directed system of affine schemes over \(T_i\). Hence (1) implies that \(x_T\) the image of a unique element \(x\) of \(\colim_{i' \geq i} F_{y_i}(T_{i'})\). Thus \(x\) is the unique element of \(\colim F(T_i)\) which maps to the pair \((y, x_T)\). This proves that (2) holds.

Assume (2). Let \(T\) be a scheme and \(y_T \in G(T)\). We have to show that \(F_{y_T}\) is limit preserving. Let \(T' = \lim_{i \in I} T'_i\) be an affine scheme over \(T\) which is the directed limit of affine scheme \(T'_i\) over \(T\). Let \(x_{T'} \in F_{y_T}\). Pick \(i \in I\) which is possible as \(I\) is a directed set. Denote \(y_i \in F(T'_i)\) the image of \(y_{T'}\). Then we see that \((y_i, x_{T'})\) is an element of the fibre product \(\colim_i G(T'_i) \times_{G(T')} F(T')\). Hence by (2) we get a unique element \(x\) of \(\colim_i F(T'_i)\) mapping to \((y_i, x_{T'})\). It is clear that \(x\) defines an element of \(\colim_i F_y(T'_i)\) mapping to \(x_{T'}\) and we win.

Lemma

Let \(S\) be a scheme contained in \(\Sch_{fppf}\). Let \(F, G, H : (\Sch/S)_{fppf}^{opp} \to \textit{Sets}\). Let \(a : F \to G\), \(b : G \to H\) be transformations of functors. If \(a\) and \(b\) are limit preserving, then \[b \circ a : F \longrightarrow H\] is limit preserving.

Proof

Let \(T = \lim_{i \in I} T_i\) as in characterization (2) of Lemma 06BC. Consider the diagram of sets \[\xymatrix{ \colim_i F(T_i) \ar[r] \ar[d]_a & F(T) \ar[d]^a \\ \colim_i G(T_i) \ar[r] \ar[d]_b & G(T) \ar[d]^b \\ \colim_i H(T_i) \ar[r] & H(T) }\] By assumption the two squares are fibre product squares. Hence the outer rectangle is a fibre product diagram too which proves the lemma.

Lemma

Let \(S\) be a scheme contained in \(\Sch_{fppf}\). Let \(F, G, H : (\Sch/S)_{fppf}^{opp} \to \textit{Sets}\). Let \(a : F \to G\), \(b : G \to H\) be transformations of functors. If \(b \circ a\) and \(b\) are limit preserving, then \(a\) is limit preserving.

Proof

Let \(T = \lim_{i \in I} T_i\) as in characterization (2) of Lemma 06BC. Consider the diagram of sets \[\xymatrix{ \colim_i F(T_i) \ar[r] \ar[d]_a & F(T) \ar[d]^a \\ \colim_i G(T_i) \ar[r] \ar[d]_b & G(T) \ar[d]^b \\ \colim_i H(T_i) \ar[r] & H(T) }\] By assumption the lower square and the outer rectangle are fibre products of sets. Hence the upper square is a fibre product square too which proves the lemma.

Lemma

Let \(S\) be a scheme contained in \(\Sch_{fppf}\). Let \(F, G, H : (\Sch/S)_{fppf}^{opp} \to \textit{Sets}\). Let \(a : F \to G\), \(b : H \to G\) be transformations of functors. Consider the fibre product diagram \[\xymatrix{ H \times_{b, G, a} F \ar[r]_-{b'} \ar[d]_{a'} & F \ar[d]^a \\ H \ar[r]^b & G }\] If \(a\) is limit preserving, then the base change \(a'\) is limit preserving.

Proof

Omitted. Hint: This is formal.

Lemma

Let \(S\) be a scheme contained in \(\Sch_{fppf}\). Let \(E, F, G, H : (\Sch/S)_{fppf}^{opp} \to \textit{Sets}\). Let \(a : F \to G\), \(b : H \to G\), and \(c : G \to E\) be transformations of functors. If \(c\), \(c \circ a\), and \(c \circ b\) are limit preserving, then \(F \times_G H \to E\) is too.

Proof

Let \(T = \lim_{i \in I} T_i\) as in characterization (2) of Lemma 06BC. Then we have \[\colim (F \times_G H)(T_i) = \colim F(T_i) \times_{\colim G(T_i)} \colim H(T_i)\] as filtered colimits commute with finite products. Our goal is thus to show that \[\xymatrix{ \colim F(T_i) \times_{\colim G(T_i)} \colim H(T_i) \ar[r] \ar[d] & F(T) \times_{G(T)} H(T) \ar[d] \\ \colim_i E(T_i) \ar[r] & E(T) }\] is a fibre product diagram. This follows from the observation that given maps of sets \(E' \to E\), \(F \to G\), \(H \to G\), and \(G \to E\) we have \[E' \times_E (F \times_G H) = (E' \times_E F) \times_{(E' \times_E G)} (E' \times_E H)\] Some details omitted.

Lemma

Let \(S\) be a scheme contained in \(\Sch_{fppf}\). Let \(F : (\Sch/S)_{fppf}^{opp} \to \textit{Sets}\) be a functor. If \(F\) is limit preserving then its sheafification \(F^\#\) is limit preserving.

Proof

Assume \(F\) is limit preserving. It suffices to show that \(F^+\) is limit preserving, since \(F^\# = (F^+)^+\), see Sites, Theorem 00WB. Let \(T\) be an affine scheme over \(S\), and let \(T = \lim T_i\) be written as the directed limit of an inverse system of affine \(S\) schemes. Recall that \(F^+(T)\) is the colimit of \(\check H^0(\mathcal{V}, F)\) where the limit is over all coverings of \(T\) in \((\Sch/S)_{fppf}\). Any fppf covering of an affine scheme can be refined by a standard fppf covering, see Topologies, Lemma 021P. Hence we can write \[F^+(T) = \colim_{\mathcal{V}\text{ standard covering }T} \check H^0(\mathcal{V}, F).\] Any \(\mathcal{V} = \{T_k \to T\}_{k = 1, \ldots, n}\) in the colimit may be written as \(\mathcal{V}_i \times_{T_i} T\) for some \(i\) and some standard fppf covering \(\mathcal{V}_i = \{T_{i, k} \to T_i\}_{k = 1, \ldots, n}\) of \(T_i\). Denote \(\mathcal{V}_{i'} = \{T_{i', k} \to T_{i'}\}_{k = 1, \ldots, n}\) the base change for \(i' \geq i\). Then we see that \[\begin{align*} \colim_{i' \geq i} \check H^0(\mathcal{V}_i, F) & = \colim_{i' \geq i} \text{Equalizer} \left( \xymatrix{ \prod F(T_{i', k}) \ar@<1ex>[r] \ar@<-1ex>[r] & \prod F(T_{i', k} \times_{T_{i'}} T_{i', l}) } \right) \\ & = \text{Equalizer} \left( \xymatrix{ \colim_{i' \geq i} \prod F(T_{i', k}) \ar@<1ex>[r] \ar@<-1ex>[r] & \colim_{k' \geq k} \prod F(T_{i', k} \times_{T_{i'}} T_{i', l}) } \right) \\ & = \text{Equalizer} \left( \xymatrix{ \prod F(T_k) \ar@<1ex>[r] \ar@<-1ex>[r] & \prod F(T_k \times_T T_l) } \right) \\ & = \check H^0(\mathcal{V}, F) \end{align*}\] Here the second equality holds because filtered colimits are exact. The third equality holds because \(F\) is limit preserving and because \(\lim_{i' \geq i} T_{i', k} = T_k\) and \(\lim_{i' \geq i} T_{i', k} \times_{T_{i'}} T_{i', l} = T_k \times_T T_l\) by Limits, Lemma 01YZ. If we use this for all coverings at the same time we obtain \[\begin{align*} F^+(T) & = \colim_{\mathcal{V}\text{ standard covering }T} \check H^0(\mathcal{V}, F) \\ & = \colim_{i \in I} \colim_{\mathcal{V}_i\text{ standard covering }T_i} \check H^0(T \times_{T_i}\mathcal{V}_i, F) \\ & = \colim_{i \in I} F^+(T_i) \end{align*}\] The switch of the order of the colimits is allowed by Categories, Lemma 002M.

Lemma

Let \(S\) be a scheme. Let \(F : (\Sch/S)_{fppf}^{opp} \to \textit{Sets}\) be a functor. Assume that

1. \(F\) is a sheaf, and

2. there exists an fppf covering \(\{U_j \to S\}_{j \in J}\) such that \(F|_{(\Sch/U_j)_{fppf}}\) is limit preserving.

Then \(F\) is limit preserving.

Proof

Let \(T\) be an affine scheme over \(S\). Let \(I\) be a directed set, and let \(T_i\) be an inverse system of affine schemes over \(S\) such that \(T = \lim T_i\). We have to show that the canonical map \(\colim F(T_i) \to F(T)\) is bijective.

Choose some \(0 \in I\) and choose a standard fppf covering \(\{V_{0, k} \to T_{0}\}_{k = 1, \ldots, m}\) which refines the pullback \(\{U_j \times_S T_0 \to T_0\}\) of the given fppf covering of \(S\). For each \(i \geq 0\) we set \(V_{i, k} = T_i \times_{T_0} V_{0, k}\), and we set \(V_k = T \times_{T_0} V_{0, k}\). Note that \(V_k = \lim_{i \geq 0} V_{i, k}\), see Limits, Lemma 01YZ.

Suppose that \(x, x' \in \colim F(T_i)\) map to the same element of \(F(T)\). Say \(x, x'\) are given by elements \(x_i, x'_i \in F(T_i)\) for some \(i \in I\) (we may choose the same \(i\) for both as \(I\) is directed). By assumption (2) and the fact that \(x_i, x'_i\) map to the same element of \(F(T)\) this implies that \[x_i|_{V_{i', k}} = x'_i|_{V_{i', k}}\] for some suitably large \(i' \in I\). We can choose the same \(i'\) for each \(k\) as \(k \in \{1, \ldots, m\}\) ranges over a finite set. Since \(\{V_{i', k} \to T_{i'}\}\) is an fppf covering and \(F\) is a sheaf this implies that \(x_i|_{T_{i'}} = x'_i|_{T_{i'}}\) as desired. This proves that the map \(\colim F(T_i) \to F(T)\) is injective.

To show surjectivity we argue in a similar fashion. Let \(x \in F(T)\). By assumption (2) for each \(k\) we can choose a \(i\) such that \(x|_{V_k}\) comes from an element \(x_{i, k} \in F(V_{i, k})\). As before we may choose a single \(i\) which works for all \(k\). By the injectivity proved above we see that \[x_{i, k}|_{V_{i', k} \times_{T_{i'}} V_{i', l}} = x_{i, l}|_{V_{i', k} \times_{T_{i'}} V_{i', l}}\] for some large enough \(i'\). Hence by the sheaf condition of \(F\) the elements \(x_{i, k}|_{V_{i', k}}\) glue to an element \(x_{i'} \in F(T_{i'})\) as desired.

Lemma

Let \(S\) be a scheme contained in \(\Sch_{fppf}\). Let \(F, G : (\Sch/S)_{fppf}^{opp} \to \textit{Sets}\) be functors. If \(a : F \to G\) is a transformation which is limit preserving, then the induced transformation of sheaves \(F^\# \to G^\#\) is limit preserving.

Proof

Suppose that \(T\) is a scheme and \(y \in G^\#(T)\). We have to show the functor \(F^\#_y : (\Sch/T)_{fppf}^{opp} \to \textit{Sets}\) constructed from \(F^\# \to G^\#\) and \(y\) as in Definition 049J is limit preserving. By Equation (049K) we see that \(F^\#_y\) is a sheaf. Choose an fppf covering \(\{V_j \to T\}_{j \in J}\) such that \(y|_{V_j}\) comes from an element \(y_j \in F(V_j)\). Note that the restriction of \(F^\#\) to \((\Sch/V_j)_{fppf}\) is just \(F^\#_{y_j}\). If we can show that \(F^\#_{y_j}\) is limit preserving then Lemma 049P guarantees that \(F^\#_y\) is limit preserving and we win. This reduces us to the case \(y \in G(T)\).

Let \(y \in G(T)\). In this case we claim that \(F^\#_y = (F_y)^\#\). This follows from Equation (049K). Thus this case follows from Lemma 049O.

Proposition

Let \(S\) be a scheme. Let \(f : X \to Y\) be a morphism of algebraic spaces over \(S\). The following are equivalent:

1. The morphism \(f\) is a morphism of algebraic spaces which is locally of finite presentation, see Morphisms of Spaces, Definition 03XP.

2. The morphism \(f : X \to Y\) is limit preserving as a transformation of functors, see Definition 049J.

Proof

Assume (1). Let \(T\) be a scheme and let \(y \in Y(T)\). We have to show that \(T \times_Y X\) is limit preserving over \(T\) in the sense of Definition 049J. Hence we are reduced to proving that if \(X\) is an algebraic space which is locally of finite presentation over \(S\) as an algebraic space, then it is limit preserving as a functor \(X : (\Sch/S)_{fppf}^{opp} \to \textit{Sets}\). To see this choose a presentation \(X = U/R\), see Spaces, Definition 0263. It follows from Morphisms of Spaces, Definition 03XP that both \(U\) and \(R\) are schemes which are locally of finite presentation over \(S\). Hence by Limits, Proposition 01ZC we have \[U(T) = \colim U(T_i), \quad R(T) = \colim R(T_i)\] whenever \(T = \lim_i T_i\) in \((\Sch/S)_{fppf}\). It follows that the presheaf \[(\Sch/S)_{fppf}^{opp} \longrightarrow \textit{Sets}, \quad W \longmapsto U(W)/R(W)\] is limit preserving. Hence by Lemma 049O its sheafification \(X = U/R\) is limit preserving too.

Assume (2). Choose a scheme \(V\) and a surjective étale morphism \(V \to Y\). Next, choose a scheme \(U\) and a surjective étale morphism \(U \to V \times_Y X\). By Lemma 049M the transformation of functors \(V \times_Y X \to V\) is limit preserving. By Morphisms of Spaces, Lemma 0468 the morphism of algebraic spaces \(U \to V \times_Y X\) is locally of finite presentation, hence limit preserving as a transformation of functors by the first part of the proof. By Lemma 049L the composition \(U \to V \times_Y X \to V\) is limit preserving as a transformation of functors. Hence the morphism of schemes \(U \to V\) is locally of finite presentation by Limits, Proposition 01ZC (modulo a set theoretic remark, see last paragraph of the proof). This means, by definition, that (1) holds.

Set theoretic remark. Let \(U \to V\) be a morphism of \((\Sch/S)_{fppf}\). In the statement of Limits, Proposition 01ZC we characterize \(U \to V\) as being locally of finite presentation if for all directed inverse systems \((T_i, f_{ii'})\) of affine schemes over \(V\) we have \(U(T) = \colim V(T_i)\), but in the current setting we may only consider affine schemes \(T_i\) over \(V\) which are (isomorphic to) an object of \((\Sch/S)_{fppf}\). So we have to make sure that there are enough affines in \((\Sch/S)_{fppf}\) to make the proof work. Inspecting the proof of (2) \(\Rightarrow\) (1) of Limits, Proposition 01ZC we see that the question reduces to the case that \(U\) and \(V\) are affine. Say \(U = \Spec(A)\) and \(V = \Spec(B)\). By construction of \((\Sch/S)_{fppf}\) the spectrum of any ring of cardinality \(\leq |B|\) is isomorphic to an object of \((\Sch/S)_{fppf}\). Hence it suffices to observe that in the "only if" part of the proof of Algebra, Lemma 00QO only \(A\)-algebras of cardinality \(\leq |B|\) are used.

Remark

Here is an important special case of Proposition 04AK. Let \(S\) be a scheme. Let \(X\) be an algebraic space over \(S\). Then \(X\) is locally of finite presentation over \(S\) if and only if \(X\), as a functor \((\Sch/S)^{opp} \to \textit{Sets}\), is limit preserving. Compare with Limits, Remark 05LX. In fact, we will see in Lemma 0CM6 below that it suffices if the map \[\colim X(T_i) \longrightarrow X(T)\] is surjective whenever \(T = \lim T_i\) is a directed limit of affine schemes over \(S\).

Lemma

Let \(S\) be a scheme. Let \(f : X \to Y\) be a morphism of algebraic spaces over \(S\). If for every directed limit \(T = \lim_{i \in I} T_i\) of affine schemes over \(S\) the map \[\colim X(T_i) \longrightarrow X(T) \times_{Y(T)} \colim Y(T_i)\] is surjective, then \(f\) is locally of finite presentation. In other words, in Proposition 04AK part (2) it suffices to check surjectivity in the criterion of Lemma 06BC.

Proof

Choose a scheme \(V\) and a surjective étale morphism \(g : V \to Y\). Next, choose a scheme \(U\) and a surjective étale morphism \(h : U \to V \times_Y X\). It suffices to show for \(T = \lim T_i\) as in the lemma that the map \[\colim U(T_i) \longrightarrow U(T) \times_{V(T)} \colim V(T_i)\] is surjective, because then \(U \to V\) will be locally of finite presentation by Limits, Lemma 0CM0 (modulo a set theoretic remark exactly as in the proof of Proposition 04AK). Thus we take \(a : T \to U\) and \(b_i : T_i \to V\) which determine the same morphism \(T \to V\). Picture \[\xymatrix{ T \ar[d]_a \ar[rr]_{p_i} & & T_i \ar[d]^{b_i} \ar@{..>}[ld] \\ U \ar[r]^-h & X \times_Y V \ar[d] \ar[r] & V \ar[d]^g \\ & X \ar[r]^f & Y }\] By the assumption of the lemma after increasing \(i\) we can find a morphism \(c_i : T_i \to X\) such that \(h \circ a = (b_i, c_i) \circ p_i : T_i \to V \times_Y X\) and such that \(f \circ c_i = g \circ b_i\). Since \(h\) is an étale morphism of algebraic spaces (and hence locally of finite presentation), we have the surjectivity of \[\colim U(T_i) \longrightarrow U(T) \times_{(X \times_Y V)(T)} \colim (X \times_Y V)(T_i)\] by Proposition 04AK. Hence after increasing \(i\) again we can find the desired morphism \(a_i : T_i \to U\) with \(a = a_i \circ p_i\) and \(b_i = (U \to V) \circ a_i\).

Limits of algebraic spaces

The following lemma explains how we think of limits of algebraic spaces in this chapter. We will use (without further mention) that the base change of an affine morphism of algebraic spaces is affine (see Morphisms of Spaces, Lemma 03WI).

Lemma

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 \(X = \lim_i X_i\) (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 \[X \times_{X_i} T = \lim_{i' \geq i} X_{i'} \times_{X_i} T.\] as algebraic spaces over \(S\).

Proof

Part (2) is a formal consequence of the existence of the limit \(X = \lim X_i\) 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 01YX we see that \(U = \lim_{i \geq 0} U_i\) and \(R = \lim_{i \geq 0} R_i\) 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 [07SG]\[\begin{equation} U/R \longrightarrow \lim X_i \end{equation}\] is an isomorphism of fppf sheaves on the category of schemes over \(S\). The claim implies \(X = \lim X_i\) is an algebraic space by Spaces, Theorem 02WW.

Let \(Z\) be a scheme and let \(a : Z \to \lim X_i\) 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 \(W_0 \to \lim_{i \geq 0} U_i = U\). Since \(W_0 \to Z\) is surjective and étale, we conclude that (07SG) 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 (07SG). 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 (07SG) 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 \(R = \lim_{i \geq 0} R_i\) we see that \(c = \lim c_i : Z \to R\) 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.

Lemma

Let \(S\) be a scheme. Let \(I\) be a directed set. Let \((X_i, f_{ii'})\) be an inverse system over \(I\) of algebraic spaces over \(S\) with affine transition maps. Let \(X = \lim_i X_i\). Let \(0 \in I\). Suppose that \(T \to X_0\) is a morphism of algebraic spaces. Then \[T \times_{X_0} X = \lim_{i \geq 0} T \times_{X_0} X_i\] as algebraic spaces over \(S\).

Proof

The limit \(X\) is an algebraic space by Lemma 07SF. The equality is formal, see Categories, Lemma 002M.

Lemma

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 \(\lim X_i \to \lim Y_i\) is a closed immersion.

Proof

Observe that \(\lim X_i\) and \(\lim Y_i\) exist by Lemma 07SF. 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 \(V = \lim V_i\), \(U = \lim U_i\) 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.

Lemma

Let \(S\) be a scheme. Let \(I\) be a directed set. Let \((X_i, f_{i'i})\) be an inverse systems over \(I\) of algebraic spaces over \(S\). If \(X_i\) is reduced for all \(i\), then \(X\) is reduced.

Proof

Observe that \(\lim X_i\) exists by Lemma 07SF. Pick \(0 \in I\) and choose an affine scheme \(V_0\) and an étale morphism \(U_0 \to X_0\). Then the affine schemes \(U_i = X_i \times_{X_0} U_0\) are reduced. Hence \(U = X \times_{X_0} U_0\) is a reduced affine scheme as a limit of reduced affine schemes: a filtered colimit of reduced rings is reduced. Since the étale morphisms \(U \to X\) form an étale covering of \(X\) as we vary our choice of \(U_0 \to X_0\) we see that the lemma is true.

Lemma

Let \(S\) be a scheme. Let \(X \to Y\) be a morphism of algebraic spaces over \(S\). The equivalent conditions (1) and (2) of Proposition 04AK are also equivalent to

1. for every directed limit \(T = \lim T_i\) of quasi-compact and quasi-separated algebraic spaces \(T_i\) over \(S\) with affine transition morphisms the diagram of sets \[\xymatrix{ \colim_i \Mor(T_i, X) \ar[r] \ar[d] & \Mor(T, X) \ar[d] \\ \colim_i \Mor(T_i, Y) \ar[r] & \Mor(T, Y) }\] is a fibre product diagram.

Proof

It is clear that (3) implies (2). We will assume (2) and prove (3). The proof is rather formal and we encourage the reader to find their own proof.

Let us first prove that (3) holds when \(T_i\) is in addition assumed separated for all \(i\). Choose \(i \in I\) and choose a surjective étale morphism \(U_i \to T_i\) where \(U_i\) is affine. Using Lemma 07SH we see that with \(U = U_i \times_{T_i} T\) and \(U_{i'} = U_i \times_{T_i} T_{i'}\) we have \(U = \lim_{i' \geq i} U_{i'}\). Of course \(U\) and \(U_{i'}\) are affine (see Lemma 07SF). Since \(T_i\) is separated, the fibre product \(V_i = U_i \times_{T_i} U_i\) is an affine scheme as well and we obtain affine schemes \(V = V_i \times_{T_i} T\) and \(V_{i'} = V_i \times_{T_i} T_{i'}\) with \(V = \lim_{i' \geq i} V_{i'}\). Observe that \(U \to T\) and \(U_i \to T_i\) are surjective étale and that \(V = U \times_T U\) and \(V_{i'} = U_{i'} \times_{T_{i'}} U_{i'}\). Note that \(\Mor(T, X)\) is the equalizer of the two maps \(\Mor(U, X) \to \Mor(V, X)\); this is true for example because \(X\) as a sheaf on \((\Sch/S)_{fppf}\) is the coequalizer of the two maps \(h_V \to h_u\). Similarly \(\Mor(T_{i'}, X)\) is the equalizer of the two maps \(\Mor(U_{i'}, X) \to \Mor(V_{i'}, X)\). And of course the same thing is true with \(X\) replaced with \(Y\). Condition (2) says that the diagrams of in (3) are fibre products in the case of \(U = \lim U_i\) and \(V = \lim V_i\). It follows formally that the same thing is true for \(T = \lim T_i\).

In the general case, choose an affine scheme \(U\), an \(i \in I\), and a surjective étale morphism \(U \to T_i\). Repeating the argument of the previous paragraph we still achieve the proof: the schemes \(V_{i'}\), \(V\) are no longer affine, but they are still quasi-compact and separated and the result of the preceding paragraph applies.

Descending properties

This section is the analogue of Limits, Section 081A.

Lemma

Let \(S\) be a scheme. Let \(X = \lim_{i \in I} X_i\) be the limit of a directed inverse system of algebraic spaces over \(S\) with affine transition morphisms (Lemma 07SF). If each \(X_i\) is decent (for example quasi-separated or locally separated) then \(|X| = \lim_i |X_i|\) as sets.

Proof

There is a canonical map \(|X| \to \lim |X_i|\). Choose \(0 \in I\). If \(W_0 \subset X_0\) is an open subspace, then we have \(f_0^{-1}W_0 = \lim_{i \geq 0} f_{i0}^{-1}W_0\), see Lemma 07SF. 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 \(U = \lim U_i\) and \(R = \lim R_i\), see proof of Lemma 07SF. Recall that \(|X| = |U|/|R|\) and \(|X_i| = |U_i|/|R_i|\). By Limits, Lemma 01YY we have \(|U| = \lim |U_i|\) and \(|R| = \lim |R_i|\).

Surjectivity of \(|X| \to \lim |X_i|\). Let \((x_i) \in \lim |X_i|\). 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 03I8). Hence \(\lim S_i\) is nonempty, see Categories, Lemma 086J. Let \((u_i) \in \lim S_i \subset \lim |U_i|\). 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 \(\lim |X_i|\).

Injectivity of \(|X| \to \lim |X_i|\). Suppose that \(x, x' \in |X|\) map to the same point of \(\lim |X_i|\). 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 03I8). Moreover \(T_i\) is nonempty as we’ve assumed that \(x\) and \(x'\) map to the same point of \(X_i\). Hence \(\lim T_i\) is nonempty, see Categories, Lemma 086J. As before let \(r \in |R| = \lim |R_i|\) be a point corresponding to an element of \(\lim T_i\). 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 03I7 (the key observation to this is Properties of Spaces, Lemma 03IJ). A locally separated algebraic space is decent by Decent Spaces, Lemma 088J.

Lemma

With same notation and assumptions as in Lemma 0CUJ we have \(|X| = \lim_i |X_i|\) as topological spaces.

Proof

We will use the criterion of Topology, Lemma 0A2Q. We have seen that \(|X| = \lim_i |X_i|\) as sets in Lemma 0CUJ. 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 \(U = \lim_{i \geq 0} U_i\) in the category of schemes by Lemma 07SF. Choose \(u \in U\) mapping to \(x\). By the result for schemes (Limits, Lemma 0CUF) 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.

Lemma

Let \(S\) be a scheme. Let \(X = \lim_{i \in I} X_i\) be the limit of a directed inverse system of algebraic spaces over \(S\) with affine transition morphisms (Lemma 07SF). 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 \(U = \lim U_i\) is nonempty (Limits, Lemma 01Z2) and thus \(X\) is nonempty.

Lemma

Let \(S\) be a scheme. Let \(X = \lim_{i \in I} X_i\) be the limit of a directed inverse system of algebraic spaces over \(S\) with affine transition morphisms (Lemma 07SF). Let \(x \in |X|\) with images \(x_i \in |X_i|\). If each \(X_i\) is decent, then \(\overline{\{x\}} = \lim_i \overline{\{x_i\}}\) 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 \(Z \to \lim Z_i\) because \(|X| = \lim |X_i|\) as sets (Lemma 0CUJ). 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 \(|X| = \lim |X_i|\) as topological spaces (Lemma 086V) 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 0A2P. 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 \(Z = \lim Z_i\) as sets.

Next, endow \(Z\) and \(Z_i\) with their reduced induced scheme structures, see Properties of Spaces, Definition 047X. 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 \(Z \to \lim Z_i \to X\) of algebraic spaces. By Lemma 0CUH we see that \(\lim Z_i \to X\) is a closed immersion. By Lemma 0CUI the algebraic space \(\lim Z_i\) is reduced. By the above \(Z \to \lim Z_i\) is bijective on points. By uniqueness of the reduced induced closed subscheme structure we find that this morphism is an isomorphism of algebraic spaces.

Situation

Let \(S\) be a scheme. Let \(X = \lim_{i \in I} X_i\) be the limit of a directed inverse system of algebraic spaces over \(S\) with affine transition morphisms (Lemma 07SF). 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

Notation and assumptions as in Situation 084R. 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 \[\Gamma(X, \mathcal{F}) = \colim_{i \geq 0} \Gamma(X_i, \mathcal{F}_i)\]

Proof

Choose a surjective étale morphism \(U_0 \to X_0\) where \(U_0\) is an affine scheme (Properties of Spaces, Lemma 03H6). 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 07SF we have seen that there exists a presentation \(X = U/R\) with \(U = \lim U_i\) and \(R = \lim R_i\). 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 01Z0 implies that \[\mathcal{F}(U) = \colim \mathcal{F}_i(U_i) \quad\text{and}\quad \mathcal{F}(R) = \colim \mathcal{F}_i(R_i).\] The lemma follows as \(\Gamma(X, \mathcal{F}) = \Ker(\mathcal{F}(U) \to \mathcal{F}(R))\) and similarly \(\Gamma(X_i, \mathcal{F}_i) = \Ker(\mathcal{F}_i(U_i) \to \mathcal{F}_i(R_i))\)

Lemma

Notation and assumptions as in Situation 084R. 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 07SF and the corresponding result for schemes: Limits, Lemma 01Z4.

The following lemma will be superseded by the stronger Lemma 0852.

Lemma

Notation and assumptions as in Situation 084R. 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 081E 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).

Lemma

Notation and assumptions as in Situation 084R. 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 081B). Fix such an \(i \in I\). Apply Limits, Lemma 01ZP 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.

Lemma

Notation and assumptions as in Situation 084R. 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 \(X = \Spec(A)\), \(U = \Spec(B)\) and \[B = A[x_1, \ldots, x_n]/(g_1, \ldots, g_n)\] such that \(\Delta = \det(\partial g_\lambda/\partial x_\mu)\) is invertible in \(B\), see Algebra, Lemma 00U9. Set \(A_i = \mathcal{O}_{X_i}(X_i)\). We have \(A = \colim A_i\) by Lemma 07SI. 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 \[B_i = A_i[x_1, \ldots, x_n]/(g_{1, i}, \ldots, g_{n, i})\] 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 \(\Spec(B_i) \to \Spec(A_i)\) is surjective, see Limits, Lemma 07RR. 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 [084U]\[\begin{equation} \vcenter{ \xymatrix{ X_i \ar[d] & U_i \ar[l] \ar[d] \\ \Spec(A_i) & \Spec(B_i) \ar[l] } } \end{equation}\] By construction \(B_i = B_0 \otimes_{A_0} A_i\) and \(B = B_0 \otimes_{A_0} A\). Consider the morphism \[f_0 : U_0 \longrightarrow X_0 \times_{\Spec(A_0)} \Spec(B_0)\] 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 084S 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 (084U) is cartesian. Thus Descent, Lemma 02W5 applied to the fppf covering \(\{\Spec(B_i) \to \Spec(A_i)\}\) combined with Descent, Lemma 0245 give that \(X_i \to \Spec(A_i)\) is representable by a scheme affine over \(\Spec(A_i)\) as desired. (Of course it then also follows that \(X_i = \Spec(A_i)\) but we don’t need this.)

Lemma

Notation and assumptions as in Situation 084R. 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 0827 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 07SQ 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 03JG).

Lemma

Let \(S\) be a scheme. Let \(B\) be an algebraic space over \(S\). Let \(X = \lim X_i\) be a directed limit of algebraic spaces over \(B\) with affine transition morphisms. Let \(Y \to X\) be a morphism of algebraic spaces over \(B\).

1. 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.

2. 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.

3. 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.

4. 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 \(V \to U = \lim U_i\) over \(W\). This follows from the case of schemes, which is Limits, Lemma 081B.

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 0827 we may assume that \(U = \lim U_i\) 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 \(U = \lim U_i\) 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 05WT). Thus we’ve reduced to the following algebra fact: If \(A = \lim A_i\) 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 03KN. We have \(\lim Z_i = Y \times_X Y = Y\) 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 0462) the morphism \(Y \to Z_0\) is locally of finite presentation (Morphisms of Spaces, Lemma 084P). 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 \(Y = \lim_{i \geq 0} Z_i\) over \(Z_0\) and we conclude \(Y = Z_i\) for some \(i\). This proves (4) and the lemma.

Lemma

Let \(S\) be a scheme. Let \(Y\) be an algebraic space over \(S\). Let \(X = \lim X_i\) be a directed limit of algebraic spaces over \(Y\) with affine transition morphisms. Assume

1. \(Y\) is quasi-separated,

2. \(X_i\) is quasi-compact and quasi-separated,

3. 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 084T.

Lemma

Let \(S\) be a scheme. Let \(Y\) be an algebraic space over \(S\). Let \(X = \lim X_i\) be a directed limit of algebraic spaces over \(Y\) with affine transition morphisms. Assume

1. \(Y\) quasi-compact and quasi-separated,

2. \(X_i\) quasi-compact and quasi-separated,

3. \(X \to Y\) 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 03WG). This follows from Lemma 07SQ.

Lemma

Let \(S\) be a scheme. Let \(Y\) be an algebraic space over \(S\). Let \(X = \lim X_i\) be a directed limit of algebraic spaces over \(Y\) with affine transition morphisms. Assume

1. \(Y\) quasi-compact and quasi-separated,

2. \(X_i\) quasi-compact and quasi-separated,

3. the transition morphisms \(X_{i'} \to X_i\) are finite,

4. \(X_i \to Y\) locally of finite type

5. \(X \to Y\) 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 03ZQ). By Lemma 07SR this reduces us to the case of schemes. In the case of schemes it follows from Limits, Lemma 09ZN.

Lemma

Let \(S\) be a scheme. Let \(Y\) be an algebraic space over \(S\). Let \(X = \lim X_i\) be a directed limit of algebraic spaces over \(Y\) with affine transition morphisms. Assume

1. \(Y\) quasi-compact and quasi-separated,

2. \(X_i\) quasi-compact and quasi-separated,

3. the transition morphisms \(X_{i'} \to X_i\) are closed immersions,

4. \(X_i \to Y\) locally of finite type

5. \(X \to Y\) 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 03M4). By Lemma 07SR this reduces us to the case of schemes. In the case of schemes it follows from Limits, Lemma 0A0N.

Descending properties of morphisms

This section is the analogue of Section 0826 for properties of morphisms. We will work in the following situation.

Situation

Let \(S\) be a scheme. Let \(B = \lim B_i\) be a limit of a directed inverse system of algebraic spaces over \(S\) with affine transition morphisms (Lemma 07SF). Let \(0 \in I\) and let \(f_0 : X_0 \to Y_0\) be a morphism of algebraic spaces over \(B_0\). Assume \(B_0\), \(X_0\), \(Y_0\) are quasi-compact and quasi-separated. Let \(f_i : X_i \to Y_i\) be the base change of \(f_0\) to \(B_i\) and let \(f : X \to Y\) be the base change of \(f_0\) to \(B\).

Lemma

With notation and assumptions as in Situation 084W. If

1. \(f\) is étale,

2. \(f_0\) is locally of finite presentation,

then \(f_i\) is étale for some \(i \geq 0\).

Proof

Choose an affine scheme \(V_0\) and a surjective étale morphism \(V_0 \to Y_0\). Choose an affine scheme \(U_0\) and a surjective étale morphism \(U_0 \to V_0 \times_{Y_0} X_0\). Diagram \[\xymatrix{ U_0 \ar[d] \ar[r] & V_0 \ar[d] \\ X_0 \ar[r] & Y_0 }\] The vertical arrows are surjective and étale by construction. We can base change this diagram to \(B_i\) or \(B\) to get \[\vcenter{ \xymatrix{ U_i \ar[d] \ar[r] & V_i \ar[d] \\ X_i \ar[r] & Y_i } } \quad\text{and}\quad \vcenter{ \xymatrix{ U \ar[d] \ar[r] & V \ar[d] \\ X \ar[r] & Y } }\] Note that \(U_i, V_i, U, V\) are affine schemes, the vertical morphisms are surjective étale, and the limit of the morphisms \(U_i \to V_i\) is \(U \to V\). Recall that \(X_i \to Y_i\) is étale if and only if \(U_i \to V_i\) is étale and similarly \(X \to Y\) is étale if and only if \(U \to V\) is étale (Morphisms of Spaces, Lemma 03XT). Since \(f_0\) is locally of finite presentation, so is the morphism \(U_0 \to V_0\). Hence the lemma follows from Limits, Lemma 07RP.

Lemma

With notation and assumptions as in Situation 084W. If

1. \(f\) is smooth,

2. \(f_0\) is locally of finite presentation,

then \(f_i\) is smooth for some \(i \geq 0\).

Proof

Choose an affine scheme \(V_0\) and a surjective étale morphism \(V_0 \to Y_0\). Choose an affine scheme \(U_0\) and a surjective étale morphism \(U_0 \to V_0 \times_{Y_0} X_0\). Diagram \[\xymatrix{ U_0 \ar[d] \ar[r] & V_0 \ar[d] \\ X_0 \ar[r] & Y_0 }\] The vertical arrows are surjective and étale by construction. We can base change this diagram to \(B_i\) or \(B\) to get \[\vcenter{ \xymatrix{ U_i \ar[d] \ar[r] & V_i \ar[d] \\ X_i \ar[r] & Y_i } } \quad\text{and}\quad \vcenter{ \xymatrix{ U \ar[d] \ar[r] & V \ar[d] \\ X \ar[r] & Y } }\] Note that \(U_i, V_i, U, V\) are affine schemes, the vertical morphisms are surjective étale, and the limit of the morphisms \(U_i \to V_i\) is \(U \to V\). Recall that \(X_i \to Y_i\) is smooth if and only if \(U_i \to V_i\) is smooth and similarly \(X \to Y\) is smooth if and only if \(U \to V\) is smooth (Morphisms of Spaces, Definition 03ZC). Since \(f_0\) is locally of finite presentation, so is the morphism \(U_0 \to V_0\). Hence the lemma follows from Limits, Lemma 0C0C.

Lemma

With notation and assumptions as in Situation 084W. If

1. \(f\) is surjective,

2. \(f_0\) is locally of finite presentation,

then \(f_i\) is surjective for some \(i \geq 0\).

Proof

Choose an affine scheme \(V_0\) and a surjective étale morphism \(V_0 \to Y_0\). Choose an affine scheme \(U_0\) and a surjective étale morphism \(U_0 \to V_0 \times_{Y_0} X_0\). Diagram \[\xymatrix{ U_0 \ar[d] \ar[r] & V_0 \ar[d] \\ X_0 \ar[r] & Y_0 }\] The vertical arrows are surjective and étale by construction. We can base change this diagram to \(B_i\) or \(B\) to get \[\vcenter{ \xymatrix{ U_i \ar[d] \ar[r] & V_i \ar[d] \\ X_i \ar[r] & Y_i } } \quad\text{and}\quad \vcenter{ \xymatrix{ U \ar[d] \ar[r] & V \ar[d] \\ X \ar[r] & Y } }\] Note that \(U_i, V_i, U, V\) are affine schemes, the vertical morphisms are surjective étale, the limit of the morphisms \(U_i \to V_i\) is \(U \to V\), and the morphisms \(U_i \to X_i \times_{Y_i} V_i\) and \(U \to X \times_Y V\) are surjective (as base changes of \(U_0 \to X_0 \times_{Y_0} V_0\)). In particular, we see that \(X_i \to Y_i\) is surjective if and only if \(U_i \to V_i\) is surjective and similarly \(X \to Y\) is surjective if and only if \(U \to V\) is surjective. Since \(f_0\) is locally of finite presentation, so is the morphism \(U_0 \to V_0\). Hence the lemma follows from the case of schemes (Limits, Lemma 07RR).

Lemma

Notation and assumptions as in Situation 084W. If

1. \(f\) is universally injective,

2. \(f_0\) is locally of finite type,

then \(f_i\) is universally injective for some \(i \geq 0\).

Proof

Recall that a morphism \(X \to Y\) is universally injective if and only if the diagonal \(X \to X \times_Y X\) is surjective (Morphisms of Spaces, Definition 03MV and Lemma 040X). Observe that \(X_0 \to X_0 \times_{Y_0} X_0\) is of locally of finite presentation (Morphisms of Spaces, Lemma 084P). Hence the lemma follows from Lemma 07SN by considering the morphism \(X_0 \to X_0 \times_{Y_0} X_0\).

Lemma

Notation and assumptions as in Situation 084W. If \(f\) is affine, then \(f_i\) is affine for some \(i \geq 0\).

Proof

Choose an affine scheme \(V_0\) and a surjective étale morphism \(V_0 \to Y_0\). Set \(V_i = V_0 \times_{Y_0} Y_i\) and \(V = V_0 \times_{Y_0} Y\). Since \(f\) is affine we see that \(V \times_Y X = \lim V_i \times_{Y_i} X_i\) is affine. By Lemma 07SQ we see that \(V_i \times_{Y_i} X_i\) is affine for some \(i \geq 0\). For this \(i\) the morphism \(f_i\) is affine (Morphisms of Spaces, Lemma 03WG).

Lemma

Notation and assumptions as in Situation 084W. If

1. \(f\) is finite,

2. \(f_0\) is locally of finite type,

then \(f_i\) is finite for some \(i \geq 0\).

Proof

Choose an affine scheme \(V_0\) and a surjective étale morphism \(V_0 \to Y_0\). Set \(V_i = V_0 \times_{Y_0} Y_i\) and \(V = V_0 \times_{Y_0} Y\). Since \(f\) is finite we see that \(V \times_Y X = \lim V_i \times_{Y_i} X_i\) is a scheme finite over \(V\). By Lemma 07SQ we see that \(V_i \times_{Y_i} X_i\) is affine for some \(i \geq 0\). Increasing \(i\) if necessary we find that \(V_i \times_{Y_i} X_i \to V_i\) is finite by Limits, Lemma 01ZO. For this \(i\) the morphism \(f_i\) is finite (Morphisms of Spaces, Lemma 03ZQ).

Lemma

Notation and assumptions as in Situation 084W. If

1. \(f\) is a closed immersion,

2. \(f_0\) is locally of finite type,

then \(f_i\) is a closed immersion for some \(i \geq 0\).

Proof

Choose an affine scheme \(V_0\) and a surjective étale morphism \(V_0 \to Y_0\). Set \(V_i = V_0 \times_{Y_0} Y_i\) and \(V = V_0 \times_{Y_0} Y\). Since \(f\) is a closed immersion we see that \(V \times_Y X = \lim V_i \times_{Y_i} X_i\) is a closed subscheme of the affine scheme \(V\). By Lemma 07SQ we see that \(V_i \times_{Y_i} X_i\) is affine for some \(i \geq 0\). Increasing \(i\) if necessary we find that \(V_i \times_{Y_i} X_i \to V_i\) is a closed immersion by Limits, Lemma 01ZP. For this \(i\) the morphism \(f_i\) is a closed immersion (Morphisms of Spaces, Lemma 03ZQ).

Lemma

Notation and assumptions as in Situation 084W. If \(f\) is separated, then \(f_i\) is separated for some \(i \geq 0\).

Proof

Apply Lemma 0850 to the diagonal morphism \(\Delta_{X_0/Y_0} : X_0 \to X_0 \times_{Y_0} X_0\). (Diagonal morphisms are locally of finite type and the fibre product \(X_0 \times_{Y_0} X_0\) is quasi-compact and quasi-separated. Some details omitted.)

Lemma

Notation and assumptions as in Situation 084W. If

1. \(f\) is an isomorphism,

2. \(f_0\) is locally of finite presentation,

then \(f_i\) is an isomorphism for some \(i \geq 0\).

Proof

Being an isomorphism is equivalent to being étale, universally injective, and surjective, see Morphisms of Spaces, Lemma 05W5. Thus the lemma follows from Lemmas 07SL, 07SN, and 084X.

Lemma

Notation and assumptions as in Situation 084W. If

1. \(f\) is a monomorphism,

2. \(f_0\) is locally of finite type,

then \(f_i\) is a monomorphism for some \(i \geq 0\).

Proof

Recall that a morphism is a monomorphism if and only if the diagonal is an isomorphism. The morphism \(X_0 \to X_0 \times_{Y_0} X_0\) is locally of finite presentation by Morphisms of Spaces, Lemma 084P. Since \(X_0 \times_{Y_0} X_0\) is quasi-compact and quasi-separated we conclude from Lemma 0852 that \(\Delta_i : X_i \to X_i \times_{Y_i} X_i\) is an isomorphism for some \(i \geq 0\). For this \(i\) the morphism \(f_i\) is a monomorphism.

Lemma

Notation and assumptions as in Situation 084W. Let \(\mathcal{F}_0\) be a quasi-coherent \(\mathcal{O}_{X_0}\)-module and denote \(\mathcal{F}_i\) the pullback to \(X_i\) and \(\mathcal{F}\) the pullback to \(X\). If

1. \(\mathcal{F}\) is flat over \(Y\),

2. \(\mathcal{F}_0\) is of finite presentation, and

3. \(f_0\) is locally of finite presentation,

then \(\mathcal{F}_i\) is flat over \(Y_i\) for some \(i \geq 0\). In particular, if \(f_0\) is locally of finite presentation and \(f\) is flat, then \(f_i\) is flat for some \(i \geq 0\).

Proof

Choose an affine scheme \(V_0\) and a surjective étale morphism \(V_0 \to Y_0\). Choose an affine scheme \(U_0\) and a surjective étale morphism \(U_0 \to V_0 \times_{Y_0} X_0\). Diagram \[\xymatrix{ U_0 \ar[d] \ar[r] & V_0 \ar[d] \\ X_0 \ar[r] & Y_0 }\] The vertical arrows are surjective and étale by construction. We can base change this diagram to \(B_i\) or \(B\) to get \[\vcenter{ \xymatrix{ U_i \ar[d] \ar[r] & V_i \ar[d] \\ X_i \ar[r] & Y_i } } \quad\text{and}\quad \vcenter{ \xymatrix{ U \ar[d] \ar[r] & V \ar[d] \\ X \ar[r] & Y } }\] Note that \(U_i, V_i, U, V\) are affine schemes, the vertical morphisms are surjective étale, and the limit of the morphisms \(U_i \to V_i\) is \(U \to V\). Recall that \(\mathcal{F}_i\) is flat over \(Y_i\) if and only if \(\mathcal{F}_i|_{U_i}\) is flat over \(V_i\) and similarly \(\mathcal{F}\) is flat over \(Y\) if and only if \(\mathcal{F}|_U\) is flat over \(V\) (Morphisms of Spaces, Definition 03ML). Since \(f_0\) is locally of finite presentation, so is the morphism \(U_0 \to V_0\). Hence the lemma follows from Limits, Lemma 05LY.

Lemma

Assumptions and notation as in Situation 084W. If

1. \(f\) is proper, and

2. \(f_0\) is locally of finite type,

then there exists an \(i\) such that \(f_i\) is proper.

Proof

Choose an affine scheme \(V_0\) and a surjective étale morphism \(V_0 \to Y_0\). Set \(V_i = Y_i \times_{Y_0} V_0\) and \(V = Y \times_{Y_0} V_0\). It suffices to prove that the base change of \(f_i\) to \(V_i\) is proper, see Morphisms of Spaces, Lemma 083R. Thus we may assume \(Y_0\) is affine.

By Lemma 0851 we see that \(f_i\) is separated for some \(i \geq 0\). Replacing \(0\) by \(i\) we may assume that \(f_0\) is separated. Observe that \(f_0\) is quasi-compact. Thus \(f_0\) is separated and of finite type. By Cohomology of Spaces, Lemma 089J we can choose a diagram \[\xymatrix{ X_0 \ar[rd] & X_0' \ar[d] \ar[l]^\pi \ar[r] & \mathbf{P}^n_{Y_0} \ar[dl] \\ & Y_0 & }\] where \(X_0' \to \mathbf{P}^n_{Y_0}\) is an immersion, and \(\pi : X_0' \to X_0\) is proper and surjective. Introduce \(X' = X_0' \times_{Y_0} Y\) and \(X_i' = X_0' \times_{Y_0} Y_i\). By Morphisms of Spaces, Lemmas 04XY and 04WP we see that \(X' \to Y\) is proper. Hence \(X' \to \mathbf{P}^n_Y\) is a closed immersion (Morphisms of Spaces, Lemma 04NX). By Morphisms of Spaces, Lemma 08AJ it suffices to prove that \(X'_i \to Y_i\) is proper for some \(i\). By Lemma 0850 we find that \(X'_i \to \mathbf{P}^n_{Y_i}\) is a closed immersion for \(i\) large enough. Then \(X'_i \to Y_i\) is proper and we win.

Lemma

Assumptions and notation as in Situation 084W. Let \(d \geq 0\). If

1. \(f\) has relative dimension \(\leq d\) (Morphisms of Spaces, Definition 06LR), and

2. \(f_0\) is locally of finite type,

then there exists an \(i\) such that \(f_i\) has relative dimension \(\leq d\).

Proof

Choose an affine scheme \(V_0\) and a surjective étale morphism \(V_0 \to Y_0\). Choose an affine scheme \(U_0\) and a surjective étale morphism \(U_0 \to V_0 \times_{Y_0} X_0\). Diagram \[\xymatrix{ U_0 \ar[d] \ar[r] & V_0 \ar[d] \\ X_0 \ar[r] & Y_0 }\] The vertical arrows are surjective and étale by construction. We can base change this diagram to \(B_i\) or \(B\) to get \[\vcenter{ \xymatrix{ U_i \ar[d] \ar[r] & V_i \ar[d] \\ X_i \ar[r] & Y_i } } \quad\text{and}\quad \vcenter{ \xymatrix{ U \ar[d] \ar[r] & V \ar[d] \\ X \ar[r] & Y } }\] Note that \(U_i, V_i, U, V\) are affine schemes, the vertical morphisms are surjective étale, and the limit of the morphisms \(U_i \to V_i\) is \(U \to V\). In this situation \(X_i \to Y_i\) has relative dimension \(\leq d\) if and only if \(U_i \to V_i\) has relative dimension \(\leq d\) (as defined in Morphisms, Definition 02NJ). To see the equivalence, use that the definition for morphisms of algebraic spaces involves Morphisms of Spaces, Definition 04NM which uses étale localization. The same is true for \(X \to Y\) and \(U \to V\). Since \(f_0\) is locally of finite type, so is the morphism \(U_0 \to V_0\). Hence the lemma follows from the more general Limits, Lemma 05M5.

Descending relative objects

The following lemma is typical of the type of results in this section.

Lemma

Let \(S\) be a scheme. Let \(I\) be a directed set. Let \((X_i, f_{ii'})\) be an inverse system over \(I\) of algebraic spaces over \(S\). Assume

1. the morphisms \(f_{ii'} : X_i \to X_{i'}\) are affine,

2. the spaces \(X_i\) are quasi-compact and quasi-separated.

Let \(X = \lim_i X_i\). Then the category of algebraic spaces of finite presentation over \(X\) is the colimit over \(I\) of the categories of algebraic spaces of finite presentation over \(X_i\).

Proof

Pick \(0 \in I\). Choose a surjective étale morphism \(U_0 \to X_0\) where \(U_0\) is an affine scheme (Properties of Spaces, Lemma 03H6). Set \(U_i = X_i \times_{X_0} U_0\). Set \(R_0 = U_0 \times_{X_0} U_0\) and \(R_i = R_0 \times_{X_0} X_i\). Denote \(s_i, t_i : R_i \to U_i\) and \(s, t : R \to U\) the two projections. In the proof of Lemma 07SF we have seen that there exists a presentation \(X = U/R\) with \(U = \lim U_i\) and \(R = \lim R_i\). 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). Let \(Y\) be an algebraic space over \(S\) and let \(Y \to X\) be a morphism of finite presentation. Set \(V = U \times_X Y\). This is an algebraic space of finite presentation over \(U\). Choose an affine scheme \(W\) and a surjective étale morphism \(W \to V\). Then \(W \to Y\) is surjective étale as well. Set \(R' = W \times_Y W\) so that \(Y = W/R'\) (see Spaces, Section 0261). Note that \(W\) is a scheme of finite presentation over \(U\) and that \(R'\) is a scheme of finite presentation over \(R\) (details omitted). By Limits, Lemma 01ZM we can find an index \(i\) and a morphism of schemes \(W_i \to U_i\) of finite presentation whose base change to \(U\) gives \(W \to U\). Similarly we can find, after possibly increasing \(i\), a scheme \(R'_i\) of finite presentation over \(R_i\) whose base change to \(R\) is \(R'\). The projection morphisms \(s', t' : R' \to W\) are morphisms over the projection morphisms \(s, t : R \to U\). Hence we can view \(s'\), resp. \(t'\) as a morphism between schemes of finite presentation over \(U\) (with structure morphism \(R' \to U\) given by \(R' \to R\) followed by \(s\), resp. \(t\)). Hence we can apply Limits, Lemma 01ZM again to see that, after possibly increasing \(i\), there exist morphisms \(s'_i, t'_i : R'_i \to W_i\), whose base change to \(U\) is \(S', t'\). By Limits, Lemmas 07RP and 07RQ we may assume that \(s'_i, t'_i\) are étale and that \(j'_i : R'_i \to W_i \times_{X_i} W_i\) is a monomorphism (here we view \(j'_i\) as a morphism of schemes of finite presentation over \(U_i\) via one of the projections – it doesn’t matter which one). Setting \(Y_i = W_i/R'_i\) (see Spaces, Theorem 02WW) we obtain an algebraic space of finite presentation over \(X_i\) whose base change to \(X\) is isomorphic to \(Y\).

This shows that every algebraic space of finite presentation over \(X\) comes from an algebraic space of finite presentation over some \(X_i\), i.e., it shows that the functor of the lemma is essentially surjective. To show that it is fully faithful, consider an index \(0 \in I\) and two algebraic spaces \(Y_0, Z_0\) of finite presentation over \(X_0\). Set \(Y_i = X_i \times_{X_0} Y_0\), \(Y = X \times_{X_0} Y_0\), \(Z_i = X_i \times_{X_0} Z_0\), and \(Z = X \times_{X_0} Z_0\). Let \(\alpha : Y \to Z\) be a morphism of algebraic spaces over \(X\). Choose a surjective étale morphism \(V_0 \to Y_0\) where \(V_0\) is an affine scheme. Set \(V_i = V_0 \times_{Y_0} Y_i\) and \(V = V_0 \times_{Y_0} Y\) which are affine schemes endowed with surjective étale morphisms to \(Y_i\) and \(Y\). The composition \(V \to Y \to Z \to Z_0\) comes from a (essentially unique) morphism \(V_i \to Z_0\) for some \(i \geq 0\) by Proposition 04AK (applied to \(Z_0 \to X_0\) which is of finite presentation by assumption). After increasing \(i\) the two compositions \[V_i \times_{Y_i} V_i \to V_i \to Z_0\] are equal as this is true in the limit. Hence we obtain a (essentially unique) morphism \(Y_i \to Z_0\). Since this is a morphism over \(X_0\) it induces a morphism into \(Z_i = Z_0 \times_{X_0} X_i\) as desired.

Lemma

With notation and assumptions as in Lemma 07SK. The category of \(\mathcal{O}_X\)-modules of finite presentation is the colimit over \(I\) of the categories \(\mathcal{O}_{X_i}\)-modules of finite presentation.

Proof

Choose \(0 \in I\). Choose an affine scheme \(U_0\) and a surjective étale morphism \(U_0 \to X_0\). Set \(U_i = X_i \times_{X_0} U_0\). Set \(R_0 = U_0 \times_{X_0} U_0\) and \(R_i = R_0 \times_{X_0} X_i\). Denote \(s_i, t_i : R_i \to U_i\) and \(s, t : R \to U\) the two projections. In the proof of Lemma 07SF we have seen that there exists a presentation \(X = U/R\) with \(U = \lim U_i\) and \(R = \lim R_i\). 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). Moreover, it is also true that \(R \times_{s, U, t} R = \colim R_i \times_{s_i, U_i, t_i} R_i\). Thus we know that \(\QCoh(\mathcal{O}_U) = \colim \QCoh(\mathcal{O}_{U_i})\), \(\QCoh(\mathcal{O}_R) = \colim \QCoh(\mathcal{O}_{R_i})\), and \(\QCoh(\mathcal{O}_{R \times_{s, U, t} R}) = \colim \QCoh(\mathcal{O}_{R_i \times_{s_i, U_i, t_i} R_i})\) by Limits, Lemma 01ZR. We have \(\QCoh(\mathcal{O}_X) = \QCoh(U, R, s, t, c)\) and \(\QCoh(\mathcal{O}_{X_i}) = \QCoh(U_i, R_i, s_i, t_i, c_i)\), see Properties of Spaces, Proposition 03M3. Thus the result follows formally.

Lemma

With notation and assumptions as in Lemma 07SK. Then

1. any finite locally free \(\mathcal{O}_X\)-module is the pullback of a finite locally free \(\mathcal{O}_{X_i}\)-module for some \(i\),

2. any invertible \(\mathcal{O}_X\)-module is the pullback of an invertible \(\mathcal{O}_{X_i}\)-module for some \(i\).

Proof

Proof of (2). Let \(\mathcal{L}\) be an invertible \(\mathcal{O}_X\)-module. Since invertible modules are of finite presentation we can find an \(i\) and modules \(\mathcal{L}_i\) and \(\mathcal{N}_i\) of finite presentation over \(X_i\) such that \(f_i^*\mathcal{L}_i \cong \mathcal{L}\) and \(f_i^*\mathcal{N}_i \cong \mathcal{L}^{\otimes -1}\), see Lemma 07V7. Since pullback commutes with tensor product we see that \(f_i^*(\mathcal{L}_i \otimes_{\mathcal{O}_{X_i}} \mathcal{N}_i)\) is isomorphic to \(\mathcal{O}_X\). Since the tensor product of finitely presented modules is finitely presented, the same lemma implies that \(f_{i'i}^*\mathcal{L}_i \otimes_{\mathcal{O}_{X_{i'}}} f_{i'i}^*\mathcal{N}_i\) is isomorphic to \(\mathcal{O}_{X_{i'}}\) for some \(i' \geq i\). It follows that \(f_{i'i}^*\mathcal{L}_i\) is invertible (Modules on Sites, Lemma 0B8N) and the proof is complete.

Proof of (1). Omitted. Hint: argue as in the proof of (2) using that a module (on a locally ringed site) is finite locally free if and only if it has a dual, see Modules on Sites, Section 0FNX. Alternatively, argue as in the proof for schemes, see Limits, Lemma 0B8W.

Absolute Noetherian approximation

The following result is [CLO, Theorem 1.2.2]. A key ingredient in the proof is Decent Spaces, Lemma 07ST.

Proposition

Let \(X\) be a quasi-compact and quasi-separated algebraic space over \(\Spec(\mathbf{Z})\). There exist a directed set \(I\) and an inverse system of algebraic spaces \((X_i, f_{ii'})\) over \(I\) such that

1. the transition morphisms \(f_{ii'}\) are affine

2. each \(X_i\) is quasi-separated and of finite type over \(\mathbf{Z}\), and

3. \(X = \lim X_i\).

Proof

We apply Decent Spaces, Lemma 07ST to get open subspaces \(U_p \subset X\), schemes \(V_p\), and morphisms \(f_p : V_p \to U_p\) with properties as stated. Note that \(f_n : V_n \to U_n\) is an étale morphism of algebraic spaces whose restriction to the inverse image of \(T_n = (V_n)_{red}\) is an isomorphism. Hence \(f_n\) is an isomorphism, for example by Morphisms of Spaces, Lemma 05W5. In particular \(U_n\) is a quasi-compact and separated scheme. Thus we can write \(U_n = \lim U_{n, i}\) as a directed limit of schemes of finite type over \(\mathbf{Z}\) with affine transition morphisms, see Limits, Proposition 01ZA. Thus, applying descending induction on \(p\), we see that we have reduced to the problem posed in the following paragraph.

Here we have \(U \subset X\), \(U = \lim U_i\), \(Z \subset X\), and \(f : V \to X\) with the following properties

1. \(X\) is a quasi-compact and quasi-separated algebraic space,

2. \(V\) is a quasi-compact and separated scheme,

3. \(U \subset X\) is a quasi-compact open subspace,

4. \((U_i, g_{ii'})\) is a directed inverse system of quasi-separated algebraic spaces of finite type over \(\mathbf{Z}\) with affine transition morphisms whose limit is \(U\),

5. \(Z \subset X\) is a closed subspace such that \(|X| = |U| \amalg |Z|\),

6. \(f : V \to X\) is a surjective étale morphism such that \(f^{-1}(Z) \to Z\) is an isomorphism.

Problem: Show that the conclusion of the proposition holds for \(X\).

Note that \(W = f^{-1}(U) \subset V\) is a quasi-compact open subscheme étale over \(U\). Hence we may apply Lemmas 07SK and 07SL to find an index \(0 \in I\) and an étale morphism \(W_0 \to U_0\) of finite presentation whose base change to \(U\) produces \(W\). Setting \(W_i = W_0 \times_{U_0} U_i\) we see that \(W = \lim_{i \geq 0} W_i\). After increasing \(0\) we may assume the \(W_i\) are schemes, see Lemma 07SR. Moreover, \(W_i\) is of finite type over \(\mathbf{Z}\).

Apply Limits, Lemma 07RN to \(W = \lim_{i \geq 0} W_i\) and the inclusion \(W \subset V\). Replace \(I\) by the directed set \(J\) found in that lemma. This allows us to write \(V\) as a directed limit \(V = \lim V_i\) of finite type schemes over \(\mathbf{Z}\) with affine transition maps such that each \(V_i\) contains \(W_i\) as an open subscheme (compatible with transition morphisms). For each \(i\) we can form the push out \[\xymatrix{ W_i \ar[r] \ar[d]_\Delta & V_i \ar[d] \\ W_i \times_{U_i} W_i \ar[r] & R_i }\] in the category of schemes. Namely, the left vertical and upper horizontal arrows are open immersions of schemes. In other words, we can construct \(R_i\) as the glueing of \(V_i\) and \(W_i \times_{U_i} W_i\) along the common open \(W_i\) (see Schemes, Section 01JA). Note that the étale projection maps \(W_i \times_{U_i} W_i \to W_i\) extend to étale morphisms \(s_i, t_i : R_i \to V_i\). It is clear that the morphism \(j_i = (t_i, s_i) : R_i \to V_i \times V_i\) is an étale equivalence relation on \(V_i\). Note that \(W_i \times_{U_i} W_i\) is quasi-compact (as \(U_i\) is quasi-separated and \(W_i\) quasi-compact) and \(V_i\) is quasi-compact, hence \(R_i\) is quasi-compact. For \(i \geq i'\) the diagram [07SV]\[\begin{equation} \vcenter{ \xymatrix{ R_i \ar[r] \ar[d]_{s_i} & R_{i'} \ar[d]^{s_{i'}} \\ V_i \ar[r] & V_{i'} } } \end{equation}\] is cartesian because \[(W_{i'} \times_{U_{i'}} W_{i'}) \times_{U_{i'}} U_i = W_{i'} \times_{U_{i'}} U_i \times_{U_i} U_i \times_{U_{i'}} W_{i'} = W_i \times_{U_i} W_i.\] Consider the algebraic space \(X_i = V_i/R_i\) (see Spaces, Theorem 02WW). As \(V_i\) is of finite type over \(\mathbf{Z}\) and \(R_i\) is quasi-compact we see that \(X_i\) is quasi-separated and of finite type over \(\mathbf{Z}\) (see Properties of Spaces, Lemma 07S4 and Morphisms of Spaces, Lemmas 040W and 040Y). As the construction of \(R_i\) above is compatible with transition morphisms, we obtain morphisms of algebraic spaces \(X_i \to X_{i'}\) for \(i \geq i'\). The commutative diagrams \[\xymatrix{ V_i \ar[r] \ar[d] & V_{i'} \ar[d] \\ X_i \ar[r] & X_{i'} }\] are cartesian as (07SV) is cartesian, see Groupoids, Lemma 07S3. Since \(V_i \to V_{i'}\) is affine, this implies that \(X_i \to X_{i'}\) is affine, see Morphisms of Spaces, Lemma 03WG. Thus we can form the limit \(X' = \lim X_i\) by Lemma 07SF. We claim that \(X \cong X'\) which finishes the proof of the proposition.

Proof of the claim. Set \(R = \lim R_i\). By construction the algebraic space \(X'\) comes equipped with a surjective étale morphism \(V \to X'\) such that \[V \times_{X'} V \cong R\] (use Lemma 07SF). By construction \(\lim W_i \times_{U_i} W_i = W \times_U W\) and \(V = \lim V_i\) so that \(R\) is the union of \(W \times_U W\) and \(V\) glued along \(W\). Property (6) implies the projections \(V \times_X V \to V\) are isomorphisms over \(f^{-1}(Z) \subset V\). Hence the scheme \(V \times_X V\) is the union of the opens \(\Delta_{V/X}(V)\) and \(W \times_U W\) which intersect along \(\Delta_{W/X}(W)\). We conclude that there exists a unique isomorphism \(R \cong V \times_X V\) compatible with the projections to \(V\). Since \(V \to X\) and \(V \to X'\) are surjective étale we see that \[X = V/ V \times_X V = V/R = V/V \times_{X'} V = X'\] by Spaces, Lemma 0262 and we win.

Applications

The following lemma can also be deduced directly from Decent Spaces, Lemma 07ST without passing through absolute Noetherian approximation.

Lemma

Let \(S\) be a scheme. Let \(X\) be a quasi-compact and quasi-separated algebraic space over \(S\). Every quasi-coherent \(\mathcal{O}_X\)-module is a filtered colimit of finitely presented \(\mathcal{O}_X\)-modules.

Proof

We may view \(X\) as an algebraic space over \(\Spec(\mathbf{Z})\), see Spaces, Definition 03I5 and Properties of Spaces, Definition 03BS. Thus we may apply Proposition 07SU and write \(X = \lim X_i\) with \(X_i\) of finite presentation over \(\mathbf{Z}\). Thus \(X_i\) is a Noetherian algebraic space, see Morphisms of Spaces, Lemma 04ZL. The morphism \(X \to X_i\) is affine, see Lemma 07SF. Conclusion by Cohomology of Spaces, Lemma 07UW.

The rest of this section consists of straightforward applications of Lemma 07V9.

Lemma

Let \(S\) be a scheme. Let \(X\) be a quasi-compact and quasi-separated algebraic space over \(S\). Let \(\mathcal{F}\) be a quasi-coherent \(\mathcal{O}_X\)-module. Then \(\mathcal{F}\) is the directed colimit of its finite type quasi-coherent submodules.

Proof

If \(\mathcal{G}, \mathcal{H} \subset \mathcal{F}\) are finite type quasi-coherent \(\mathcal{O}_X\)-submodules then the image of \(\mathcal{G} \oplus \mathcal{H} \to \mathcal{F}\) is another finite type quasi-coherent \(\mathcal{O}_X\)-submodule which contains both of them. In this way we see that the system is directed. To show that \(\mathcal{F}\) is the colimit of this system, write \(\mathcal{F} = \colim_i \mathcal{F}_i\) as a directed colimit of finitely presented quasi-coherent sheaves as in Lemma 07V9. Then the images \(\mathcal{G}_i = \Im(\mathcal{F}_i \to \mathcal{F})\) are finite type quasi-coherent subsheaves of \(\mathcal{F}\). Since \(\mathcal{F}\) is the colimit of these the result follows.

Lemma

Let \(S\) be a scheme. Let \(X\) be a quasi-compact and quasi-separated algebraic space over \(S\). Let \(\mathcal{F}\) be a finite type quasi-coherent \(\mathcal{O}_X\)-module. Then we can write \(\mathcal{F} = \lim \mathcal{F}_i\) where each \(\mathcal{F}_i\) is an \(\mathcal{O}_X\)-module of finite presentation and all transition maps \(\mathcal{F}_i \to \mathcal{F}_{i'}\) surjective.

Proof

Write \(\mathcal{F} = \colim \mathcal{G}_i\) as a filtered colimit of finitely presented \(\mathcal{O}_X\)-modules (Lemma 07V9). We claim that \(\mathcal{G}_i \to \mathcal{F}\) is surjective for some \(i\). Namely, choose an étale surjection \(U \to X\) where \(U\) is an affine scheme. Choose finitely many sections \(s_k \in \mathcal{F}(U)\) generating \(\mathcal{F}|_U\). Since \(U\) is affine we see that \(s_k\) is in the image of \(\mathcal{G}_i \to \mathcal{F}\) for \(i\) large enough. Hence \(\mathcal{G}_i \to \mathcal{F}\) is surjective for \(i\) large enough. Choose such an \(i\) and let \(\mathcal{K} \subset \mathcal{G}_i\) be the kernel of the map \(\mathcal{G}_i \to \mathcal{F}\). Write \(\mathcal{K} = \colim \mathcal{K}_a\) as the filtered colimit of its finite type quasi-coherent submodules (Lemma 0829). Then \(\mathcal{F} = \colim \mathcal{G}_i/\mathcal{K}_a\) is a solution to the problem posed by the lemma.

Let \(X\) be an algebraic space. In the following lemma we use the notion of a finitely presented quasi-coherent \(\mathcal{O}_X\)-algebra \(\mathcal{A}\). This means that for every affine \(U = \Spec(R)\) étale over \(X\) we have \(\mathcal{A}|_U = \widetilde{A}\) where \(A\) is a (commutative) \(R\)-algebra which is of finite presentation as an \(R\)-algebra.

Lemma

Let \(S\) be a scheme. Let \(X\) be a quasi-compact and quasi-separated algebraic space over \(S\). Let \(\mathcal{A}\) be a quasi-coherent \(\mathcal{O}_X\)-algebra. Then \(\mathcal{A}\) is a directed colimit of finitely presented quasi-coherent \(\mathcal{O}_X\)-algebras.

Proof

First we write \(\mathcal{A} = \colim_i \mathcal{F}_i\) as a directed colimit of finitely presented quasi-coherent sheaves as in Lemma 07V9. For each \(i\) let \(\mathcal{B}_i = \text{Sym}(\mathcal{F}_i)\) be the symmetric algebra on \(\mathcal{F}_i\) over \(\mathcal{O}_X\). Write \(\mathcal{I}_i = \Ker(\mathcal{B}_i \to \mathcal{A})\). Write \(\mathcal{I}_i = \colim_j \mathcal{F}_{i, j}\) where \(\mathcal{F}_{i, j}\) is a finite type quasi-coherent submodule of \(\mathcal{I}_i\), see Lemma 0829. Set \(\mathcal{I}_{i, j} \subset \mathcal{I}_i\) equal to the \(\mathcal{B}_i\)-ideal generated by \(\mathcal{F}_{i, j}\). Set \(\mathcal{A}_{i, j} = \mathcal{B}_i/\mathcal{I}_{i, j}\). Then \(\mathcal{A}_{i, j}\) is a quasi-coherent finitely presented \(\mathcal{O}_X\)-algebra. Define \((i, j) \leq (i', j')\) if \(i \leq i'\) and the map \(\mathcal{B}_i \to \mathcal{B}_{i'}\) maps the ideal \(\mathcal{I}_{i, j}\) into the ideal \(\mathcal{I}_{i', j'}\). Then it is clear that \(\mathcal{A} = \colim_{i, j} \mathcal{A}_{i, j}\).

Let \(X\) be an algebraic space. In the following lemma we use the notion of a quasi-coherent \(\mathcal{O}_X\)-algebra \(\mathcal{A}\) of finite type. This means that for every affine \(U = \Spec(R)\) étale over \(X\) we have \(\mathcal{A}|_U = \widetilde{A}\) where \(A\) is a (commutative) \(R\)-algebra which is of finite type as an \(R\)-algebra.

Lemma

Let \(S\) be a scheme. Let \(X\) be a quasi-compact and quasi-separated algebraic space over \(S\). Let \(\mathcal{A}\) be a quasi-coherent \(\mathcal{O}_X\)-algebra. Then \(\mathcal{A}\) is the directed colimit of its finite type quasi-coherent \(\mathcal{O}_X\)-subalgebras.

Proof

Omitted. Hint: Compare with the proof of Lemma 0829.

Let \(X\) be an algebraic space. In the following lemma we use the notion of a finite (resp. integral) quasi-coherent \(\mathcal{O}_X\)-algebra \(\mathcal{A}\). This means that for every affine \(U = \Spec(R)\) étale over \(X\) we have \(\mathcal{A}|_U = \widetilde{A}\) where \(A\) is a (commutative) \(R\)-algebra which is finite (resp. integral) as an \(R\)-algebra.

Lemma

Let \(S\) be a scheme. Let \(X\) be a quasi-compact and quasi-separated algebraic space over \(S\). Let \(\mathcal{A}\) be a finite quasi-coherent \(\mathcal{O}_X\)-algebra. Then \(\mathcal{A} = \colim \mathcal{A}_i\) is a directed colimit of finite and finitely presented quasi-coherent \(\mathcal{O}_X\)-algebras with surjective transition maps.

Proof

By Lemma 086Y there exists a finitely presented \(\mathcal{O}_X\)-module \(\mathcal{F}\) and a surjection \(\mathcal{F} \to \mathcal{A}\). Using the algebra structure we obtain a surjection \[\text{Sym}^*_{\mathcal{O}_X}(\mathcal{F}) \longrightarrow \mathcal{A}\] Denote \(\mathcal{J}\) the kernel. Write \(\mathcal{J} = \colim \mathcal{E}_i\) as a filtered colimit of finite type \(\mathcal{O}_X\)-submodules \(\mathcal{E}_i\) (Lemma 0829). Set \[\mathcal{A}_i = \text{Sym}^*_{\mathcal{O}_X}(\mathcal{F})/(\mathcal{E}_i)\] where \((\mathcal{E}_i)\) indicates the ideal sheaf generated by the image of \(\mathcal{E}_i \to \text{Sym}^*_{\mathcal{O}_X}(\mathcal{F})\). Then each \(\mathcal{A}_i\) is a finitely presented \(\mathcal{O}_X\)-algebra, the transition maps are surjective, and \(\mathcal{A} = \colim \mathcal{A}_i\). To finish the proof we still have to show that \(\mathcal{A}_i\) is a finite \(\mathcal{O}_X\)-algebra for \(i\) sufficiently large. To do this we choose an étale surjective map \(U \to X\) where \(U\) is an affine scheme. Take generators \(f_1, \ldots, f_m \in \Gamma(U, \mathcal{F})\). As \(\mathcal{A}(U)\) is a finite \(\mathcal{O}_X(U)\)-algebra we see that for each \(j\) there exists a monic polynomial \(P_j \in \mathcal{O}(U)[T]\) such that \(P_j(f_j)\) is zero in \(\mathcal{A}(U)\). Since \(\mathcal{A} = \colim \mathcal{A}_i\) by construction, we have \(P_j(f_j) = 0\) in \(\mathcal{A}_i(U)\) for all sufficiently large \(i\). For such \(i\) the algebras \(\mathcal{A}_i\) are finite.

Lemma

Let \(S\) be a scheme. Let \(X\) be a quasi-compact and quasi-separated algebraic space over \(S\). Let \(\mathcal{A}\) be an integral quasi-coherent \(\mathcal{O}_X\)-algebra. Then

1. \(\mathcal{A}\) is the directed colimit of its finite quasi-coherent \(\mathcal{O}_X\)-subalgebras, and

2. \(\mathcal{A}\) is a directed colimit of finite and finitely presented \(\mathcal{O}_X\)-algebras.

Proof

By Lemma 082B we have \(\mathcal{A} = \colim \mathcal{A}_i\) where \(\mathcal{A}_i \subset \mathcal{A}\) runs through the quasi-coherent \(\mathcal{O}_X\)-sub algebras of finite type. Any finite type quasi-coherent \(\mathcal{O}_X\)-subalgebra of \(\mathcal{A}\) is finite (use Algebra, Lemma 02JJ on affine schemes étale over \(X\)). This proves (1).

To prove (2), write \(\mathcal{A} = \colim \mathcal{F}_i\) as a colimit of finitely presented \(\mathcal{O}_X\)-modules using Lemma 07V9. For each \(i\), let \(\mathcal{J}_i\) be the kernel of the map \[\text{Sym}^*_{\mathcal{O}_X}(\mathcal{F}_i) \longrightarrow \mathcal{A}\] For \(i' \geq i\) there is an induced map \(\mathcal{J}_i \to \mathcal{J}_{i'}\) and we have \(\mathcal{A} = \colim \text{Sym}^*_{\mathcal{O}_X}(\mathcal{F}_i)/\mathcal{J}_i\). Moreover, the quasi-coherent \(\mathcal{O}_X\)-algebras \(\text{Sym}^*_{\mathcal{O}_X}(\mathcal{F}_i)/\mathcal{J}_i\) are finite (see above). Write \(\mathcal{J}_i = \colim \mathcal{E}_{ik}\) as a colimit of finitely presented \(\mathcal{O}_X\)-modules. Given \(i' \geq i\) and \(k\) there exists a \(k'\) such that we have a map \(\mathcal{E}_{ik} \to \mathcal{E}_{i'k'}\) making \[\xymatrix{ \mathcal{J}_i \ar[r] & \mathcal{J}_{i'} \\ \mathcal{E}_{ik} \ar[u] \ar[r] & \mathcal{E}_{i'k'} \ar[u] }\] commute. This follows from Cohomology of Spaces, Lemma 07U7. This induces a map \[\mathcal{A}_{ik} = \text{Sym}^*_{\mathcal{O}_X}(\mathcal{F}_i)/(\mathcal{E}_{ik}) \longrightarrow \text{Sym}^*_{\mathcal{O}_X}(\mathcal{F}_{i'})/(\mathcal{E}_{i'k'}) = \mathcal{A}_{i'k'}\] where \((\mathcal{E}_{ik})\) denotes the ideal generated by \(\mathcal{E}_{ik}\). The quasi-coherent \(\mathcal{O}_X\)-algebras \(\mathcal{A}_{ki}\) are of finite presentation and finite for \(k\) large enough (see proof of Lemma 086Z). Finally, we have \[\colim \mathcal{A}_{ik} = \colim \mathcal{A}_i = \mathcal{A}\] Namely, the first equality was shown in the proof of Lemma 086Z and the second equality because \(\mathcal{A}\) is the colimit of the modules \(\mathcal{F}_i\).

Lemma

Let \(S\) be a scheme. Let \(X\) be a quasi-compact and quasi-separated algebraic space over \(S\). Let \(U \subset X\) be a quasi-compact open. Let \(\mathcal{F}\) be a quasi-coherent \(\mathcal{O}_X\)-module. Let \(\mathcal{G} \subset \mathcal{F}|_U\) be a quasi-coherent \(\mathcal{O}_U\)-submodule which is of finite type. Then there exists a quasi-coherent submodule \(\mathcal{G}' \subset \mathcal{F}\) which is of finite type such that \(\mathcal{G}'|_U = \mathcal{G}\).

Proof

Denote \(j : U \to X\) the inclusion morphism. As \(X\) is quasi-separated and \(U\) quasi-compact, the morphism \(j\) is quasi-compact. Hence \(j_*\mathcal{G} \subset j_*\mathcal{F}|_U\) are quasi-coherent modules on \(X\) (Morphisms of Spaces, Lemma 03M9). Let \(\mathcal{H} = \Ker(j_*\mathcal{G} \oplus \mathcal{F} \to j_*\mathcal{F}|_U)\). Then \(\mathcal{H}|_U = \mathcal{G}\). By Lemma 0829 we can find a finite type quasi-coherent submodule \(\mathcal{H}' \subset \mathcal{H}\) such that \(\mathcal{H}'|_U = \mathcal{H}|_U = \mathcal{G}\). Set \(\mathcal{G}' = \Im(\mathcal{H}' \to \mathcal{F})\) to conclude.

Relative approximation

We discuss variants of Proposition 07SU over a base.

Lemma

Let \(f : X \to Y\) be a morphism of quasi-compact and quasi-separated algebraic spaces over \(\mathbf{Z}\). Then there exists a direct set \(I\) and an inverse system \((f_i : X_i \to Y_i)\) of morphisms algebraic spaces over \(I\), such that the transition morphisms \(X_i \to X_{i'}\) and \(Y_i \to Y_{i'}\) are affine, such that \(X_i\) and \(Y_i\) are quasi-separated and of finite type over \(\mathbf{Z}\), and such that \((X \to Y) = \lim (X_i \to Y_i)\).

Proof

Write \(X = \lim_{a \in A} X_a\) and \(Y = \lim_{b \in B} Y_b\) as in Proposition 07SU, i.e., with \(X_a\) and \(Y_b\) quasi-separated and of finite type over \(\mathbf{Z}\) and with affine transition morphisms.

Fix \(b \in B\). By Lemma 0CP4 applied to \(Y_b\) and \(X = \lim X_a\) over \(\mathbf{Z}\) we find there exists an \(a \in A\) and a morphism \(f_{a, b} : X_a \to Y_b\) making the diagram \[\xymatrix{ X \ar[d] \ar[r] & Y \ar[d] \\ X_a \ar[r] & Y_b }\] commute. Let \(I\) be the set of triples \((a, b, f_{a, b})\) we obtain in this manner.

Let \((a, b, f_{a, b})\) and \((a', b', f_{a', b'})\) be in \(I\). Let \(b'' \leq \min(b, b')\). By Lemma 0CP4 again, there exists an \(a'' \geq \max(a, a')\) such that the compositions \(X_{a''} \to X_a \to Y_b \to Y_{b''}\) and \(X_{a''} \to X_{a'} \to Y_{b'} \to Y_{b''}\) are equal. We endow \(I\) with the preorder \[(a, b, f_{a, b}) \geq (a', b', f_{a', b'}) \Leftrightarrow a \geq a',\ b \geq b',\text{ and } g_{b, b'} \circ f_{a, b} = f_{a', b'} \circ h_{a, a'}\] where \(h_{a, a'} : X_a \to X_{a'}\) and \(g_{b, b'} : Y_b \to Y_{b'}\) are the transition morphisms. The remarks above show that \(I\) is directed and that the maps \(I \to A\), \((a, b, f_{a, b}) \mapsto a\) and \(I \to B\), \((a, b, f_{a, b})\) are cofinal. If for \(i = (a, b, f_{a, b})\) we set \(X_i = X_a\), \(Y_i = Y_b\), and \(f_i = f_{a, b}\), then we get an inverse system of morphisms over \(I\) and we have \[\lim_{i \in I} X_i = \lim_{a \in A} X_a = X \quad\text{and}\quad \lim_{i \in I} S_i = \lim_{b \in B} Y_b = Y\] by Categories, Lemma 002R (recall that limits over \(I\) are really limits over the opposite category associated to \(I\) and hence cofinal turns into initial). This finishes the proof.

Lemma

Let \(S\) be a scheme. Let \(f : X \to Y\) be a morphism of algebraic spaces over \(S\). Assume that

1. \(X\) is quasi-compact and quasi-separated, and

2. \(Y\) is quasi-separated.

Then \(X = \lim X_i\) is a limit of a directed inverse system of algebraic spaces \(X_i\) of finite presentation over \(Y\) with affine transition morphisms over \(Y\).

Proof

Since \(|f|(|X|)\) is quasi-compact we may replace \(Y\) by a quasi-compact open subspace whose set of points contains \(|f|(|X|)\). Hence we may assume \(Y\) is quasi-compact as well. By Lemma 0GS3 we can write \((X \to Y) = \lim (X_i \to Y_i)\) for some directed inverse system of morphisms of finite type schemes over \(\mathbf{Z}\) with affine transition morphisms. Since limits commute with limits (Categories, Lemma 002M) we have \(X = \lim X_i \times_{Y_i} Y\). For \(i \geq i'\) the transition morphism \(X_i \times_{Y_i} Y \to X_{i'} \times_{Y_{i'}} Y\) is affine as the composition \[X_i \times_{Y_i} Y \to X_i \times_{Y_{i'}} Y \to X_{i'} \times_{Y_{i'}} Y\] where the first morphism is a closed immersion (by Morphisms of Spaces, Lemma 03KN) and the second is a base change of an affine morphism (Morphisms of Spaces, Lemma 03WI) and the composition of affine morphisms is affine (Morphisms of Spaces, Lemma 03WH). The morphisms \(f_i\) are of finite presentation (Morphisms of Spaces, Lemmas 06G4 and 05WT) and hence the base changes \(X_i \times_{f_i, Y_i} Y \to Y\) are of finite presentation (Morphisms of Spaces, Lemma 03XR).

Finite type closed in finite presentation

This section is the analogue of Limits, Section 01ZD.

Lemma

Let \(S\) be a scheme. Let \(f : X \to Y\) be an affine morphism of algebraic spaces over \(S\). If \(Y\) quasi-compact and quasi-separated, then \(X\) is a directed limit \(X = \lim X_i\) with each \(X_i\) affine and of finite presentation over \(Y\).

Proof

Consider the quasi-coherent \(\mathcal{O}_Y\)-module \(\mathcal{A} = f_*\mathcal{O}_X\). By Lemma 082A we can write \(\mathcal{A} = \colim \mathcal{A}_i\) as a directed colimit of finitely presented \(\mathcal{O}_Y\)-algebras \(\mathcal{A}_i\). Set \(X_i = \underline{\Spec}_Y(\mathcal{A}_i)\), see Morphisms of Spaces, Definition 081W. By construction \(X_i \to Y\) is affine and of finite presentation and \(X = \lim X_i\).

Lemma

Let \(S\) be a scheme. Let \(f : X \to Y\) be an integral morphism of algebraic spaces over \(S\). Assume \(Y\) quasi-compact and quasi-separated. Then \(X\) can be written as a directed limit \(X = \lim X_i\) where \(X_i\) are finite and of finite presentation over \(Y\).

Proof

Consider the quasi-coherent \(\mathcal{O}_Y\)-module \(\mathcal{A} = f_*\mathcal{O}_X\). By Lemma 082C we can write \(\mathcal{A} = \colim \mathcal{A}_i\) as a directed colimit of finite and finitely presented \(\mathcal{O}_Y\)-algebras \(\mathcal{A}_i\). Set \(X_i = \underline{\Spec}_Y(\mathcal{A}_i)\), see Morphisms of Spaces, Definition 081W. By construction \(X_i \to Y\) is finite and of finite presentation and \(X = \lim X_i\).

Lemma

Let \(S\) be a scheme. Let \(f : X \to Y\) be a finite morphism of algebraic spaces over \(S\). Assume \(Y\) quasi-compact and quasi-separated. Then \(X\) can be written as a directed limit \(X = \lim X_i\) where the transition maps are closed immersions and the objects \(X_i\) are finite and of finite presentation over \(Y\).

Proof

Consider the finite quasi-coherent \(\mathcal{O}_Y\)-module \(\mathcal{A} = f_*\mathcal{O}_X\). By Lemma 086Z we can write \(\mathcal{A} = \colim \mathcal{A}_i\) as a directed colimit of finite and finitely presented \(\mathcal{O}_Y\)-algebras \(\mathcal{A}_i\) with surjective transition maps. Set \(X_i = \underline{\Spec}_Y(\mathcal{A}_i)\), see Morphisms of Spaces, Definition 081W. By construction \(X_i \to Y\) is finite and of finite presentation, the transition maps are closed immersions, and \(X = \lim X_i\).

Lemma

Let \(S\) be a scheme. Let \(f : X \to Y\) be a closed immersion of algebraic spaces over \(S\). Assume \(Y\) quasi-compact and quasi-separated. Then \(X\) can be written as a directed limit \(X = \lim X_i\) where the transition maps are closed immersions and the morphisms \(X_i \to Y\) are closed immersions of finite presentation.

Proof

Let \(\mathcal{I} \subset \mathcal{O}_Y\) be the quasi-coherent sheaf of ideals defining \(X\) as a closed subspace of \(Y\). By Lemma 0829 we can write \(\mathcal{I} = \colim \mathcal{I}_i\) as the filtered colimit of its finite type quasi-coherent submodules. Let \(X_i\) be the closed subspace of \(X\) cut out by \(\mathcal{I}_i\). Then \(X_i \to Y\) is a closed immersion of finite presentation, and \(X = \lim X_i\). Some details omitted.

Lemma

Let \(S\) be a scheme. Let \(f : X \to Y\) be a morphism of algebraic spaces over \(S\). Assume

1. \(f\) is locally of finite type and quasi-affine, and

2. \(Y\) is quasi-compact and quasi-separated.

Then there exists a morphism of finite presentation \(f' : X' \to Y\) and a closed immersion \(X \to X'\) over \(Y\).

Proof

By Morphisms of Spaces, Lemma 086S we can find a factorization \(X \to Z \to Y\) where \(X \to Z\) is a quasi-compact open immersion and \(Z \to Y\) is affine. Write \(Z = \lim Z_i\) with \(Z_i\) affine and of finite presentation over \(Y\) (Lemma 0870). For some \(0 \in I\) we can find a quasi-compact open \(U_0 \subset Z_0\) such that \(X\) is isomorphic to the inverse image of \(U_0\) in \(Z\) (Lemma 0827). Let \(U_i\) be the inverse image of \(U_0\) in \(Z_i\), so \(U = \lim U_i\). By Lemma 0828 we see that \(X \to U_i\) is a closed immersion for some \(i\) large enough. Setting \(X' = U_i\) finishes the proof.

Lemma

Let \(S\) be a scheme. Let \(f : X \to Y\) be a morphism of algebraic spaces over \(S\). Assume:

1. \(f\) is of locally of finite type.

2. \(X\) is quasi-compact and quasi-separated, and

3. \(Y\) is quasi-compact and quasi-separated.

Then there exists a morphism of finite presentation \(f' : X' \to Y\) and a closed immersion \(X \to X'\) of algebraic spaces over \(Y\).

Proof

By Proposition 07SU we can write \(X = \lim_i X_i\) with \(X_i\) quasi-separated of finite type over \(\mathbf{Z}\) and with transition morphisms \(f_{ii'} : X_i \to X_{i'}\) affine. Consider the commutative diagram \[\xymatrix{ X \ar[r] \ar[rd] & X_{i, Y} \ar[r] \ar[d] & X_i \ar[d] \\ & Y \ar[r] & \Spec(\mathbf{Z}) }\] Note that \(X_i\) is of finite presentation over \(\Spec(\mathbf{Z})\), see Morphisms of Spaces, Lemma 06G4. Hence the base change \(X_{i, Y} \to Y\) is of finite presentation by Morphisms of Spaces, Lemma 03XR. Observe that \(\lim X_{i, Y} = X \times Y\) and that \(X \to X \times Y\) is a monomorphism. By Lemma 0828 we see that \(X \to X_{i, Y}\) is a monomorphism for \(i\) large enough. Fix such an \(i\). Note that \(X \to X_{i, Y}\) is locally of finite type (Morphisms of Spaces, Lemma 0462) and a monomorphism, hence separated and locally quasi-finite (Morphisms of Spaces, Lemma 0463). Hence \(X \to X_{i, Y}\) is representable. Hence \(X \to X_{i, Y}\) is quasi-affine because we can use the principle Spaces, Lemma 02YO and the result for morphisms of schemes More on Morphisms, Lemma 02LR. Thus Lemma 0871 gives a factorization \(X \to X' \to X_{i, Y}\) with \(X \to X'\) a closed immersion and \(X' \to X_{i, Y}\) of finite presentation. Finally, \(X' \to Y\) is of finite presentation as a composition of morphisms of finite presentation (Morphisms of Spaces, Lemma 03XQ).

Proposition

Let \(S\) be a scheme. \(f : X \to Y\) be a morphism of algebraic spaces over \(S\). Assume

1. \(f\) is of finite type and separated, and

2. \(Y\) is quasi-compact and quasi-separated.

Then there exists a separated morphism of finite presentation \(f' : X' \to Y\) and a closed immersion \(X \to X'\) over \(Y\).

Proof

By Lemma 0872 there is a closed immersion \(X \to Z\) with \(Z/Y\) of finite presentation. Let \(\mathcal{I} \subset \mathcal{O}_Z\) be the quasi-coherent sheaf of ideals defining \(X\) as a closed subscheme of \(Y\). By Lemma 0829 we can write \(\mathcal{I}\) as a directed colimit \(\mathcal{I} = \colim_{a \in A} \mathcal{I}_a\) of its quasi-coherent sheaves of ideals of finite type. Let \(X_a \subset Z\) be the closed subspace defined by \(\mathcal{I}_a\). These form an inverse system indexed by \(A\). The transition morphisms \(X_a \to X_{a'}\) are affine because they are closed immersions. Each \(X_a\) is quasi-compact and quasi-separated since it is a closed subspace of \(Z\) and \(Z\) is quasi-compact and quasi-separated by our assumptions. We have \(X = \lim_a X_a\) as follows directly from the fact that \(\mathcal{I} = \colim_{a \in A} \mathcal{I}_a\). Each of the morphisms \(X_a \to Z\) is of finite presentation, see Morphisms, Lemma 01TV. Hence the morphisms \(X_a \to Y\) are of finite presentation. Thus it suffices to show that \(X_a \to Y\) is separated for some \(a \in A\). This follows from Lemma 086X as we have assumed that \(X \to Y\) is separated.

Approximating proper morphisms

Lemma

Let \(S\) be a scheme. Let \(f : X \to Y\) be a proper morphism of algebraic spaces over \(S\) with \(Y\) quasi-compact and quasi-separated. Then \(X = \lim X_i\) is a directed limit of algebraic spaces \(X_i\) proper and of finite presentation over \(Y\) and with transition morphisms and morphisms \(X \to X_i\) closed immersions.

Proof

By Proposition 0873 we can find a closed immersion \(X \to X'\) with \(X'\) separated and of finite presentation over \(Y\). By Lemma 0A0U we can write \(X = \lim X_i\) with \(X_i \to X'\) a closed immersion of finite presentation. We claim that for all \(i\) large enough the morphism \(X_i \to Y\) is proper which finishes the proof.

To prove this we may assume that \(Y\) is an affine scheme, see Morphisms of Spaces, Lemma 083R. Next, we use the weak version of Chow’s lemma, see Cohomology of Spaces, Lemma 089J, to find a diagram \[\xymatrix{ X' \ar[rd] & X'' \ar[d] \ar[l]^\pi \ar[r] & \mathbf{P}^n_Y \ar[dl] \\ & Y & }\] where \(X'' \to \mathbf{P}^n_Y\) is an immersion, and \(\pi : X'' \to X'\) is proper and surjective. Denote \(X'_i \subset X''\), resp. \(\pi^{-1}(X)\) the scheme theoretic inverse image of \(X_i \subset X'\), resp. \(X \subset X'\). Then \(\lim X'_i = \pi^{-1}(X)\). Since \(\pi^{-1}(X) \to Y\) is proper (Morphisms of Spaces, Lemmas 04XY), we see that \(\pi^{-1}(X) \to \mathbf{P}^n_Y\) is a closed immersion (Morphisms of Spaces, Lemmas 04NX and 04CD). Hence for \(i\) large enough we find that \(X'_i \to \mathbf{P}^n_Y\) is a closed immersion by Lemma 0A0T. Thus \(X'_i\) is proper over \(Y\). For such \(i\) the morphism \(X_i \to Y\) is proper by Morphisms of Spaces, Lemma 08AJ.

Lemma

Let \(f : X \to Y\) be a proper morphism of algebraic spaces over \(\mathbf{Z}\) with \(Y\) quasi-compact and quasi-separated. Then there exists a directed set \(I\), an inverse system \((f_i : X_i \to Y_i)\) of morphisms of algebraic spaces over \(I\), such that the transition morphisms \(X_i \to X_{i'}\) and \(Y_i \to Y_{i'}\) are affine, such that \(f_i\) is proper and of finite presentation, such that \(Y_i\) is of finite presentation over \(\mathbf{Z}\), and such that \((X \to Y) = \lim (X_i \to Y_i)\).

Proof

By Lemma 0A0W we can write \(X = \lim_{k \in K} X_k\) with \(X_k \to Y\) proper and of finite presentation. Next, by absolute Noetherian approximation (Proposition 07SU) we can write \(Y = \lim_{j \in J} Y_j\) with \(Y_j\) of finite presentation over \(\mathbf{Z}\). For each \(k\) there exists a \(j\) and a morphism \(X_{k, j} \to Y_j\) of finite presentation with \(X_k \cong Y \times_{Y_j} X_{k, j}\) as algebraic spaces over \(Y\), see Lemma 07SK. After increasing \(j\) we may assume \(X_{k, j} \to Y_j\) is proper, see Lemma 08K1. The set \(I\) will be consist of these pairs \((k, j)\) and the corresponding morphism is \(X_{k, j} \to Y_j\). For every \(k' \geq k\) we can find a \(j' \geq j\) and a morphism \(X_{j', k'} \to X_{j, k}\) over \(Y_{j'} \to Y_j\) whose base change to \(Y\) gives the morphism \(X_{k'} \to X_k\) (follows again from Lemma 07SK). These morphisms form the transition morphisms of the system. Some details omitted.

Recall the scheme theoretic support of a finite type quasi-coherent module, see Morphisms of Spaces, Definition 07U1.

Lemma

Assumptions and notation as in Situation 084W. Let \(\mathcal{F}_0\) be a quasi-coherent \(\mathcal{O}_{X_0}\)-module. Denote \(\mathcal{F}\) and \(\mathcal{F}_i\) the pullbacks of \(\mathcal{F}_0\) to \(X\) and \(X_i\). Assume

1. \(f_0\) is locally of finite type,

2. \(\mathcal{F}_0\) is of finite type,

3. the scheme theoretic support of \(\mathcal{F}\) is proper over \(Y\).

Then the scheme theoretic support of \(\mathcal{F}_i\) is proper over \(Y_i\) for some \(i\).

Proof

We may replace \(X_0\) by the scheme theoretic support of \(\mathcal{F}_0\). By Morphisms of Spaces, Lemma 07TZ this guarantees that \(X_i\) is the support of \(\mathcal{F}_i\) and \(X\) is the support of \(\mathcal{F}\). Then, if \(Z \subset X\) denotes the scheme theoretic support of \(\mathcal{F}\), we see that \(Z \to X\) is a universal homeomorphism. We conclude that \(X \to Y\) is proper as this is true for \(Z \to Y\) by assumption, see Morphisms, Lemma 03GN. By Lemma 08K1 we see that \(X_i \to Y\) is proper for some \(i\). Then it follows that the scheme theoretic support \(Z_i\) of \(\mathcal{F}_i\) is proper over \(Y\) by Morphisms of Spaces, Lemmas 04XZ and 04XY.

Embedding into affine space

Some technical lemmas to be used in the proof of Chow’s lemma later.

Lemma

Let \(S\) be a scheme. Let \(f : U \to X\) be a morphism of algebraic spaces over \(S\). Assume \(U\) is an affine scheme, \(f\) is locally of finite type, and \(X\) quasi-separated and locally separated. Then there exists an immersion \(U \to \mathbf{A}^n_X\) over \(X\).

Proof

Say \(U = \Spec(A)\). Write \(A = \colim A_i\) as a filtered colimit of finite type \(\mathbf{Z}\)-subalgebras. For each \(i\) the morphism \(U \to U_i = \Spec(A_i)\) induces a morphism \[U \longrightarrow X \times U_i\] over \(X\). In the limit the morphism \(U \to X \times U\) is an immersion as \(X\) is locally separated, see Morphisms of Spaces, Lemma 03KO. By Lemma 0828 we see that \(U \to X \times U_i\) is an immersion for some \(i\). Since \(U_i\) is isomorphic to a closed subscheme of \(\mathbf{A}^n_{\mathbf{Z}}\) the lemma follows.

Remark

We have seen in Examples, Section 088V that Lemma 088L does not hold if we drop the assumption that \(X\) be locally separated. This raises the question: Does Lemma 088L hold if we drop the assumption that \(X\) be quasi-separated? If you know the answer, please email stacks.project@gmail.com.

Lemma

Let \(S\) be a scheme. Let \(f : Y \to X\) be a morphism of algebraic spaces over \(S\). Assume \(X\) Noetherian and \(f\) of finite presentation. Then there exists a dense open \(V \subset Y\) and an immersion \(V \to \mathbf{A}^n_X\).

Proof

The assumptions imply that \(Y\) is Noetherian (Morphisms of Spaces, Lemma 04ZL). Then \(Y\) is quasi-separated, hence has a dense open subscheme (Properties of Spaces, Proposition 06NH). Thus we may assume that \(Y\) is a Noetherian scheme. By removing intersections of irreducible components of \(Y\) (use Topology, Lemma 0052 and Properties, Lemma 01OZ) we may assume that \(Y\) is a disjoint union of irreducible Noetherian schemes. Since there is an immersion \[\mathbf{A}^n_X \amalg \mathbf{A}^m_X \longrightarrow \mathbf{A}^{\max(n, m) + 1}_X\] (details omitted) we see that it suffices to prove the result in case \(Y\) is irreducible.

Assume \(Y\) is an irreducible scheme. Let \(T \subset |X|\) be the closure of the image of \(f : Y \to X\). Note that since \(|Y|\) and \(|X|\) are sober topological spaces (Properties of Spaces, Lemma 06NJ) \(T\) is irreducible with a unique generic point \(\xi\) which is the image of the generic point \(\eta\) of \(Y\). Let \(\mathcal{I} \subset X\) be a quasi-coherent sheaf of ideals cutting out the reduced induced space structure on \(T\) (Properties of Spaces, Definition 047X). Since \(\mathcal{O}_{Y, \eta}\) is an Artinian local ring we see that for some \(n > 0\) we have \(f^{-1}\mathcal{I}^n \mathcal{O}_{Y, \eta} = 0\). As \(f^{-1}\mathcal{I}\mathcal{O}_Y\) is a finite type quasi-coherent ideal we conclude that \(f^{-1}\mathcal{I}^n\mathcal{O}_V = 0\) for some nonempty open \(V \subset Y\). Let \(Z \subset X\) be the closed subspace cut out by \(\mathcal{I}^n\). By construction \(V \to Y \to X\) factors through \(Z\). Because \(\mathbf{A}^n_Z \to \mathbf{A}^n_X\) is an immersion, we may replace \(X\) by \(Z\) and \(Y\) by \(V\). Hence we reach the situation where \(Y\) and \(X\) are irreducible and \(Y \to X\) maps the generic point of \(Y\) onto the generic point of \(X\).

Assume \(Y\) and \(X\) are irreducible, \(Y\) is a scheme, and \(Y \to X\) maps the generic point of \(Y\) onto the generic point of \(X\). By Properties of Spaces, Proposition 06NH \(X\) has a dense open subscheme \(U \subset X\). Choose a nonempty affine open \(V \subset Y\) whose image in \(X\) is contained in \(U\). By Morphisms, Lemma 04II we may factor \(V \to U\) as \(V \to \mathbf{A}^n_U \to U\). Composing with \(\mathbf{A}^n_U \to \mathbf{A}^n_X\) we obtain the desired immersion.

Sections with support in a closed subset

This section is the analogue of Properties, Section 07ZM.

Lemma

Let \(S\) be a scheme. Let \(X\) be a quasi-compact and quasi-separated algebraic space. Let \(U \subset X\) be an open subspace. The following are equivalent:

1. \(U \to X\) is quasi-compact,

2. \(U\) is quasi-compact, and

3. there exists a finite type quasi-coherent sheaf of ideals \(\mathcal{I} \subset \mathcal{O}_X\) such that \(|X| \setminus |U| = |V(\mathcal{I})|\).

Proof

Let \(W\) be an affine scheme and let \(\varphi : W \to X\) be a surjective étale morphism, see Properties of Spaces, Lemma 03H6. If (1) holds, then \(\varphi^{-1}(U) \to W\) is quasi-compact, hence \(\varphi^{-1}(U)\) is quasi-compact, hence \(U\) is quasi-compact (as \(|\varphi^{-1}(U)| \to |U|\) is surjective). If (2) holds, then \(\varphi^{-1}(U)\) is quasi-compact because \(\varphi\) is quasi-compact since \(X\) is quasi-separated (Morphisms of Spaces, Lemma 073B). Hence \(\varphi^{-1}(U) \to W\) is a quasi-compact morphism of schemes by Properties, Lemma 01PH. It follows that \(U \to X\) is quasi-compact by Morphisms of Spaces, Lemma 03KG. Thus (1) and (2) are equivalent.

Assume (1) and (2). By Properties of Spaces, Lemma 03IQ there exists a unique quasi-coherent sheaf of ideals \(\mathcal{J}\) cutting out the reduced induced closed subspace structure on \(|X| \setminus |U|\). Note that \(\mathcal{J}|_U = \mathcal{O}_U\) which is an \(\mathcal{O}_U\)-modules of finite type. As \(U\) is quasi-compact it follows from Lemma 0829 that there exists a quasi-coherent subsheaf \(\mathcal{I} \subset \mathcal{J}\) which is of finite type and has the property that \(\mathcal{I}|_U = \mathcal{J}|_U\). Then \(|X| \setminus |U| = |V(\mathcal{I})|\) and we obtain (3). Conversely, if \(\mathcal{I}\) is as in (3), then \(\varphi^{-1}(U) \subset W\) is a quasi-compact open by the lemma for schemes (Properties, Lemma 01PH) applied to \(\varphi^{-1}\mathcal{I}\) on \(W\). Thus (2) holds.

Lemma

Let \(S\) be a scheme. Let \(X\) be an algebraic space over \(S\). Let \(\mathcal{I} \subset \mathcal{O}_X\) be a quasi-coherent sheaf of ideals. Let \(\mathcal{F}\) be a quasi-coherent \(\mathcal{O}_X\)-module. Consider the sheaf of \(\mathcal{O}_X\)-modules \(\mathcal{F}'\) which associates to every object \(U\) of \(X_\etale\) the module \[\mathcal{F}'(U) = \{s \in \mathcal{F}(U) \mid \mathcal{I}s = 0\}\] Assume \(\mathcal{I}\) is of finite type. Then

1. \(\mathcal{F}'\) is a quasi-coherent sheaf of \(\mathcal{O}_X\)-modules,

2. for affine \(U\) in \(X_\etale\) we have \(\mathcal{F}'(U) = \{s \in \mathcal{F}(U) \mid \mathcal{I}(U)s = 0\}\), and

3. \(\mathcal{F}'_x = \{s \in \mathcal{F}_x \mid \mathcal{I}_x s = 0\}\).

Proof

It is clear that the rule defining \(\mathcal{F}'\) gives a subsheaf of \(\mathcal{F}\). Hence we may work étale locally on \(X\) to verify the other statements. Thus the lemma reduces to the case of schemes which is Properties, Lemma 01PO.

Definition

Let \(S\) be a scheme. Let \(X\) be an algebraic space over \(S\). Let \(\mathcal{I} \subset \mathcal{O}_X\) be a quasi-coherent sheaf of ideals of finite type. Let \(\mathcal{F}\) be a quasi-coherent \(\mathcal{O}_X\)-module. The subsheaf \(\mathcal{F}' \subset \mathcal{F}\) defined in Lemma 0856 above is called the subsheaf of sections annihilated by \(\mathcal{I}\).

Lemma

Let \(S\) be a scheme. Let \(f : X \to Y\) be a quasi-compact and quasi-separated morphism of algebraic spaces over \(S\). Let \(\mathcal{I} \subset \mathcal{O}_Y\) be a quasi-coherent sheaf of ideals of finite type. Let \(\mathcal{F}\) be a quasi-coherent \(\mathcal{O}_X\)-module. Let \(\mathcal{F}' \subset \mathcal{F}\) be the subsheaf of sections annihilated by \(f^{-1}\mathcal{I}\mathcal{O}_X\). Then \(f_*\mathcal{F}' \subset f_*\mathcal{F}\) is the subsheaf of sections annihilated by \(\mathcal{I}\).

Proof

Omitted. Hint: The assumption that \(f\) is quasi-compact and quasi-separated implies that \(f_*\mathcal{F}\) is quasi-coherent (Morphisms of Spaces, Lemma 03M9) so that Lemma 0856 applies to \(\mathcal{I}\) and \(f_*\mathcal{F}\).

Next we come to the sheaf of sections supported in a closed subset. Again this isn’t always a quasi-coherent sheaf, but if the complement of the closed is “retrocompact” in the given algebraic space, then it is.

Lemma

Let \(S\) be a scheme. Let \(X\) be an algebraic space over \(S\). Let \(T \subset |X|\) be a closed subset and let \(U \subset X\) be the open subspace such that \(T \amalg |U| = |X|\). Let \(\mathcal{F}\) be a quasi-coherent \(\mathcal{O}_X\)-module. Consider the sheaf of \(\mathcal{O}_X\)-modules \(\mathcal{F}'\) which associates to every object \(\varphi : W \to X\) of \(X_\etale\) the module \[\mathcal{F}'(W) = \{s \in \mathcal{F}(W) \mid \text{the support of }s\text{ is contained in }|\varphi|^{-1}(T)\}\] If \(U \to X\) is quasi-compact, then

1. for \(W\) affine there exist a finitely generated ideal \(I \subset \mathcal{O}_X(W)\) such that \(|\varphi|^{-1}(T) = V(I)\),

2. for \(W\) and \(I\) as in (1) we have \(\mathcal{F}'(W) = \{x \in \mathcal{F}(W) \mid I^nx = 0 \text{ for some } n\}\),

3. \(\mathcal{F}'\) is a quasi-coherent sheaf of \(\mathcal{O}_X\)-modules.

Proof

It is clear that the rule defining \(\mathcal{F}'\) gives a subsheaf of \(\mathcal{F}\). Hence we may work étale locally on \(X\) to verify the other statements. Thus the lemma reduces to the case of schemes which is Properties, Lemma 07ZP.

Definition

Let \(S\) be a scheme. Let \(X\) be an algebraic space over \(S\). Let \(T \subset |X|\) be a closed subset whose complement corresponds to an open subspace \(U \subset X\) with quasi-compact inclusion morphism \(U \to X\). Let \(\mathcal{F}\) be a quasi-coherent \(\mathcal{O}_X\)-module. The quasi-coherent subsheaf \(\mathcal{F}' \subset \mathcal{F}\) defined in Lemma 0859 above is called the subsheaf of sections supported on \(T\).

Lemma

Let \(S\) be a scheme. Let \(f : X \to Y\) be a quasi-compact and quasi-separated morphism of algebraic spaces over \(S\). Let \(T \subset |Y|\) be a closed subset. Assume \(|Y| \setminus T\) corresponds to an open subspace \(V \subset Y\) such that \(V \to Y\) is quasi-compact. Let \(\mathcal{F}\) be a quasi-coherent \(\mathcal{O}_X\)-module. Let \(\mathcal{F}' \subset \mathcal{F}\) be the subsheaf of sections supported on \(|f|^{-1}T\). Then \(f_*\mathcal{F}' \subset f_*\mathcal{F}\) is the subsheaf of sections supported on \(T\).

Proof

Omitted. Hints: \(|X| \setminus |f|^{-1}T\) is the support of the open subspace \(U = f^{-1}V \subset X\). Since \(V \to Y\) is quasi-compact, so is \(U \to X\) (by base change). The assumption that \(f\) is quasi-compact and quasi-separated implies that \(f_*\mathcal{F}\) is quasi-coherent. Hence Lemma 0859 applies to \(T\) and \(f_*\mathcal{F}\) as well as to \(|f|^{-1}T\) and \(\mathcal{F}\). The equality of the given quasi-coherent modules is immediate from the definitions.

Characterizing affine spaces

This section is the analogue of Limits, Section 01ZS.

Lemma

Let \(S\) be a scheme. Let \(f : X \to Y\) be a morphism of algebraic spaces over \(S\). Assume that \(f\) is surjective and finite, and assume that \(X\) is affine. Then \(Y\) is affine.

Proof

We may and do view \(f : X \to Y\) as a morphism of algebraic spaces over \(\Spec(\mathbf{Z})\) (see Spaces, Definition 03I5). Note that a finite morphism is affine and universally closed, see Morphisms of Spaces, Lemma 0415. By Morphisms of Spaces, Lemma 05Z2 we see that \(Y\) is a separated algebraic space. As \(f\) is surjective and \(X\) is quasi-compact we see that \(Y\) is quasi-compact.

By Lemma 07VR we can write \(X = \lim X_a\) with each \(X_a \to Y\) finite and of finite presentation. By Lemma 07SQ we see that \(X_a\) is affine for \(a\) large enough. Hence we may and do assume that \(f : X \to Y\) is finite, surjective, and of finite presentation.

By Proposition 07SU we may write \(Y = \lim Y_i\) as a directed limit of algebraic spaces of finite presentation over \(\mathbf{Z}\). By Lemma 07SK we can find \(0 \in I\) and a morphism \(X_0 \to Y_0\) of finite presentation such that \(X_i = X_0 \times_{Y_0} Y_i\) for \(i \geq 0\) and such that \(X = \lim_i X_i\). By Lemma 084Z we see that \(X_i \to Y_i\) is finite for \(i\) large enough. By Lemma 07SN we see that \(X_i \to Y_i\) is surjective for \(i\) large enough. By Lemma 07SQ we see that \(X_i\) is affine for \(i\) large enough. Hence for \(i\) large enough we can apply Cohomology of Spaces, Lemma 07VP to conclude that \(Y_i\) is affine. This implies that \(Y\) is affine and we conclude.

Proposition

Let \(S\) be a scheme. Let \(f : X \to Y\) be a morphism of algebraic spaces over \(S\). Assume that \(X\) is affine and \(f\) is surjective and universally closed². Then \(Y\) is affine.

Proof

We may and do view \(f : X \to Y\) as a morphism of algebraic spaces over \(\Spec(\mathbf{Z})\) (see Spaces, Definition 03I5). By Morphisms of Spaces, Lemma 05Z2 we see that \(Y\) is a separated algebraic space. Then by Morphisms of Spaces, Lemma 08GB we find that \(f\) is affine. Whereupon by Morphisms of Spaces, Lemma 0415 we see that \(f\) is integral.

By the preceding paragraph, we may assume \(f : X \to Y\) is surjective and integral, \(X\) is affine, and \(Y\) is separated. Since \(f\) is surjective and \(X\) is quasi-compact we also deduce that \(Y\) is quasi-compact.

Consider the sheaf \(\mathcal{A} = f_*\mathcal{O}_X\). This is a quasi-coherent sheaf of \(\mathcal{O}_Y\)-algebras, see Morphisms of Spaces, Lemma 03M9. By Lemma 07V9 we can write \(\mathcal{A} = \colim_i \mathcal{F}_i\) as a filtered colimit of finite type \(\mathcal{O}_Y\)-modules. Let \(\mathcal{A}_i \subset \mathcal{A}\) be the \(\mathcal{O}_Y\)-subalgebra generated by \(\mathcal{F}_i\). Since the map of algebras \(\mathcal{O}_Y \to \mathcal{A}\) is integral, we see that each \(\mathcal{A}_i\) is a finite quasi-coherent \(\mathcal{O}_Y\)-algebra. Hence \[X_i = \underline{\Spec}_Y(\mathcal{A}_i) \longrightarrow Y\] is a finite morphism of algebraic spaces. Here \(\underline{\Spec}\) is the construction of Morphisms of Spaces, Lemma 081V. It is clear that \(X = \lim_i X_i\). Hence by Lemma 07SQ we see that for \(i\) sufficiently large the scheme \(X_i\) is affine. Moreover, since \(X \to Y\) factors through each \(X_i\) we see that \(X_i \to Y\) is surjective. Hence we conclude that \(Y\) is affine by Lemma 07VS.

The following corollary of the result above can be found in [CLO].

Lemma

Let \(S\) be a scheme. Let \(X\) be an algebraic space over \(S\). If \(X_{red}\) is a scheme, then \(X\) is a scheme.

Proof

Let \(U' \subset X_{red}\) be an open affine subscheme. Let \(U \subset X\) be the open subspace corresponding to the open \(|U'| \subset |X_{red}| = |X|\). Then \(U' \to U\) is surjective and integral. Hence \(U\) is affine by Proposition 07VT. Thus every point is contained in an open subscheme of \(X\), i.e., \(X\) is a scheme.

Lemma

Let \(S\) be a scheme. Let \(f : X \to Y\) be a morphism of algebraic spaces over \(S\). Assume \(f\) is integral and induces a bijection \(|X| \to |Y|\). Then \(X\) is a scheme if and only if \(Y\) is a scheme.

Proof

An integral morphism is representable by definition, hence if \(Y\) is a scheme, so is \(X\). Conversely, assume that \(X\) is a scheme. Let \(U \subset X\) be an affine open. An integral morphism is closed and \(|f|\) is bijective, hence \(|f|(|U|) \subset |Y|\) is open as the complement of \(|f|(|X| \setminus |U|)\). Let \(V \subset Y\) be the open subspace with \(|V| = |f|(|U|)\), see Properties of Spaces, Lemma 03BZ. Then \(U \to V\) is integral and surjective, hence \(V\) is an affine scheme by Proposition 07VT. This concludes the proof.

Lemma

Let \(S\) be a scheme. Let \(f : X \to B\) and \(B' \to B\) be morphisms of algebraic spaces over \(S\). Assume

1. \(B' \to B\) is a closed immersion,

2. \(|B'| \to |B|\) is bijective,

3. \(X \times_B B' \to B'\) is a closed immersion, and

4. \(X \to B\) is of finite type or \(B' \to B\) is of finite presentation.

Then \(f : X \to B\) is a closed immersion.

Proof

Assumptions (1) and (2) imply that \(B_{red} = B'_{red}\). Set \(X' = X \times_B B'\). Then \(X' \to X\) is closed immersion and \(X'_{red} = X_{red}\). Let \(U \to B\) be an étale morphism with \(U\) affine. Then \(X' \times_B U \to X \times_B U\) is a closed immersion of algebraic spaces inducing an isomorphism on underlying reduced spaces. Since \(X' \times_B U\) is a scheme (as \(B' \to B\) and \(X' \to B'\) are representable) so is \(X \times_B U\) by Lemma 07VU. Hence \(X \to B\) is representable too. Thus we reduce to the case of schemes, see Morphisms, Lemma 0896.

Finite cover by a scheme

As an application of the limit results of this chapter, we prove that given any quasi-compact and quasi-separated algebraic space \(X\), there is a scheme \(Y\) and a surjective, finite morphism \(Y \to X\). We will rely on the already proven result that we can find a finite integral cover by a scheme, which was proved in Decent Spaces, Section 0D2T.

Proposition

Let \(S\) be a scheme. Let \(X\) be a quasi-compact and quasi-separated algebraic space over \(S\).

1. There exists a surjective finite morphism \(Y \to X\) of finite presentation where \(Y\) is a scheme,

2. given a surjective étale morphism \(U \to X\) we may choose \(Y \to X\) such that for every \(y \in Y\) there is an open neighbourhood \(V \subset Y\) such that \(V \to X\) factors through \(U\).

Proof

Part (1) is the special case of (2) with \(U = X\). Let \(Y \to X\) be as in Decent Spaces, Lemma 09YB. Choose a finite affine open covering \(Y = \bigcup V_j\) such that \(V_j \to X\) factors through \(U\). We can write \(Y = \lim Y_i\) with \(Y_i \to X\) finite and of finite presentation, see Lemma 09YA. For large enough \(i\) the algebraic space \(Y_i\) is a scheme, see Lemma 07SR. For large enough \(i\) we can find affine opens \(V_{i, j} \subset Y_i\) whose inverse image in \(Y\) recovers \(V_j\), see Lemma 0827. For even larger \(i\) the morphisms \(V_j \to U\) over \(X\) come from morphisms \(V_{i, j} \to U\) over \(X\), see Proposition 04AK. This finishes the proof.

Lemma

Let \(S\) be a scheme. Let \(f : X \to Y\) be an integral morphism of algebraic spaces over \(S\). Assume \(Y\) quasi-compact and quasi-separated. Let \(V \subset Y\) be a quasi-compact open subspace such that \(f^{-1}(V) \to V\) is finite and of finite presentation. Then \(X\) can be written as a directed limit \(X = \lim X_i\) where \(f_i : X_i \to Y\) are finite and of finite presentation such that \(f^{-1}(V) \to f_i^{-1}(V)\) is an isomorphism for all \(i\).

Proof

This lemma is a slight refinement of Proposition 09YC. Consider the integral quasi-coherent \(\mathcal{O}_Y\)-algebra \(\mathcal{A} = f_*\mathcal{O}_X\). In the next paragraph, we will write \(\mathcal{A} = \colim \mathcal{A}_i\) as a directed colimit of finite and finitely presented \(\mathcal{O}_Y\)-algebras \(\mathcal{A}_i\) such that \(\mathcal{A}_i|_V = \mathcal{A}|_V\). Having done this we set \(X_i = \underline{\Spec}_Y(\mathcal{A}_i)\), see Morphisms of Spaces, Definition 081W. By construction \(X_i \to Y\) is finite and of finite presentation, \(X = \lim X_i\), and \(f_i^{-1}(V) = f^{-1}(V)\).

The proof of the assertion on algebras is similar to the proof of part (2) of Lemma 082C. First, write \(\mathcal{A} = \colim \mathcal{F}_i\) as a colimit of finitely presented \(\mathcal{O}_Y\)-modules using Lemma 07V9. Since \(\mathcal{A}|_V\) is a finite type \(\mathcal{O}_V\)-module we may and do assume that \(\mathcal{F}_i|_V \to \mathcal{A}|_V\) is surjective for all \(i\). For each \(i\), let \(\mathcal{J}_i\) be the kernel of the map \[\text{Sym}^*_{\mathcal{O}_X}(\mathcal{F}_i) \longrightarrow \mathcal{A}\] For \(i' \geq i\) there is an induced map \(\mathcal{J}_i \to \mathcal{J}_{i'}\). We have \(\mathcal{A} = \colim \text{Sym}^*_{\mathcal{O}_X}(\mathcal{F}_i)/\mathcal{J}_i\). Moreover, the quasi-coherent \(\mathcal{O}_X\)-algebras \(\text{Sym}^*_{\mathcal{O}_X}(\mathcal{F}_i)/\mathcal{J}_i\) are finite (as finite type quasi-coherent subalgebras of the integral quasi-coherent \(\mathcal{O}_Y\)-algebra \(\mathcal{A}\) over \(\mathcal{O}_X\)). The restriction of \(\text{Sym}^*_{\mathcal{O}_X}(\mathcal{F}_i)/\mathcal{J}_i\) to \(V\) is \(\mathcal{A}|_V\) by the surjectivity above. Hence \(\mathcal{J}_i|_V\) is finitely generated as an ideal sheaf of \(\text{Sym}^*_{\mathcal{O}_X}(\mathcal{F}_i)|_V\) due to the fact that \(\mathcal{A}|_V\) is finitely presented as an \(\mathcal{O}_Y\)-algebra. Write \(\mathcal{J}_i = \colim \mathcal{E}_{ik}\) as a colimit of finitely presented \(\mathcal{O}_X\)-modules. We may and do assume that \(\mathcal{E}_{ik}|_V\) generates \(\mathcal{J}_i|_V\) as a sheaf of ideal of \(\text{Sym}^*_{\mathcal{O}_X}(\mathcal{F}_i)|_V\) by the statement on finite generation above. Given \(i' \geq i\) and \(k\) there exists a \(k'\) such that we have a map \(\mathcal{E}_{ik} \to \mathcal{E}_{i'k'}\) making \[\xymatrix{ \mathcal{J}_i \ar[r] & \mathcal{J}_{i'} \\ \mathcal{E}_{ik} \ar[u] \ar[r] & \mathcal{E}_{i'k'} \ar[u] }\] commute. This follows from Cohomology of Spaces, Lemma 07U7. This induces a map \[\mathcal{A}_{ik} = \text{Sym}^*_{\mathcal{O}_X}(\mathcal{F}_i)/(\mathcal{E}_{ik}) \longrightarrow \text{Sym}^*_{\mathcal{O}_X}(\mathcal{F}_{i'})/(\mathcal{E}_{i'k'}) = \mathcal{A}_{i'k'}\] where \((\mathcal{E}_{ik})\) denotes the ideal generated by \(\mathcal{E}_{ik}\). The quasi-coherent \(\mathcal{O}_X\)-algebras \(\mathcal{A}_{ki}\) are of finite presentation and finite for \(k\) large enough (see proof of Lemma 086Z). Moreover we have \(\mathcal{A}_{ik}|_V = \mathcal{A}|_V\) by construction. Finally, we have \[\colim \mathcal{A}_{ik} = \colim \mathcal{A}_i = \mathcal{A}\] Namely, the first equality was shown in the proof of Lemma 086Z and the second equality because \(\mathcal{A}\) is the colimit of the modules \(\mathcal{F}_i\).

Lemma

Let \(S\) be a scheme. Let \(X\) be a quasi-compact and quasi-separated algebraic space over \(S\) such that \(|X|\) has finitely many irreducible components.

1. There exists a surjective finite morphism \(f : Y \to X\) of finite presentation where \(Y\) is a scheme such that \(f\) is finite étale over a quasi-compact dense open \(U \subset X\),

2. given a surjective étale morphism \(V \to X\) we may choose \(Y \to X\) such that for every \(y \in Y\) there is an open neighbourhood \(W \subset Y\) such that \(W \to X\) factors through \(V\).

Proof

Part (1) is the special case of (2) with \(V = X\).

Proof of (2). Let \(\pi : Y \to X\) be as in Decent Spaces, Lemma 0GUL and let \(U \subset X\) be a quasi-compact dense open such that \(\pi^{-1}(U) \to U\) is finite étale. Choose a finite affine open covering \(Y = \bigcup W_j\) such that \(W_j \to X\) factors through \(V\). We can write \(Y = \lim Y_i\) with \(\pi_i : Y_i \to X\) finite and of finite presentation such that \(\pi^{-1}(U) \to \pi_i^{-1}(U)\) is an isomorphism, see Lemma 0GUM. For large enough \(i\) the algebraic space \(Y_i\) is a scheme, see Lemma 07SR. For large enough \(i\) we can find affine opens \(W_{i, j} \subset Y_i\) whose inverse image in \(Y\) recovers \(W_j\), see Lemma 0827. For even larger \(i\) the morphisms \(W_j \to V\) over \(X\) come from morphisms \(W_{i, j} \to U\) over \(X\), see Proposition 04AK. This finishes the proof.

Lemma

Let \(S\) be a scheme. Let \(X\) be a quasi-compact and quasi-separated algebraic space over \(S\). There exists a \(t \geq 0\) and closed subspaces \[X \supset Z_0 \supset Z_1 \supset \ldots \supset Z_t = \emptyset\] such that \(Z_i \to X\) is of finite presentation, \(Z_0 \subset X\) is a thickening, and for each \(i = 0, \ldots t - 1\) there exists a scheme \(Y_i\), a surjective, finite, and finitely presented morphism \(Y_i \to Z_i\) which is finite étale over \(Z_i \setminus Z_{i + 1}\).

Proof

We may view \(X\) as an algebraic space over \(\Spec(\mathbf{Z})\), see Spaces, Definition 03I5 and Properties of Spaces, Definition 03BS. Thus we may apply Proposition 07SU. It follows that we can find an affine morphism \(X \to X_0\) with \(X_0\) of finite presentation over \(\mathbf{Z}\). If we can prove the lemma for \(X_0\), then we can pull back the stratification and the morphisms to \(X\) and get the result for \(X\); some details omitted. This reduces us to the case discussed in the next paragraph.

Assume \(X\) is of finite presentation over \(\mathbf{Z}\). Then \(X\) is Noetherian and \(|X|\) is a Noetherian topological space (with finitely many irreducible components) of finite dimension. Hence we may use induction on \(\dim(|X|)\). Any finite morphism towards \(X\) is of finite presentation, so we can ignore that requirement in the rest of the proof. By Lemma 0GUN there exists a surjective finite morphism \(Y \to X\) which is finite étale over a dense open \(U \subset X\). Set \(Z_0 = X\) and let \(Z_1 \subset X\) be the reduced closed subspace with \(|Z_1| = |X| \setminus |U|\). By induction we find an integer \(t \geq 0\) and a filtration \[Z_1 \supset Z_{1, 0} \supset Z_{1, 1} \supset \ldots \supset Z_{1, t} = \emptyset\] by closed subspaces, where \(Z_{1, 0} \to Z_1\) is a thickening and there exist finite surjective morphisms \(Y_{1, i} \to Z_{1, i}\) which are finite étale over \(Z_{1, i} \setminus Z_{1, i + 1}\). Since \(Z_1\) is reduced, we have \(Z_1 = Z_{1, 0}\). Hence we can set \(Z_i = Z_{1, i - 1}\) and \(Y_i = Y_{1, i - 1}\) for \(i \geq 1\) and the lemma is proved.

Obtaining schemes

A few more techniques to show an algebraic space is a scheme. The first is that we can show there is a minimal closed subspace which is not a scheme.

Lemma

Let \(S\) be a scheme. Let \(X\) be a quasi-compact and quasi-separated algebraic space over \(S\). If \(X\) is not a scheme, then there exists a closed subspace \(Z \subset X\) such that \(Z\) is not a scheme, but every proper closed subspace \(Z' \subset Z\) is a scheme.

Proof

We prove this by Zorn’s lemma. Let \(\mathcal{Z}\) be the set of closed subspaces \(Z\) which are not schemes ordered by inclusion. By assumption \(\mathcal{Z}\) contains \(X\), hence is nonempty. If \(Z_\alpha\) is a totally ordered subset of \(\mathcal{Z}\), then \(Z = \bigcap Z_\alpha\) is in \(\mathcal{Z}\). Namely, \[Z = \lim Z_\alpha\] and the transition morphisms are affine. Thus we may apply Lemma 07SR to see that if \(Z\) were a scheme, then so would one of the \(Z_\alpha\). (This works even if \(Z = \emptyset\), but note that by Lemma 086W this cannot happen.) Thus \(\mathcal{Z}\) has minimal elements by Zorn’s lemma.

Now we can prove a little bit about these minimal non-schemes.

Lemma

Let \(S\) be a scheme. Let \(X\) be a quasi-compact and quasi-separated algebraic space over \(S\). Assume that every proper closed subspace \(Z \subset X\) is a scheme, but \(X\) is not a scheme. Then \(X\) is reduced and irreducible.

Proof

We see that \(X\) is reduced by Lemma 07VU. Choose closed subsets \(T_1 \subset |X|\) and \(T_2 \subset |X|\) such that \(|X| = T_1 \cup T_2\). If \(T_1\) and \(T_2\) are proper closed subsets, then the corresponding reduced induced closed subspaces \(Z_1, Z_2 \subset X\) (Properties of Spaces, Definition 047X) are schemes and so is \(Z = Z_1 \times_X Z_2 = Z_1 \cap Z_2\) as a closed subscheme of either \(Z_1\) or \(Z_2\). Observe that the coproduct \(Z_1 \amalg_Z Z_2\) exists in the category of schemes, see More on Morphisms, Lemma 0B7M. One way to proceed, is to show that \(Z_1 \amalg_Z Z_2\) is isomorphic to \(X\), but we cannot use this here as the material on pushouts of algebraic spaces comes later in the theory. Instead we will use Lemma 07VS to find an affine neighbourhood of every point. Namely, let \(x \in |X|\). If \(x \not \in Z_1\), then \(x\) has a neighbourhood which is a scheme, namely, \(X \setminus Z_1\). Similarly if \(x \not \in Z_2\). If \(x \in Z = Z_1 \cap Z_2\), then we choose an affine open \(U \subset Z_1 \amalg_Z Z_2\) containing \(z\). Then \(U_1 = Z_1 \cap U\) and \(U_2 = Z_2 \cap U\) are affine opens whose intersections with \(Z\) agree. Since \(|Z_1| = T_1\) and \(|Z_2| = T_2\) are closed subsets of \(|X|\) which intersect in \(|Z|\), we find an open \(W \subset |X|\) with \(W \cap T_1 = |U_1|\) and \(W \cap T_2 = |U_2|\). Let \(W\) denote the corresponding open subspace of \(X\). Then \(x \in |W|\) and the morphism \(U_1 \amalg U_2 \to W\) is a surjective finite morphism whose source is an affine scheme. Thus \(W\) is an affine scheme by Lemma 07VS.

A key point in the following lemma is that we only need to check the condition in the images of points of \(X\).

Lemma

Let \(f: X \to S\) be a quasi-compact and quasi-separated morphism from an algebraic space to a scheme \(S\). If for every \(x \in |X|\) with image \(s = f(x) \in S\) the algebraic space \(X \times_S \Spec(\mathcal{O}_{S,s})\) is a scheme, then \(X\) is a scheme.

Proof

Let \(x \in |X|\). It suffices to find an open neighbourhood \(U\) of \(s = f(x)\) such that \(X \times_S U\) is a scheme. As \(X \times_S \Spec(\mathcal{O}_{S, s})\) is a scheme, then, since \(\mathcal{O}_{S, s} = \colim \mathcal{O}_S(U)\) where the colimit is over affine open neighbourhoods of \(s\) in \(S\) we see that \[X \times_S \Spec(\mathcal{O}_{S, s}) = \lim X \times_S U\] By Lemma 07SR we see that \(X \times_S U\) is a scheme for some \(U\).

Instead of restricting to local rings as in Lemma 0B80, we can restrict to closed subschemes of the base.

Lemma

Let \(\varphi : X \to \Spec(A)\) be a quasi-compact and quasi-separated morphism from an algebraic space to an affine scheme. If \(X\) is not a scheme, then there exists an ideal \(I \subset A\) such that the base change \(X_{A/I}\) is not a scheme, but for every \(I \subset I'\), \(I \not = I'\) the base change \(X_{A/I'}\) is a scheme.

Proof

We prove this by Zorn’s lemma. Let \(\mathcal{I}\) be the set of ideals \(I\) such that \(X_{A/I}\) is not a scheme. By assumption \(\mathcal{I}\) contains \((0)\). If \(I_\alpha\) is a chain of ideals in \(\mathcal{I}\), then \(I = \bigcup I_\alpha\) is in \(\mathcal{I}\). Namely, \(A/I = \colim A/I_\alpha\), hence \[X_{A/I} = \lim X_{A/I_\alpha}\] Thus we may apply Lemma 07SR to see that if \(X_{A/I}\) were a scheme, then so would be one of the \(X_{A/I_\alpha}\). Thus \(\mathcal{I}\) has maximal elements by Zorn’s lemma.

Glueing in closed fibres

Applying our theory above to the spectrum of a local ring we obtain a few pleasing glueing results for relative algebraic spaces. We first prove a helper lemma (which will be vastly generalized in Bootstrap, Section 04SJ).

Lemma

Let \(S = U \cup W\) be an open covering of a scheme. Then the functor \[FP_S \longrightarrow FP_U \times_{FP_{U \cap W}} FP_W\] given by base change is an equivalence where \(FP_T\) is the category of algebraic spaces of finite presentation over the scheme \(T\).

Proof

First, since \(S = U \cup W\) is a Zariski covering, we see that the category of sheaves on \((\Sch/S)_{fppf}\) is equivalent to the category of triples \((\mathcal{F}_U, \mathcal{F}_W, \varphi)\) where \(\mathcal{F}_U\) is a sheaf on \((\Sch/U)_{fppf}\), \(\mathcal{F}_W\) is a sheaf on \((\Sch/W)_{fppf}\), and \[\varphi : \mathcal{F}_U|_{(\Sch/U \cap W)_{fppf}} \longrightarrow \mathcal{F}_W|_{(\Sch/U \cap W)_{fppf}}\] is an isomorphism. See Sites, Lemma 04TS (note that no other gluing data are necessary because \(U \times_S U = U\), \(W \times_S W = W\) and that the cocycle condition is automatic for the same reason). Now, if the sheaf \(\mathcal{F}\) on \((\Sch/S)_{fppf}\) maps to \((\mathcal{F}_U, \mathcal{F}_W, \varphi)\) via this equivalence, then \(\mathcal{F}\) is an algebraic space if and only if \(\mathcal{F}_U\) and \(\mathcal{F}_W\) are algebraic spaces. This follows immediately from Algebraic Spaces, Lemma 02WR as \(\mathcal{F}_U \to \mathcal{F}\) and \(\mathcal{F}_W \to \mathcal{F}\) are representable by open immersions and cover \(\mathcal{F}\). Finally, in this case the algebraic space \(\mathcal{F}\) is of finite presentation over \(S\) if and only if \(\mathcal{F}_U\) is of finite presentation over \(U\) and \(\mathcal{F}_W\) is of finite presentation over \(W\) by Morphisms of Spaces, Lemmas 03KG, 03KM, and 0410.

Lemma

Let \(S\) be a scheme. Let \(s \in S\) be a closed point such that \(U = S \setminus \{s\} \to S\) is quasi-compact. With \(V = \Spec(\mathcal{O}_{S, s}) \setminus \{s\}\) there is an equivalence of categories \[FP_S \longrightarrow FP_U \times_{FP_V} FP_{\Spec(\mathcal{O}_{S, s})}\] where \(FP_T\) is the category of algebraic spaces of finite presentation over \(T\).

Proof

Let \(W \subset S\) be an open neighbourhood of \(s\). The functor \[FP_S \to FP_U \times_{FP_{W \setminus \{s\}}} FP_W\] is an equivalence of categories by Lemma 0E8Z. We have \(\mathcal{O}_{S, s} = \colim \mathcal{O}_W(W)\) where \(W\) runs over the affine open neighbourhoods of \(s\). Hence \(\Spec(\mathcal{O}_{S, s}) = \lim W\) where \(W\) runs over the affine open neighbourhoods of \(s\). Thus the category of algebraic spaces of finite presentation over \(\Spec(\mathcal{O}_{S, s})\) is the limit of the category of algebraic spaces of finite presentation over \(W\) where \(W\) runs over the affine open neighbourhoods of \(s\), see Lemma 07SK. For every affine open \(s \in W\) we see that \(U \cap W\) is quasi-compact as \(U \to S\) is quasi-compact. Hence \(V = \lim W \cap U = \lim W \setminus \{s\}\) is a limit of quasi-compact and quasi-separated schemes (see Limits, Lemma 01YX). Thus also the category of algebraic spaces of finite presentation over \(V\) is the limit of the categories of algebraic spaces of finite presentation over \(W \cap U\) where \(W\) runs over the affine open neighbourhoods of \(s\). The lemma follows formally from a combination of these results.

Lemma

Let \(S\) be a scheme. Let \(U \subset S\) be a retrocompact open. Let \(s \in S\) be a point in the complement of \(U\). With \(V = \Spec(\mathcal{O}_{S, s}) \cap U\) there is an equivalence of categories \[\colim_{s \in U' \supset U\text{ open}} FP_{U'} \longrightarrow FP_U \times_{FP_V} FP_{\Spec(\mathcal{O}_{S, s})}\] where \(FP_T\) is the category of algebraic spaces of finite presentation over \(T\).

Proof

Let \(W \subset S\) be an open neighbourhood of \(s\). By Lemma 0E8Z the functor \[FP_{U \cup W} \longrightarrow FP_U \times_{FP_{U \cap W}} FP_W\] is an equivalence of categories. We have \(\mathcal{O}_{S, s} = \colim \mathcal{O}_W(W)\) where \(W\) runs over the affine open neighbourhoods of \(s\). Hence \(\Spec(\mathcal{O}_{S, s}) = \lim W\) where \(W\) runs over the affine open neighbourhoods of \(s\). Thus the category of algebraic spaces of finite presentation over \(\Spec(\mathcal{O}_{S, s})\) is the limit of the category of algebraic spaces of finite presentation over \(W\) where \(W\) runs over the affine open neighbourhoods of \(s\), see Lemma 07SK. For every affine open \(s \in W\) we see that \(U \cap W\) is quasi-compact as \(U \to S\) is quasi-compact. Hence \(V = \lim W \cap U\) is a limit of quasi-compact and quasi-separated schemes (see Limits, Lemma 01YX). Thus also the category of algebraic spaces of finite presentation over \(V\) is the limit of the categories of algebraic spaces of finite presentation over \(W \cap U\) where \(W\) runs over the affine open neighbourhoods of \(s\). The lemma follows formally from a combination of these results.

Lemma

Let \(S\) be a scheme. Let \(s_1, \ldots, s_n \in S\) be pairwise distinct closed points such that \(U = S \setminus \{s_1, \ldots, s_n\} \to S\) is quasi-compact. With \(S_i = \Spec(\mathcal{O}_{S, s_i})\) and \(U_i = S_i \setminus \{s_i\}\) there is an equivalence of categories \[FP_S \longrightarrow FP_U \times_{(FP_{U_1} \times \ldots \times FP_{U_n})} (FP_{S_1} \times \ldots \times FP_{S_n})\] where \(FP_T\) is the category of algebraic spaces of finite presentation over \(T\).

Proof

For \(n = 1\) this is Lemma 0E90. For \(n > 1\) the lemma can be proved in exactly the same way or it can be deduced from it. For example, suppose that \(f_i : X_i \to S_i\) are objects of \(FP_{S_i}\) and \(f : X \to U\) is an object of \(FP_U\) and we’re given isomorphisms \(X_i \times_{S_i} U_i = X \times_U U_i\). By Lemma 0E90 we can find a morphism \(f' : X' \to U' = S \setminus \{s_1, \ldots, s_{n - 1}\}\) which is of finite presentation, which is isomorphic to \(X_i\) over \(S_i\), which is isomorphic to \(X\) over \(U\), and these isomorphisms are compatible with the given isomorphism \(X_i \times_{S_n} U_n = X \times_U U_n\). Then we can apply induction to \(f_i : X_i \to S_i\), \(i \leq n - 1\), \(f' : X' \to U'\), and the induced isomorphisms \(X_i \times_{S_i} U_i = X' \times_{U'} U_i\), \(i \leq n - 1\). This shows essential surjectivity. We omit the proof of fully faithfulness.

Application to modifications

Using limits we can describe the category of modifications of a decent algebraic space over a closed point in terms of the henselian local ring.

Lemma

Let \(S\) be a scheme. Consider a separated étale morphism \(f : V \to W\) of algebraic spaces over \(S\). Assume there exists a closed subspace \(T \subset W\) such that \(f^{-1}T \to T\) is an isomorphism. Then, with \(W^0 = W \setminus T\) and \(V^0 = f^{-1}W^0\) the base change functor \[\left\{ \begin{matrix} g : X \to W\text{ morphism of algebraic spaces} \\ g^{-1}(W^0) \to W^0\text{ is an isomorphism} \end{matrix} \right\} \longrightarrow \left\{ \begin{matrix} h : Y \to V\text{ morphism of algebraic spaces} \\ h^{-1}(V^0) \to V^0\text{ is an isomorphism} \end{matrix} \right\}\] is an equivalence of categories.

Proof

Since \(V \to W\) is separated we see that \(V \times_W V = \Delta(V) \amalg U\) for some open and closed subspace \(U\) of \(V \times_W V\). By the assumption that \(f^{-1}T \to T\) is an isomorphism we see that \(U \times_W T = \emptyset\), i.e., the two projections \(U \to V\) maps into \(V^0\).

Given \(h : Y \to V\) in the right hand category, consider the contravariant functor \(X\) on \((\Sch/S)_{fppf}\) defined by the rule \[X(T) = \{(w, y) \mid w : T \to W,\ y : T \times_{w, W} V \to Y\text{ morphism over }V\}\] Denote \(g : X \to W\) the map sending \((w, y) \in X(T)\) to \(w \in W(T)\). Since \(h^{-1}V^0 \to V^0\) is an isomorphism, we see that if \(w : T \to W\) maps into \(W^0\), then there is a unique choice for \(h\). In other words \(X \times_{g, W} W^0 = W^0\). On the other hand, consider a \(T\)-valued point \((w, y, v)\) of \(X \times_{g, W, f} V\). Then \(w = f \circ v\) and \[y : T \times_{f \circ v, W} V \longrightarrow V\] is a morphism over \(V\). Consider the morphism \[T \times_{f \circ v, W} V \xrightarrow{(v, \text{id}_V)} V \times_W V = V \amalg U\] The inverse image of \(V\) is \(T\) embedded via \((\text{id}_T, v) : T \to T \times_{f \circ v, W} V\). The composition \(y' = y \circ (\text{id}_T, v) : T \to Y\) is a morphism with \(v = h \circ y'\) which determines \(y\) because the restriction of \(y\) to the other part is uniquely determined as \(U\) maps into \(V^0\) by the second projection. It follows that \(X \times_{g, W, f} V \to Y\), \((w, y, v) \mapsto y'\) is an isomorphism.

Thus if we can show that \(X\) is an algebraic space, then we are done. Since \(V \to W\) is separated and étale it is representable by Morphisms of Spaces, Lemma 0418 (and Morphisms of Spaces, Lemma 03XU). Of course \(W^0 \to W\) is representable and étale as it is an open immersion. Thus \[W^0 \amalg Y = X \times_{g, W} W^0 \amalg X \times_{g, W, f} V = X \times_{g, W} (W^0 \amalg V) \longrightarrow X\] is representable, surjective, and étale by Spaces, Lemmas 02WB and 02WL. Thus \(X\) is an algebraic space by Spaces, Lemma 0BGR.

Lemma

Notation and assumptions as in Lemma 0BGY. Let \(g : X \to W\) correspond to \(h : Y \to V\) via the equivalence. Then \(g\) is quasi-compact, quasi-separated, separated, locally of finite presentation, of finite presentation, locally of finite type, of finite type, proper, integral, finite, and add more here if and only if \(h\) is so.

Proof

If \(g\) is quasi-compact, quasi-separated, separated, locally of finite presentation, of finite presentation, locally of finite type, of finite type, proper, finite, so is \(h\) as a base change of \(g\) by Morphisms of Spaces, Lemmas 03HF, 03KL, 03XR, 03XH, 04WP, 03ZS. Conversely, let \(P\) be a property of morphisms of algebraic spaces which is étale local on the base and which holds for the identity morphism of any algebraic space. Since \(\{W^0 \to W, V \to W\}\) is an étale covering, to prove that \(g\) has \(P\) it suffices to show that \(h\) has \(P\). Thus we conclude using Morphisms of Spaces, Lemmas 03KG, 03KM, 0410, 040Y, 083R, 03ZQ.

Lemma

Let \(S\) be a scheme. Let \(X\) be a decent algebraic space over \(S\). Let \(x \in |X|\) be a closed point such that \(U = X \setminus \{x\} \to X\) is quasi-compact. With \(V = \Spec(\mathcal{O}_{X, x}^h) \setminus \{\mathfrak m_x^h\}\) the base change functor \[\left\{ \begin{matrix} f : Y \to X\text{ of finite presentation} \\ f^{-1}(U) \to U\text{ is an isomorphism} \end{matrix} \right\} \longrightarrow \left\{ \begin{matrix} g : Y \to \Spec(\mathcal{O}_{X, x}^h)\text{ of finite presentation} \\ g^{-1}(V) \to V\text{ is an isomorphism} \end{matrix} \right\}\] is an equivalence of categories.

Proof

Let \(a : (W, w) \to (X, x)\) be an elementary étale neighbourhood of \(x\) with \(W\) affine as in Decent Spaces, Lemma 0BBP. Since \(x\) is a closed point of \(X\) and \(w\) is the unique point of \(W\) lying over \(x\), we see that \(w\) is a closed point of \(W\). Since \(a\) is étale and identifies residue fields at \(x\) and \(w\), it follows that \(a\) induces an isomorphism \(a^{-1}x \to x\) (as closed subspaces of \(X\) and \(W\)). Thus we may apply Lemma 0BGY and 0BGZ to reduce the problem to the case where \(X\) is an affine scheme.

Assume \(X\) is an affine scheme. Recall that \(\mathcal{O}_{X, x}^h\) is the colimit of \(\Gamma(U, \mathcal{O}_U)\) over affine elementary étale neighbourhoods \((U, u) \to (X, x)\). Recall that the category of these neighbourhoods is cofiltered, see Decent Spaces, Lemma 0BGV or More on Morphisms, Lemma 057B. Then \(\Spec(\mathcal{O}_{X, x}^h) = \lim U\) and \(V = \lim U \setminus \{u\}\) (Lemma 07SF) where the limits are taken over the same category. Thus by Lemma 07SK The category on the right is the colimit of the categories for the pairs \((U, u)\). And by the material in the first paragraph, each of these categories is equivalent to the category for the pair \((X, x)\). This finishes the proof.

Universally closed morphisms

In this section we discuss when a quasi-compact (but not necessarily separated) morphism is universally closed. We first prove a lemma which will allow us to check universal closedness after a base change which is locally of finite presentation.

Lemma

Let \(S\) be a scheme. Let \(f : X \to Y\) and \(g : Z \to Y\) be morphisms of algebraic spaces over \(S\). Let \(z \in |Z|\) and let \(T \subset |X \times_Y Z|\) be a closed subset with \(z \not \in \Im(T \to |Z|)\). If \(f\) is quasi-compact, then there exists an étale neighbourhood \((V, v) \to (Z, z)\), a commutative diagram \[\xymatrix{ V \ar[d] \ar[r]_a & Z' \ar[d]^b \\ Z \ar[r]^g & Y, }\] and a closed subset \(T' \subset |X \times_Y Z'|\) such that

1. the morphism \(b : Z' \to Y\) is locally of finite presentation,

2. with \(z' = a(v)\) we have \(z' \not \in \Im(T' \to |Z'|)\), and

3. the inverse image of \(T\) in \(|X \times_Y V|\) maps into \(T'\) via \(|X \times_Y V| \to |X \times_Y Z'|\).

Moreover, we may assume \(V\) and \(Z'\) are affine schemes and if \(Z\) is a scheme we may assume \(V\) is an affine open neighbourhood of \(z\).

Proof

We will deduce this from the corresponding result for morphisms of schemes. Let \(y \in |Y|\) be the image of \(z\). First we choose an affine étale neighbourhood \((U, u) \to (Y, y)\) and then we choose an affine étale neighbourhood \((V, v) \to (Z, z)\) such that the morphism \(V \to Y\) factors through \(U\). Then we may replace

1. \(X \to Y\) by \(X \times_Y U \to U\),

2. \(Z \to Y\) by \(V \to U\),

3. \(z\) by \(v\), and

4. \(T\) by its inverse image in \(|(X \times_Y U) \times_U V| = |X \times_Y V|\).

In fact, below we will show that after replacing \(V\) by an affine open neighbourhood of \(v\) there will be a morphism \(a : V \to Z'\) for some \(Z' \to U\) of finite presentation and a closed subset \(T'\) of \(|(X \times_Y U) \times_U Z'| = |X \times_Y Z'|\) such that \(T\) maps into \(T'\) and \(a(v) \not \in \Im(T' \to |Z'|)\). Thus we may and do assume that \(Z\) and \(Y\) are affine schemes with the proviso that we need to find a solution where \(V\) is an open neighbourhood of \(z\).

Since \(f\) is quasi-compact and \(Y\) is affine, the algebraic space \(X\) is quasi-compact. Choose an affine scheme \(W\) and a surjective étale morphism \(W \to X\). Let \(T_W \subset |W \times_Y Z|\) be the inverse image of \(T\). Then \(z\) is not in the image of \(T_W\). By the schemes case (Limits, Lemma 05BD) we can find an open neighbourhood \(V \subset Z\) of \(z\) a commutative diagram of schemes \[\xymatrix{ V \ar[d] \ar[r]_a & Z' \ar[d]^b \\ Z \ar[r]^g & Y, }\] and a closed subset \(T' \subset |W \times_Y Z'|\) such that

1. the morphism \(b : Z' \to Y\) is locally of finite presentation,

2. with \(z' = a(z)\) we have \(z' \not \in \Im(T' \to Z')\), and

3. \(T_1 = T_W \cap |W \times_Y V|\) maps into \(T'\) via \(|W \times_Y V| \to |W \times_Y Z'|\).

The commutative diagram \[\xymatrix{ W \times_Y Z \ar[d] & W \times_Y V \ar[l] \ar[rr]_{a_1} \ar[d]_c & & W \times_Y Z' \ar[d]^q \\ X \times_Y Z & X \times_Y V \ar[l] \ar[rr]^{a_2} & & X \times_Y Z' }\] has cartesian squares and the vertical maps are, surjective, étale and a fortiori open. Looking at the left hand square we see that \(T_1 = T_W \cap |W \times_Y V|\) is the inverse image of \(T_2 = T \cap |X \times_Y V|\) by \(c\). By Properties of Spaces, Lemma 03H4 we get \(a_1(T_1) = q^{-1}(a_2(T_2))\). By Topology, Lemma 02YB we get \[q^{-1}\left(\overline{a_2(T_2)}\right) = \overline{q^{-1}(a_2(T_2))} = \overline{a_1(T_1)} \subset T'\] As \(q\) is surjective the image of \(\overline{a_2(T_2)} \to |Z'|\) does not contain \(z'\) since the same is true for \(T'\). Thus we can take the diagram with \(Z', V, a, b\) above and the closed subset \(\overline{a_2(T_2)} \subset |X \times_Y Z'|\) as a solution to the problem posed by the lemma.

Lemma

Let \(S\) be a scheme. Let \(f : X \to Y\) be a quasi-compact morphism of algebraic spaces over \(S\). The following are equivalent

1. \(f\) is universally closed,

2. for every morphism \(Z \to Y\) which is locally of finite presentation the map \(|X \times_Y Z| \to |Z|\) is closed, and

3. there exists a scheme \(V\) and a surjective étale morphism \(V \to Y\) such that \(|\mathbf{A}^n \times (X \times_Y V)| \to |\mathbf{A}^n \times V|\) is closed for all \(n \geq 0\).

Proof

It is clear that (1) implies (2). Suppose that \(|X \times_Y Z| \to |Z|\) is not closed for some morphism of algebraic spaces \(Z \to Y\) over \(S\). This means that there exists some closed subset \(T \subset |X \times_Y Z|\) such that \(\Im(T \to |Z|)\) is not closed. Pick \(z \in |Z|\) in the closure of the image of \(T\) but not in the image. Apply Lemma 0CM8. We find an étale neighbourhood \((V, v) \to (Z, z)\), a commutative diagram \[\xymatrix{ V \ar[d] \ar[r]_a & Z' \ar[d]^b \\ Z \ar[r]^g & Y, }\] and a closed subset \(T' \subset |X \times_Y Z'|\) such that

1. the morphism \(b : Z' \to Y\) is locally of finite presentation,

2. with \(z' = a(v)\) we have \(z' \not \in \Im(T' \to |Z'|)\), and

3. the inverse image of \(T\) in \(|X \times_Y V|\) maps into \(T'\) via \(|X \times_Y V| \to |X \times_Y Z'|\).

We claim that \(z'\) is in the closure of \(\Im(T' \to |Z'|)\) which implies that \(|X \times_Y Z'| \to |Z'|\) is not closed. The claim shows that (2) implies (1). To see the claim is true we contemplate following commutative diagram \[\xymatrix{ X \times_Y Z \ar[d] & X \times_Y V \ar[l] \ar[d] \ar[r] & X \times_Y Z' \ar[d] \\ Z & V \ar[l] \ar[r]^a & Z' }\] Let \(T_V \subset |X \times_Y V|\) be the inverse image of \(T\). By Properties of Spaces, Lemma 03H4 the image of \(T_V\) in \(|V|\) is the inverse image of the image of \(T\) in \(|Z|\). Then since \(z\) is in the closure of the image of \(T \to |Z|\) and since \(|V| \to |Z|\) is open, we see that \(v\) is in the closure of the image of \(T_V \to |V|\). Since the image of \(T_V\) in \(|X \times_Y Z'|\) is contained in \(|T'|\) it follows immediately that \(z' = a(v)\) is in the closure of the image of \(T'\).

It is clear that (1) implies (3). Let \(V \to Y\) be as in (3). If we can show that \(X \times_Y V \to V\) is universally closed, then \(f\) is universally closed by Morphisms of Spaces, Lemma 03IT. Thus it suffices to show that \(f : X \to Y\) satisfies (2) if \(f\) is a quasi-compact morphism of algebraic spaces, \(Y\) is a scheme, and \(|\mathbf{A}^n \times X| \to |\mathbf{A}^n \times Y|\) is closed for all \(n\). Let \(Z \to Y\) be locally of finite presentation. We have to show the map \(|X \times_Y Z| \to |Z|\) is closed. This question is étale local on \(Z\) hence we may assume \(Z\) is affine (some details omitted). Since \(Y\) is a scheme, \(Z\) is affine, and \(Z \to Y\) is locally of finite presentation we can find an immersion \(Z \to \mathbf{A}^n \times Y\), see Morphisms, Lemma 04II. Consider the cartesian diagram \[\vcenter{ \xymatrix{ X \times_Y Z \ar[d] \ar[r] & \mathbf{A}^n \times X \ar[d] \\ Z \ar[r] & \mathbf{A}^n \times Y } } \quad \begin{matrix} \text{inducing the} \\ \text{cartesian square} \end{matrix} \quad \vcenter{ \xymatrix{ |X \times_Y Z| \ar[d] \ar[r] & |\mathbf{A}^n \times X| \ar[d] \\ |Z| \ar[r] & |\mathbf{A}^n \times Y| } }\] of topological spaces whose horizontal arrows are homeomorphisms onto locally closed subsets (Properties of Spaces, Lemma 0ABJ). Thus every closed subset \(T\) of \(|X \times_Y Z|\) is the pullback of a closed subset \(T'\) of \(|\mathbf{A}^n \times Y|\). Since the assumption is that the image of \(T'\) in \(|\mathbf{A}^n \times X|\) is closed we conclude that the image of \(T\) in \(|Z|\) is closed as desired.

Lemma

Let \(S\) be a scheme. Let \(f : X \to Y\) be a morphism of algebraic spaces over \(S\). Assume \(f\) separated and of finite type. The following are equivalent

1. The morphism \(f\) is proper.

2. For any morphism \(Y \to Z\) which is locally of finite presentation the map \(|X \times_Y Z| \to |Z|\) is closed, and

3. there exists a scheme \(V\) and a surjective étale morphism \(V \to Y\) such that \(|\mathbf{A}^n \times (X \times_Y V)| \to |\mathbf{A}^n \times V|\) is closed for all \(n \geq 0\).

Proof

In view of the fact that a proper morphism is the same thing as a separated, finite type, and universally closed morphism, this lemma is a special case of Lemma 0CM9.

Noetherian valuative criterion

We have already proved some results in Cohomology of Spaces, Section 0ARI. The corresponding section for schemes is Limits, Section 0CM1.

Many of the results in this section can (and perhaps should) be proved by appealing to the following lemma, although we have not always done so.

Lemma

Let \(S\) be a scheme. Let \(f : X \to Y\) be a morphism of algebraic spaces over \(S\). Assume \(f\) finite type and \(Y\) locally Noetherian. Let \(y \in |Y|\) be a point in the closure of the image of \(|f|\). Then there exists a commutative diagram \[\xymatrix{ \Spec(K) \ar[r] \ar[d] & X \ar[d]^f \\ \Spec(A) \ar[r] & Y }\] where \(A\) is a discrete valuation ring and \(K\) is its field of fractions mapping the closed point of \(\Spec(A)\) to \(y\). Moreover, we can assume that the point \(x \in |X|\) corresponding to \(\Spec(K) \to X\) is a codimension \(0\) point³ and that \(K\) is the residue field of a point on a scheme étale over \(X\).

Proof

Choose an affine scheme \(V\), a point \(v \in V\) and an étale morphism \(V \to Y\) mapping \(v\) to \(y\). The map \(|V| \to |Y|\) is open and by Properties of Spaces, Lemma 03H4 the image of \(|X \times_Y V| \to |V|\) is the inverse image of the image of \(|f|\). We conclude that the point \(v\) is in the closure of the image of \(|X \times_Y V| \to |V|\). If we prove the lemma for \(X \times_Y V \to V\) and the point \(v\), then the lemma follows for \(f\) and \(y\). In this way we reduce to the situation described in the next paragraph.

Assume we have \(f : X \to Y\) and \(y \in |Y|\) as in the lemma where \(Y\) is an affine scheme. Since \(f\) is quasi-compact, we conclude that \(X\) is quasi-compact. Hence we can choose an affine scheme \(W\) and a surjective étale morphism \(W \to X\). Then the image of \(|f|\) is the same as the image of \(W \to Y\). In this way we reduce to the case of schemes which is Limits, Lemma 0CM2.

First we state the result concerning separation. We will often use solid commutative diagrams of morphisms of algebraic spaces over a base scheme \(S\) having the following shape [0H1V]\[\begin{equation} \vcenter{ \xymatrix{ \Spec(K) \ar[r] \ar[d] & X \ar[d] \\ \Spec(A) \ar[r] \ar@{-->}[ru] & Y } } \end{equation}\] with \(A\) a valuation ring and \(K\) its field of fractions.

Lemma

Let \(S\) be a scheme. Let \(f : X \to Y\) be a morphism of algebraic spaces over \(S\). Assume \(f\) is quasi-separated and locally of finite type and \(Y\) is locally Noetherian. The following are equivalent:

1. The morphism \(f\) is separated.

2. For any diagram (0H1V) there is at most one dotted arrow.

3. For all diagrams (0H1V) with \(A\) a discrete valuation ring there is at most one dotted arrow.

4. For all diagrams (0H1V) where \(A\) is a discrete valuation ring and where the image of \(\Spec(K) \to X\) is a point of codimension \(0\) on \(X\) there is at most one dotted arrow.

Proof

We have (1) \(\Rightarrow\) (2) by Morphisms of Spaces, Lemma 03KU. The implications (2) \(\Rightarrow\) (3) and (3) \(\Rightarrow\) (4) are immediate. It remains to show (4) implies (1).

Assume (4). We have to show that the diagonal \(\Delta : X \to X \times_Y X\) is a closed immersion. We already know \(\Delta\) is representable, separated, a monomorphism, and locally of finite type, see Morphisms of Spaces, Lemma 03HK. Choose an affine scheme \(U\) and an étale morphism \(U \to X \times_Y X\). Set \(V = X \times_{\Delta, X \times_Y X} U\). It suffices to show that \(V \to U\) is a closed immersion (Morphisms of Spaces, Lemma 03M4). Since \(X \times_Y X\) is locally of finite type over \(Y\) we see that \(U\) is Noetherian (use Morphisms of Spaces, Lemmas 03XG, 03XH, and 04ZK). Note that \(V\) is a scheme as \(\Delta\) is representable. Also, \(V\) is quasi-compact because \(f\) is quasi-separated. Hence \(V \to U\) is separated and of finite type. Consider a commutative diagram \[\xymatrix{ \Spec(K) \ar[r] \ar[d] & V \ar[d] \\ \Spec(A) \ar[r] \ar@{-->}[ru] & U }\] of morphisms of schemes where \(A\) is a discrete valuation ring with fraction field \(K\) and where \(K\) is the residue field of a generic point of the Noetherian scheme \(V\). Since \(V \to X\) is étale (as a base change of the étale morphism \(U \to X \times_Y X\)) we see that the image of \(\Spec(K) \to V \to X\) is a point of codimension \(0\), see Properties of Spaces, Section 04N7. We can interpret the composition \(\Spec(A) \to U \to X \times_Y X\) as a pair of morphisms \(a, b : \Spec(A) \to X\) agreeing as morphisms into \(Y\) and equal when restricted to \(\Spec(K)\) and that this restriction maps to a point of codimension \(0\). Hence our assumption (4) guarantees \(a = b\) and we find the dotted arrow in the diagram. By Limits, Lemma 0208 we conclude that \(V \to U\) is proper. In other words, \(\Delta\) is proper. Since \(\Delta\) is a monomorphism, we find that \(\Delta\) is a closed immersion (Étale Morphisms, Lemma 04XV) as desired.

Lemma

Let \(S\) be a scheme. Let \(f : X \to Y\) be a morphism of algebraic spaces over \(S\). Assume \(f\) is quasi-separated and of finite type and \(Y\) is locally Noetherian. The following are equivalent:

1. \(f\) is proper,

2. \(f\) satisfies the valuative criterion, see Morphisms of Spaces, Definition 03IX,

3. for any diagram (0H1V) there exists exactly one dotted arrow,

4. for all diagrams (0H1V) with \(A\) a discrete valuation ring there exists exactly one dotted arrow, and

5. for all diagrams (0H1V) where \(A\) is a discrete valuation ring and where the image of \(\Spec(K) \to X\) is a point of codimension \(0\) on \(X\) there exists exactly one dotted arrow⁴.

Proof

We have (1) \(\Leftrightarrow\) (2) \(\Leftrightarrow\) (3) by Morphisms of Spaces, Lemma 0A40. It is clear that (3) \(\Rightarrow\) (4) \(\Rightarrow\) (5). To finish the proof we will now show (5) implies (1).

Assume (5). By Lemma 0H1W we see that \(f\) is separated. To finish the proof it suffices to show that \(f\) is universally closed. Let \(V \to Y\) be an étale morphism where \(V\) is an affine scheme. It suffices to show that the base change \(V \times_Y X \to V\) is universally closed, see Morphisms of Spaces, Lemma 03IT. Let \[\xymatrix{ \Spec(K) \ar[r] \ar[d] & V \times_Y X \ar[d] \ar[r] & X \ar[d] \\ \Spec(A) \ar[r] \ar@{-->}[ru] \ar@{..>}[rru] & V \ar[r] & Y }\] of algebraic spaces over \(S\) be a commutative diagram where \(A\) is a discrete valuation ring with fraction field \(K\) and where \(\Spec(K) \to V \times_Y X\) maps to a point of codimension \(0\) of the algebraic space \(V \times_Y X\). Since \(V \times_Y X \to X\) is étale it follows that the image of \(\Spec(K) \to X\) is a point of codimension \(0\) of \(X\). Thus by (5) we obtain the longer of the two dotted arrows fitting into the diagram. Then of course we obtain the shorter one as well. It follows that our assumptions hold for the morphism \(V \times_Y X \to V\) and we reduce to the case discussed in the next paragraph.

Aassume \(Y\) is a Noetherian affine scheme. In this case \(X\) is a separated Noetherian algebraic space (we already know \(f\) is separated) of finite type over \(Y\). (In particular, the algebraic space \(X\) has a dense open subspace which is a scheme by Properties of Spaces, Proposition 06NH although strictly speaking we will not need this.) Choose a quasi-projective scheme \(X'\) over \(Y\) and a proper surjective morphism \(X' \to X\) as in the weak form of Chow’s lemma (Cohomology of Spaces, Lemma 089J). We may replace \(X'\) by the disjoint union of the irreducible components which dominate an irreducible component of \(X\); details omitted. In particular, we may assume that generic points of the scheme \(X'\) map to points of codimension \(0\) of \(X\) (in this case these are exactly the generic points of \(X\)). We claim that \(X' \to Y\) is proper. The claim implies \(X\) is proper over \(Y\) by Morphisms of Spaces, Lemma 08AJ. To prove this, according to Limits, Lemma 0208 it suffices to prove that in every solid commutative diagram \[\xymatrix{ \Spec(K) \ar[r] \ar[d] & X' \ar[r] & X \ar[d] \\ \Spec(A) \ar[rr] \ar@{-->}[ru]^a \ar@{-->}[rru]_b & & Y }\] where \(A\) is a dvr with fraction field \(K\) and where \(K\) is the residue field of a generic point of \(X'\) we can find the dotted arrow \(a\) (we already know uniqueness as \(X'\) is separated). By assumption (5) we can find the dotted arrow \(b\). Then the morphism \(X' \times_{X, b} \Spec(A) \to \Spec(A)\) is a proper morphism of schemes and by the valuative criterion for morphisms of schemes we can lift \(b\) to the desired morphism \(a\).

Lemma

Let \(S\) be a scheme. Let \(f : X \to Y\) be a morphism of algebraic spaces over \(S\). Assume \(Y\) is locally Noetherian and \(f\) is of finite type. Then the following are equivalent

1. \(f\) is universally closed,

2. \(f\) satisfies the existence part of the valuative criterion,

3. there exists a scheme \(V\) and a surjective étale morphism \(V \to Y\) such that \(|\mathbf{A}^n \times X \times_Y V| \to |\mathbf{A}^n \times V|\) is closed for all \(n \geq 0\),

4. for all diagrams (0H1V) with \(A\) a discrete valuation ring there there exists a finite separable extension \(K'/K\) of fields, a discrete valuation ring \(A' \subset K'\) dominating \(A\), and a morphism \(\Spec(A') \to X\) such that the following diagram commutes \[\xymatrix{ \Spec(K') \ar[r] \ar[d] & \Spec(K) \ar[r] & X \ar[d] \\ \Spec(A') \ar[r] \ar[rru] & \Spec(A) \ar[r] & Y }\]

5. for all diagrams (0H1V) with \(A\) a discrete valuation ring there there exists a field extension \(K'/K\), a valuation ring \(A' \subset K'\) dominating \(A\), and a morphism \(\Spec(A') \to X\) such that the following diagram commutes \[\xymatrix{ \Spec(K') \ar[r] \ar[d] & \Spec(K) \ar[r] & X \ar[d] \\ \Spec(A') \ar[r] \ar[rru] & \Spec(A) \ar[r] & Y }\]

Proof

Parts (1), (2), and (3) are equivalent by Lemma 0CM9 and Morphisms of Spaces, Lemma 03KA. These equivalent conditions imply part (4) as Morphisms of Spaces, Lemma 03KH tells us that we may always choose \(K'/K\) finite separable in the existence part of the valuative criterion and this automatically forces \(A'\) to be a discrete valuation ring by Krull-Akizuki (Algebra, Lemma 00PG). The implication (4) \(\Rightarrow\) (5) is immediate. In the rest of the proof we show that (5) implies (1).

Assume (5). Chose an affine scheme \(V\) and an étale morphism \(V \to Y\). It suffices to show that the base change of \(f\) to \(V\) is universally closed, see Morphisms of Spaces, Lemma 03IT. Exactly as in the proof of Lemma 0H1X we see that assumption (5) is inherited by this base change; details omitted. This reduces us to the case discussed in the next paragraph.

Assume \(Y\) is a Noetherian affine scheme and we have (5). To prove that \(f\) is universally closed it suffices to show that \(|X \times \mathbf{A}^n| \to |Y \times \mathbf{A}^n|\) is closed for all \(n\) (by the discussion above). Since assumption (5) is inherited by the product morphism \(X \times \mathbf{A}^n \to Y \times \mathbf{A}^n\) (details omitted) we reduce to proving that \(|X| \to |Y|\) is closed.

Assume \(Y\) is a Noetherian affine scheme and we have (5). Let \(T \subset |X|\) be a closed subset. We have to show that the image of \(T\) in \(|Y|\) is closed. We may replace \(X\) by the reduced induced closed subspace structure on \(T\); we omit the verification that property (5) is preserved by this replacement. Thus we reduce to proving that the image of \(|X| \to |Y|\) is closed.

Let \(y \in |Y|\) be a point in the closure of the image of \(|X| \to |Y|\). By Lemma 0CMC we may choose a commutative diagram \[\xymatrix{ \Spec(K) \ar[r] \ar[d] & X \ar[d]^f \\ \Spec(A) \ar[r] & Y }\] where \(A\) is a discrete valuation ring and \(K\) is its field of fractions mapping the closed point of \(\Spec(A)\) to \(y\). It follows immediately from property (5) that \(y\) is in the image of \(|X| \to |Y|\) and the proof is complete.

Refined Noetherian valuative criteria

This section is the analogue of Limits, Section 0H1P. One usually does not have to consider all possible diagrams with valuation rings when checking valuative criteria.

Lemma

Let \(S\) be a scheme. Let \(f : X \to Y\) and \(h : U \to X\) be morphisms of algebraic spaces over \(S\). Assume that \(Y\) is locally Noetherian, that \(f\) and \(h\) are of finite type, that \(f\) is separated, and that the image of \(|h| : |U| \to |X|\) is dense in \(|X|\). If given any commutative solid diagram \[\xymatrix{ \Spec(K) \ar[r] \ar[d] & U \ar[r]^h & X \ar[d]^f \\ \Spec(A) \ar[rr] \ar@{-->}[rru] & & Y }\] where \(A\) is a discrete valuation ring with field of fractions \(K\), there exists a dotted arrow making the diagram commute, then \(f\) is proper.

Proof

It suffices to prove that \(f\) is universally closed. Let \(V \to Y\) be an étale morphism where \(V\) is an affine scheme. By Morphisms of Spaces, Lemma 03IT it suffices to prove that the base change \(X \times_Y V \to V\) is universally closed. By Properties of Spaces, Lemma 03H4 the image \(I\) of \(|U \times_Y V| \to |X \times_Y V|\) is the inverse image of the image of \(|h|\). Since \(|X \times_Y V| \to |X|\) is open (Properties of Spaces, Lemma 03IR) we conclude that \(I\) is dense in \(|X \times_Y V|\). Therefore the assumptions of the lemma are satisfied for the morphisms \(U \times_Y V \to X \times_Y V \to V\). Hence we may assume \(Y\) is an affine scheme.

Assume \(Y\) is an affine scheme. Then \(U\) is quasi-compact. Choose an affine scheme and a surjective étale morphism \(W \to U\). Then we may and do replace \(U\) by \(W\) and assume that \(U\) is affine. By the weak version of Chow’s lemma (Cohomology of Spaces, Lemma 089J) we can choose a surjective proper morphism \(X' \to X\) where \(X'\) is a scheme. Then \(U' = X' \times_X U\) is a scheme and \(U' \to X'\) is of finite type. We may replace \(X'\) by the scheme theoretic image of \(h' : U' \to X'\) and hence \(h'(U')\) is dense in \(X'\). We claim that for every diagram \[\xymatrix{ \Spec(K) \ar[r] \ar[d] & U' \ar[r]^h & X' \ar[d]^{f'} \\ \Spec(A) \ar[rr] \ar@{-->}[rru] & & Y }\] where \(A\) is a discrete valuation ring with field of fractions \(K\), there exists a dotted arrow making the diagram commute. Namely, we first get an arrow \(\Spec(A) \to X\) by the assumption of the lemma and then we lift this to an arrow \(\Spec(A) \to X'\) using the valuative criterion for properness (Morphisms of Spaces, Lemma 0A40). The morphism \(X' \to Y\) is separated as a composition of a proper and a separated morphism. Thus by the case of schemes the morphism \(X' \to Y\) is proper (Limits, Lemma 0CM3). By Morphisms of Spaces, Lemma 08AJ we conclude that \(X \to Y\) is proper.

Lemma

Let \(S\) be a scheme. Let \(f : X \to Y\) and \(h : U \to X\) be morphisms of algebraic spaces over \(S\). Assume that \(Y\) is locally Noetherian, that \(f\) is locally of finite type and quasi-separated, that \(h\) is of finite type, and that the image of \(|h| : |U| \to |X|\) is dense in \(|X|\). If given any commutative solid diagram \[\xymatrix{ \Spec(K) \ar[r] \ar[d] & U \ar[r]^h & X \ar[d]^f \\ \Spec(A) \ar[rr] \ar@{-->}[rru] & & Y }\] where \(A\) is a discrete valuation ring with field of fractions \(K\), there exists at most one dotted arrow making the diagram commute, then \(f\) is separated.

Proof

We will apply Lemma 0CMD to the morphisms \(U \to X\) and \(\Delta : X \to X \times_Y X\). We check the conditions. Observe that \(\Delta\) is quasi-compact because \(f\) is quasi-separated. Of course \(\Delta\) is locally of finite type and separated (true for any diagonal morphism). Finally, suppose given a commutative solid diagram \[\xymatrix{ \Spec(K) \ar[r] \ar[d] & U \ar[r]^h & X \ar[d]^\Delta \\ \Spec(A) \ar[rr]^{(a, b)} \ar@{-->}[rru] & & X \times_Y X }\] where \(A\) is a discrete valuation ring with field of fractions \(K\). Then \(a\) and \(b\) give two dotted arrows in the diagram of the lemma and have to be equal. Hence as dotted arrow we can use \(a = b\) which gives existence. This finishes the proof.

Lemma

Let \(S\) be a scheme. Let \(f : X \to Y\) and \(h : U \to X\) be morphisms of algebraic spaces over \(S\). Assume that \(Y\) is locally Noetherian, that \(f\) and \(h\) are of finite type, that \(f\) is quasi-separated, and that \(h(U)\) is dense in \(X\). If given any commutative solid diagram \[\xymatrix{ \Spec(K) \ar[r] \ar[d] & U \ar[r]^h & X \ar[d]^f \\ \Spec(A) \ar[rr] \ar@{-->}[rru] & & Y }\] where \(A\) is a discrete valuation ring with field of fractions \(K\), there exists a unique dotted arrow making the diagram commute, then \(f\) is proper.

Proof

Combine Lemmas 0CME and 0CMD.

Descending finite type spaces

This section continues the theme of Section 07SP in the spirit of the results discussed in Section 07SJ. It is also the analogue of Limits, Section 0CNL for algebraic spaces.

Situation

Let \(S\) be a scheme, for example \(\Spec(\mathbf{Z})\). Let \(B = \lim_{i \in I} B_i\) be the limit of a directed inverse system of Noetherian spaces over \(S\) with affine transition morphisms \(B_{i'} \to B_i\) for \(i' \geq i\).

Lemma

In Situation 0CP6. Let \(X \to B\) be a quasi-separated and finite type morphism of algebraic spaces. Then there exists an \(i \in I\) and a diagram [0CP8]\[\begin{equation} \vcenter{ \xymatrix{ X \ar[r] \ar[d] & W \ar[d] \\ B \ar[r] & B_i } } \end{equation}\] such that \(W \to B_i\) is of finite type and such that the induced morphism \(X \to B \times_{B_i} W\) is a closed immersion.

Proof

By Lemma 0872 we can find a closed immersion \(X \to X'\) over \(B\) where \(X'\) is an algebraic space of finite presentation over \(B\). By Lemma 07SK we can find an \(i\) and a morphism of finite presentation \(X'_i \to B_i\) whose pull back is \(X'\). Set \(W = X'_i\).

Lemma

In Situation 0CP6. Let \(X \to B\) be a quasi-separated and finite type morphism of algebraic spaces. Given \(i \in I\) and a diagram \[\vcenter{ \xymatrix{ X \ar[r] \ar[d] & W \ar[d] \\ B \ar[r] & B_i } }\] as in (0CP8) for \(i' \geq i\) let \(X_{i'}\) be the scheme theoretic image of \(X \to B_{i'} \times_{B_i} W\). Then \(X = \lim_{i' \geq i} X_{i'}\).

Proof

Since \(X\) is quasi-compact and quasi-separated formation of the scheme theoretic image of \(X \to B_{i'} \times_{B_i} W\) commutes with étale localization (Morphisms of Spaces, Lemma 082Z). Hence we may and do assume \(W\) is affine and maps into an affine \(U_i\) étale over \(B_i\). Then \[B_{i'} \times_{B_i} W = B_{i'} \times_{B_i} U_i \times_{U_i} W = U_{i'} \times_{U_i} W\] where \(U_{i'} = B_{i'} \times_{B_i} U_i\) is affine as the transition morphisms are affine. Thus the lemma follows from the case of schemes which is Limits, Lemma 0CNQ.

Lemma

In Situation 0CP6. Let \(f : X \to Y\) be a morphism of algebraic spaces quasi-separated and of finite type over \(B\). Let \[\vcenter{ \xymatrix{ X \ar[r] \ar[d] & W \ar[d] \\ B \ar[r] & B_{i_1} } } \quad\text{and}\quad \vcenter{ \xymatrix{ Y \ar[r] \ar[d] & V \ar[d] \\ B \ar[r] & B_{i_2} } }\] be diagrams as in (0CP8). Let \(X = \lim_{i \geq i_1} X_i\) and \(Y = \lim_{i \geq i_2} Y_i\) be the corresponding limit descriptions as in Lemma 0CP9. Then there exists an \(i_0 \geq \max(i_1, i_2)\) and a morphism \[(f_i)_{i \geq i_0} : (X_i)_{i \geq i_0} \to (Y_i)_{i \geq i_0}\] of inverse systems over \((B_i)_{i \geq i_0}\) such that such that \(f = \lim_{i \geq i_0} f_i\). If \((g_i)_{i \geq i_0} : (X_i)_{i \geq i_0} \to (Y_i)_{i \geq i_0}\) is a second morphism of inverse systems over \((B_i)_{i \geq i_0}\) such that such that \(f = \lim_{i \geq i_0} g_i\) then \(f_i = g_i\) for all \(i \gg i_0\).

Proof

Since \(V \to B_{i_2}\) is of finite presentation and \(X = \lim_{i \geq i_1} X_i\) we can appeal to Proposition 04AK as improved by Lemma 0CP4 to find an \(i_0 \geq \max(i_1, i_2)\) and a morphism \(h : X_{i_0} \to V\) over \(B_{i_2}\) such that \(X \to X_{i_0} \to V\) is equal to \(X \to Y \to V\). For \(i \geq i_0\) we get a commutative solid diagram \[\xymatrix{ X \ar[d] \ar[r] & X_i \ar[r] \ar@{..>}[d] \ar@/_2pc/[dd] |!{[d];[ld]}\hole & X_{i_0} \ar[d]^h \\ Y \ar[r] \ar[d] & Y_i \ar[r] \ar[d] & V \ar[d] \\ B \ar[r] & B_i \ar[r] & B_{i_0} }\] Since \(X \to X_i\) has scheme theoretically dense image and since \(Y_i\) is the scheme theoretic image of \(Y \to B_i \times_{B_{i_2}} V\) we find that the morphism \(X_i \to B_i \times_{B_{i_2}} V\) induced by the diagram factors through \(Y_i\) (Morphisms of Spaces, Lemma 0CP2). This proves existence.

Uniqueness. Let \(E_i \to X_i\) be the equalizer of \(f_i\) and \(g_i\) for \(i \geq i_0\). We have \(E_i = Y_i \times_{\Delta, Y_i \times_{B_i} Y_i, (f_i, g_i)} X_i\). Hence \(E_i \to X_i\) is a monomorphism of finite presentation as a base change of the diagonal of \(Y_i\) over \(B_i\), see Morphisms of Spaces, Lemmas 03HK and 084P. Since \(X_i\) is a closed subspace of \(B_i \times_{B_{i_0}} X_{i_0}\) and similarly for \(Y_i\) we see that \[E_i = X_i \times_{(B_i \times_{B_{i_0}} X_{i_0})} (B_i \times_{B_{i_0}} E_{i_0}) = X_i \times_{X_{i_0}} E_{i_0}\] Similarly, we have \(X = X \times_{X_{i_0}} E_{i_0}\). Hence we conclude that \(E_i = X_i\) for \(i\) large enough by Lemma 0852.

Remark

In Situation 0CP6 Lemmas 0CP7, 0CP9, and 0CPA tell us that the category of algebraic spaces quasi-separated and of finite type over \(B\) is equivalent to certain types of inverse systems of algebraic spaces over \((B_i)_{i \in I}\), namely the ones produced by applying Lemma 0CP9 to a diagram of the form (0CP8). For example, given \(X \to B\) finite type and quasi-separated if we choose two different diagrams \(X \to V_1 \to B_{i_1}\) and \(X \to V_2 \to B_{i_2}\) as in (0CP8), then applying Lemma 0CPA to \(\text{id}_X\) (in two directions) we see that the corresponding limit descriptions of \(X\) are canonically isomorphic (up to shrinking the directed set \(I\)). And so on and so forth.

Lemma

Notation and assumptions as in Lemma 0CPA. If \(f\) is flat and of finite presentation, then there exists an \(i_3 > i_0\) such that for \(i \geq i_3\) we have \(f_i\) is flat, \(X_i = Y_i \times_{Y_{i_3}} X_{i_3}\), and \(X = Y \times_{Y_{i_3}} X_{i_3}\).

Proof

By Lemma 07SK we can choose an \(i \geq i_2\) and a morphism \(U \to Y_i\) of finite presentation such that \(X = Y \times_{Y_i} U\) (this is where we use that \(f\) is of finite presentation). After increasing \(i\) we may assume that \(U \to Y_i\) is flat, see Lemma 08K0. As discussed in Remark 0CPB we may and do replace the initial diagram used to define the system \((X_i)_{i \geq i_1}\) by the system corresponding to \(X \to U \to B_i\). Thus \(X_{i'}\) for \(i' \geq i\) is defined as the scheme theoretic image of \(X \to B_{i'} \times_{B_i} U\).

Because \(U \to Y_i\) is flat (this is where we use that \(f\) is flat), because \(X = Y \times_{Y_i} U\), and because the scheme theoretic image of \(Y \to Y_i\) is \(Y_i\), we see that the scheme theoretic image of \(X \to U\) is \(U\) (Morphisms of Spaces, Lemma 089E). Observe that \(Y_{i'} \to B_{i'} \times_{B_i} Y_i\) is a closed immersion for \(i' \geq i\) by construction of the system of \(Y_j\). Then the same argument as above shows that the scheme theoretic image of \(X \to B_{i'} \times_{B_i} U\) is equal to the closed subspace \(Y_{i'} \times_{Y_i} U\). Thus we see that \(X_{i'} = Y_{i'} \times_{Y_i} U\) for all \(i' \geq i\) and hence the lemma holds with \(i_3 = i\).

Lemma

Notation and assumptions as in Lemma 0CPA. If \(f\) is smooth, then there exists an \(i_3 > i_0\) such that for \(i \geq i_3\) we have \(f_i\) is smooth.

Proof

Combine Lemmas 0CPC and 0CN2.

Lemma

Notation and assumptions as in Lemma 0CPA. If \(f\) is proper, then there exists an \(i_3 \geq i_0\) such that for \(i \geq i_3\) we have \(f_i\) is proper.

Proof

By the discussion in Remark 0CPB the choice of \(i_1\) and \(W\) fitting into a diagram as in (0CP8) is immaterial for the truth of the lemma. Thus we choose \(W\) as follows. First we choose a closed immersion \(X \to X'\) with \(X' \to Y\) proper and of finite presentation, see Lemma 0A0W. Then we choose an \(i_3 \geq i_2\) and a proper morphism \(W \to Y_{i_3}\) such that \(X' = Y \times_{Y_{i_3}} W\). This is possible because \(Y = \lim_{i \geq i_2} Y_i\) and Lemmas 09NS and 08K1. With this choice of \(W\) it is immediate from the construction that for \(i \geq i_3\) the algebraic space \(X_i\) is a closed subspace of \(Y_i \times_{Y_{i_3}} W \subset B_i \times_{B_{i_3}} W\) and hence proper over \(Y_i\).

Lemma

In Situation 0CP6 suppose that we have a cartesian diagram \[\xymatrix{ X^1 \ar[r]_p \ar[d]_q & X^3 \ar[d]^a \\ X^2 \ar[r]^b & X^4 }\] of algebraic spaces quasi-separated and of finite type over \(B\). For each \(j = 1, 2, 3, 4\) choose \(i_j \in I\) and a diagram \[\xymatrix{ X^j \ar[r] \ar[d] & W^j \ar[d] \\ B \ar[r] & B_{i_j} }\] as in (0CP8). Let \(X^j = \lim_{i \geq i_j} X^j_i\) be the corresponding limit descriptions as in Lemma 0CPA. Let \((a_i)_{i \geq i_5}\), \((b_i)_{i \geq i_6}\), \((p_i)_{i \geq i_7}\), and \((q_i)_{i \geq i_8}\) be the corresponding morphisms of inverse systems constructed in Lemma 0CPA. Then there exists an \(i_9 \geq \max(i_5, i_6, i_7, i_8)\) such that for \(i \geq i_9\) we have \(a_i \circ p_i = b_i \circ q_i\) and such that \[(q_i, p_i) : X^1_i \longrightarrow X^2_i \times_{b_i, X^4_i, a_i} X^3_i\] is a closed immersion. If \(a\) and \(b\) are flat and of finite presentation, then there exists an \(i_{10} \geq \max(i_5, i_6, i_7, i_8, i_9)\) such that for \(i \geq i_{10}\) the last displayed morphism is an isomorphism.

Proof

According to the discussion in Remark 0CPB the choice of \(W^1\) fitting into a diagram as in (0CP8) is immaterial for the truth of the lemma. Thus we may choose \(W^1 = W^2 \times_{W^4} W^3\). Then it is immediate from the construction of \(X^1_i\) that \(a_i \circ p_i = b_i \circ q_i\) and that \[(q_i, p_i) : X^1_i \longrightarrow X^2_i \times_{b_i, X^4_i, a_i} X^3_i\] is a closed immersion.

If \(a\) and \(b\) are flat and of finite presentation, then so are \(p\) and \(q\) as base changes of \(a\) and \(b\). Thus we can apply Lemma 0CPC to each of \(a\), \(b\), \(p\), \(q\), and \(a \circ p = b \circ q\). It follows that there exists an \(i_9 \in I\) such that \[(q_i, p_i) : X^1_i \to X^2_i \times_{X^4_i} X^3_i\] is the base change of \((q_{i_9}, p_{i_9})\) by the morphism by the morphism \(X^4_i \to X^4_{i_9}\) for all \(i \geq i_9\). We conclude that \((q_i, p_i)\) is an isomorphism for all sufficiently large \(i\) by Lemma 0852.