Divisors on Algebraic Spaces

Introduction

In this chapter we study divisors on algebraic spaces and related topics. A basic reference for algebraic spaces is [Kn].

Associated and weakly associated points

In the case of schemes we have introduced two competing notions of associated points. Namely, the usual associated points (Divisors, Section 02OI) and the weakly associated points (Divisors, Section 056K). For a general algebraic space the notion of an associated point is basically useless and we don’t even bother to introduce it. If the algebraic space is locally Noetherian, then we allow ourselves to use the phrase “associated point” instead of “weakly associated point” as the notions are the same for Noetherian schemes (Divisors, Lemma 05AR). Before we make our definition, we need a lemma.

Lemma

Let \(S\) be a scheme. Let \(X\) be an algebraic space over \(S\). Let \(\mathcal{F}\) be a quasi-coherent \(\mathcal{O}_X\)-module. Let \(x \in |X|\). The following are equivalent

1. for some étale morphism \(f : U \to X\) with \(U\) a scheme and \(u \in U\) mapping to \(x\), the point \(u\) is weakly associated to \(f^*\mathcal{F}\),

2. for every étale morphism \(f : U \to X\) with \(U\) a scheme and \(u \in U\) mapping to \(x\), the point \(u\) is weakly associated to \(f^*\mathcal{F}\),

3. the maximal ideal of \(\mathcal{O}_{X, \overline{x}}\) is a weakly associated prime of the stalk \(\mathcal{F}_{\overline{x}}\).

If \(X\) is locally Noetherian, then these are also equivalent to

1. for some étale morphism \(f : U \to X\) with \(U\) a scheme and \(u \in U\) mapping to \(x\), the point \(u\) is associated to \(f^*\mathcal{F}\),

2. for every étale morphism \(f : U \to X\) with \(U\) a scheme and \(u \in U\) mapping to \(x\), the point \(u\) is associated to \(f^*\mathcal{F}\),

3. the maximal ideal of \(\mathcal{O}_{X, \overline{x}}\) is an associated prime of the stalk \(\mathcal{F}_{\overline{x}}\).

Proof

Choose a scheme \(U\) with a point \(u\) and an étale morphism \(f : U \to X\) mapping \(u\) to \(x\). Lift \(\overline{x}\) to a geometric point of \(U\) over \(u\). Recall that \(\mathcal{O}_{X, \overline{x}} = \mathcal{O}_{U, u}^{sh}\) where the strict henselization is with respect to our chosen lift of \(\overline{x}\), see Properties of Spaces, Lemma 04KF. Finally, we have \[\mathcal{F}_{\overline{x}} = (f^*\mathcal{F})_u \otimes_{\mathcal{O}_{U, u}} \mathcal{O}_{X, \overline{x}} = (f^*\mathcal{F})_u \otimes_{\mathcal{O}_{U, u}} \mathcal{O}_{U, u}^{sh}\] by Properties of Spaces, Lemma 05VP. Hence the equivalence of (1), (2), and (3) follows from More on Flatness, Lemma 0CTU. If \(X\) is locally Noetherian, then any \(U\) as above is locally Noetherian, hence we see that (1), resp. (2) are equivalent to (4), resp. (5) by Divisors, Lemma 05AR. On the other hand, in the locally Noetherian case the local ring \(\mathcal{O}_{X, \overline{x}}\) is Noetherian too (Properties of Spaces, Lemma 08AH). Hence the equivalence of (3) and (6) by the same lemma (or by Algebra, Lemma 058A).

Definition

Let \(S\) be a scheme. Let \(X\) be an algebraic space over \(S\). Let \(\mathcal{F}\) be a quasi-coherent sheaf on \(X\). Let \(x \in |X|\).

1. We say \(x\) is weakly associated to \(\mathcal{F}\) if the equivalent conditions (1), (2), and (3) of Lemma 0CTW are satisfied.

2. We denote \(\text{WeakAss}(\mathcal{F})\) the set of weakly associated points of \(\mathcal{F}\).

3. The weakly associated points of \(X\) are the weakly associated points of \(\mathcal{O}_X\).

If \(X\) is locally Noetherian we will say \(x\) is associated to \(\mathcal{F}\) if and only if \(x\) is weakly associated to \(\mathcal{F}\) and we set \(\text{Ass}(\mathcal{F}) = \text{WeakAss}(\mathcal{F})\). Finally (still assuming \(X\) is locally Noetherian), we will say \(x\) is an associated point of \(X\) if and only if \(x\) is a weakly associated point of \(X\).

At this point we can prove the obligatory lemmas.

Lemma

Let \(S\) be a scheme. Let \(X\) be an algebraic space over \(S\). Let \(\mathcal{F}\) be a quasi-coherent \(\mathcal{O}_X\)-module. Then \(\text{WeakAss}(\mathcal{F}) \subset \text{Supp}(\mathcal{F})\).

Proof

This is immediate from the definitions. The support of an abelian sheaf on \(X\) is defined in Properties of Spaces, Definition 04KA.

Lemma

Let \(S\) be a scheme. Let \(X\) be an algebraic space over \(S\). Let \(0 \to \mathcal{F}_1 \to \mathcal{F}_2 \to \mathcal{F}_3 \to 0\) be a short exact sequence of quasi-coherent sheaves on \(X\). Then \(\text{WeakAss}(\mathcal{F}_2) \subset \text{WeakAss}(\mathcal{F}_1) \cup \text{WeakAss}(\mathcal{F}_3)\) and \(\text{WeakAss}(\mathcal{F}_1) \subset \text{WeakAss}(\mathcal{F}_2)\).

Proof

For every geometric point \(\overline{x} \in X\) the sequence of stalks \(0 \to \mathcal{F}_{1, \overline{x}} \to \mathcal{F}_{2, \overline{x}} \to \mathcal{F}_{3, \overline{x}} \to 0\) is a short exact sequence of \(\mathcal{O}_{X, \overline{x}}\)-modules. Hence the lemma follows from Algebra, Lemma 0548.

Lemma

Let \(S\) be a scheme. Let \(X\) be an algebraic space over \(S\). Let \(\mathcal{F}\) be a quasi-coherent \(\mathcal{O}_X\)-module. Then \[\mathcal{F} = (0) \Leftrightarrow \text{WeakAss}(\mathcal{F}) = \emptyset\]

Proof

Choose a scheme \(U\) and a surjective étale morphism \(f : U \to X\). Then \(\mathcal{F}\) is zero if and only if \(f^*\mathcal{F}\) is zero. Hence the lemma follows from the definition and the lemma in the case of schemes, see Divisors, Lemma 05AP.

Lemma

Let \(S\) be a scheme. Let \(X\) be an algebraic space over \(S\). Let \(\mathcal{F}\) be a quasi-coherent \(\mathcal{O}_X\)-module. Let \(x \in |X|\). If

1. \(x \in \text{Supp}(\mathcal{F})\)

2. \(x\) is a codimension \(0\) point of \(X\) (Properties of Spaces, Definition 04NA).

Then \(x \in \text{WeakAss}(\mathcal{F})\). If \(\mathcal{F}\) is a finite type \(\mathcal{O}_X\)-module with scheme theoretic support \(Z\) (Morphisms of Spaces, Definition 07U1) and \(x\) is a codimension \(0\) point of \(Z\), then \(x \in \text{WeakAss}(\mathcal{F})\).

Proof

Since \(x \in \text{Supp}(\mathcal{F})\) the stalk \(\mathcal{F}_{\overline{x}}\) is not zero. Hence \(\text{WeakAss}(\mathcal{F}_{\overline{x}})\) is nonempty by Algebra, Lemma 0588. On the other hand, the spectrum of \(\mathcal{O}_{X, \overline{x}}\) is a singleton. Hence \(x\) is a weakly associated point of \(\mathcal{F}\) by definition. The final statement follows as \(\mathcal{O}_{X, \overline{x}} \to \mathcal{O}_{Z, \overline{z}}\) is a surjection, the spectrum of \(\mathcal{O}_{Z, \overline{z}}\) is a singleton, and \(\mathcal{F}_{\overline{x}}\) is a nonzero module over \(\mathcal{O}_{Z, \overline{z}}\).

Lemma

Let \(S\) be a scheme. Let \(X\) be an algebraic space over \(S\). Let \(\mathcal{F}\) be a quasi-coherent \(\mathcal{O}_X\)-module. Let \(x \in |X|\). If

1. \(X\) is decent (for example quasi-separated or locally separated),

2. \(x \in \text{Supp}(\mathcal{F})\)

3. \(x\) is not a specialization of another point in \(\text{Supp}(\mathcal{F})\).

Then \(x \in \text{WeakAss}(\mathcal{F})\).

Proof

(A quasi-separated algebraic space is decent, see Decent Spaces, Section 03I7. A locally separated algebraic space is decent, see Decent Spaces, Lemma 088J.) Choose a scheme \(U\), a point \(u \in U\), and an étale morphism \(f : U \to X\) mapping \(u\) to \(x\). By Decent Spaces, Lemma 03K5 if \(u' \leadsto u\) is a nontrivial specialization, then \(f(u') \not = x\). Hence we see that \(u \in \text{Supp}(f^*\mathcal{F})\) is not a specialization of another point of \(\text{Supp}(f^*\mathcal{F})\). Hence \(u \in \text{WeakAss}(f^*\mathcal{F})\) by Divisors, Lemma 0CUL.

Lemma

Let \(S\) be a scheme. Let \(X\) be a locally Noetherian algebraic space over \(S\). Let \(\mathcal{F}\) be a coherent \(\mathcal{O}_X\)-module. Then \(\text{Ass}(\mathcal{F}) \cap W\) is finite for every quasi-compact open \(W \subset |X|\).

Proof

Choose a quasi-compact scheme \(U\) and an étale morphism \(U \to X\) such that \(W\) is the image of \(|U| \to |X|\). Then \(U\) is a Noetherian scheme and we may apply Divisors, Lemma 05AF to conclude.

Lemma

Let \(S\) be a scheme. Let \(X\) be an algebraic space over \(S\). Let \(\mathcal{F}\) be a quasi-coherent \(\mathcal{O}_X\)-module. If \(U \to X\) is an étale morphism such that \(\text{WeakAss}(\mathcal{F}) \subset \Im(|U| \to |X|)\), then \(\Gamma(X, \mathcal{F}) \to \Gamma(U, \mathcal{F})\) is injective.

Proof

Let \(s \in \Gamma(X, \mathcal{F})\) be a section which restricts to zero on \(U\). Let \(\mathcal{F}' \subset \mathcal{F}\) be the image of the map \(\mathcal{O}_X \to \mathcal{F}\) defined by \(s\). Then \(\mathcal{F}'|_U = 0\). This implies that \(\text{WeakAss}(\mathcal{F}') \cap \Im(|U| \to |X|) = \emptyset\) (by the definition of weakly associated points). On the other hand, \(\text{WeakAss}(\mathcal{F}') \subset \text{WeakAss}(\mathcal{F})\) by Lemma 0CTZ. We conclude \(\text{WeakAss}(\mathcal{F}') = \emptyset\). Hence \(\mathcal{F}' = 0\) by Lemma 0CU0.

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 \(\mathcal{F}\) be a quasi-coherent \(\mathcal{O}_X\)-module. Let \(y \in |Y|\) be a point which is not in the image of \(|f|\). Then \(y\) is not weakly associated to \(f_*\mathcal{F}\).

Proof

By Morphisms of Spaces, Lemma 03M9 the \(\mathcal{O}_Y\)-module \(f_*\mathcal{F}\) is quasi-coherent hence the lemma makes sense. Choose an affine scheme \(V\), a point \(v \in V\), and an étale morphism \(V \to Y\) mapping \(v\) to \(y\). We may replace \(f : X \to Y\), \(\mathcal{F}\), \(y\) by \(X \times_Y V \to V\), \(\mathcal{F}|_{X \times_Y V}\), \(v\). Thus we may assume \(Y\) is an affine scheme. In this case \(X\) is quasi-compact, hence we can choose an affine scheme \(U\) and a surjective étale morphism \(U \to X\). Denote \(g : U \to Y\) the composition. Then \(f_*\mathcal{F} \subset g_*(\mathcal{F}|_U)\). By Lemma 0CTZ we reduce to the case of schemes which is Divisors, Lemma 0AVN.

Lemma

Let \(S\) be a scheme. Let \(X\) be an algebraic space over \(S\). Let \(\varphi : \mathcal{F} \to \mathcal{G}\) be a map of quasi-coherent \(\mathcal{O}_X\)-modules. Assume that for every \(x \in |X|\) at least one of the following happens

1. \(\mathcal{F}_{\overline{x}} \to \mathcal{G}_{\overline{x}}\) is injective, or

2. \(x \not \in \text{WeakAss}(\mathcal{F})\).

Then \(\varphi\) is injective.

Proof

The assumptions imply that \(\text{WeakAss}(\Ker(\varphi)) = \emptyset\) and hence \(\Ker(\varphi) = 0\) by Lemma 0CU0.

Lemma

Let \(S\) be a scheme. Let \(X\) be a reduced algebraic space over \(S\). Then the weakly associated point of \(X\) are exactly the codimension \(0\) points of \(X\).

Proof

Working étale locally this follows from Divisors, Lemma 0EME and Properties of Spaces, Lemma 0BAQ.

Morphisms and weakly associated points

Lemma

Let \(S\) be a scheme. Let \(f : X \to Y\) be an affine morphism of algebraic spaces over \(S\). Let \(\mathcal{F}\) be a quasi-coherent \(\mathcal{O}_X\)-module. Then we have \[\text{WeakAss}_S(f_*\mathcal{F}) \subset f(\text{WeakAss}_X(\mathcal{F}))\]

Proof

Choose a scheme \(V\) and a surjective étale morphism \(V \to Y\). Set \(U = X \times_Y V\). Then \(U \to V\) is an affine morphism of schemes. By our definition of weakly associated points the problem is reduced to the morphism of schemes \(U \to V\). This case is treated in Divisors, Lemma 05EX.

Lemma

Let \(S\) be a scheme. Let \(f : X \to Y\) be an affine morphism of algebraic spaces over \(S\). Let \(\mathcal{F}\) be a quasi-coherent \(\mathcal{O}_X\)-module. If \(X\) is locally Noetherian, then we have \[\text{WeakAss}_Y(f_*\mathcal{F}) = f(\text{WeakAss}_X(\mathcal{F}))\]

Proof

Choose a scheme \(V\) and a surjective étale morphism \(V \to Y\). Set \(U = X \times_Y V\). Then \(U \to V\) is an affine morphism of schemes and \(U\) is locally Noetherian. By our definition of weakly associated points the problem is reduced to the morphism of schemes \(U \to V\). This case is treated in Divisors, Lemma 05EY.

Lemma

Let \(S\) be a scheme. Let \(f : X \to Y\) be a finite morphism of algebraic spaces over \(S\). Let \(\mathcal{F}\) be a quasi-coherent \(\mathcal{O}_X\)-module. Then \(\text{WeakAss}(f_*\mathcal{F}) = f(\text{WeakAss}(\mathcal{F}))\).

Proof

Choose a scheme \(V\) and a surjective étale morphism \(V \to Y\). Set \(U = X \times_Y V\). Then \(U \to V\) is a finite morphism of schemes. By our definition of weakly associated points the problem is reduced to the morphism of schemes \(U \to V\). This case is treated in Divisors, Lemma 05EZ.

Lemma

Let \(S\) be a scheme. Let \(f : X \to Y\) be a morphism of algebraic spaces over \(S\). Let \(\mathcal{G}\) be a quasi-coherent \(\mathcal{O}_Y\)-module. Let \(x \in |X|\) and \(y = f(x) \in |Y|\). If

1. \(y \in \text{WeakAss}_S(\mathcal{G})\),

2. \(f\) is flat at \(x\), and

3. the dimension of the local ring of the fibre of \(f\) at \(x\) is zero (Morphisms of Spaces, Definition 04NM),

then \(x \in \text{WeakAss}(f^*\mathcal{G})\).

Proof

Choose a scheme \(V\), a point \(v \in V\), and an étale morphism \(V \to Y\) mapping \(v\) to \(y\). Choose a scheme \(U\), a point \(u \in U\), and an étale morphism \(U \to V \times_Y X\) mapping \(v\) to a point lying over \(v\) and \(x\). This is possible because there is a \(t \in |V \times_Y X|\) mapping to \((v, y)\) by Properties of Spaces, Lemma 03H4. By definition we see that the dimension of \(\mathcal{O}_{U_v, u}\) is zero. Hence \(u\) is a generic point of the fiber \(U_v\). By our definition of weakly associated points the problem is reduced to the morphism of schemes \(U \to V\). This case is treated in Divisors, Lemma 05F0.

Lemma

Let \(K/k\) be a field extension. Let \(X\) be an algebraic space over \(k\). Let \(\mathcal{F}\) be a quasi-coherent \(\mathcal{O}_X\)-module. Let \(y \in X_K\) with image \(x \in X\). If \(y\) is a weakly associated point of the pullback \(\mathcal{F}_K\), then \(x\) is a weakly associated point of \(\mathcal{F}\).

Proof

This is the translation of Divisors, Lemma 0CUC into the language of algebraic spaces. We omit the details of the translation.

Lemma

Let \(S\) be a scheme. Let \(f : X \to Y\) be a finite flat morphism of algebraic spaces. Let \(\mathcal{G}\) be a quasi-coherent \(\mathcal{O}_Y\)-module. Let \(x \in |X|\) be a point with image \(y \in |Y|\). Then \[x \in \text{WeakAss}(g^*\mathcal{G}) \Leftrightarrow y \in \text{WeakAss}(\mathcal{G})\]

Proof

Follows immediately from the case of schemes (More on Flatness, Lemma 05FN) by étale localization.

Lemma

Let \(S\) be a scheme. Let \(f : X \to Y\) be an étale morphism of algebraic spaces. Let \(\mathcal{G}\) be a quasi-coherent \(\mathcal{O}_Y\)-module. Let \(x \in |X|\) be a point with image \(y \in |Y|\). Then \[x \in \text{WeakAss}(f^*\mathcal{G}) \Leftrightarrow y \in \text{WeakAss}(\mathcal{G})\]

Proof

This is immediate from the definition of weakly associated points and in fact the corresponding lemma for the case of schemes (More on Flatness, Lemma 05FP) is the basis for our definition.

Relative weak assassin

We need a couple of lemmas to define this gadget.

Lemma

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

1. for some scheme \(V\), point \(v \in V\), and étale morphism \(V \to Y\) mapping \(v\) to \(y\), the algebraic space \(X_v\) is locally Noetherian,

2. for every scheme \(V\), point \(v \in V\), and étale morphism \(V \to Y\) mapping \(v\) to \(y\), the algebraic space \(X_v\) is locally Noetherian, and

3. there exists a field \(k\) and a morphism \(\Spec(k) \to Y\) representing \(y\) such that \(X_k\) is locally Noetherian.

If there exists a field \(k_0\) and a monomorphism \(\Spec(k_0) \to Y\) representing \(y\), then these are also equivalent to

1. the algebraic space \(X_{k_0}\) is locally Noetherian.

Proof

Observe that \(X_v = v \times_Y X = \Spec(\kappa(v)) \times_Y X\). Hence the implications (2) \(\Rightarrow\) (1) \(\Rightarrow\) (3) are clear. Assume that \(\Spec(k) \to Y\) is a morphism from the spectrum of a field such that \(X_k\) is locally Noetherian. Let \(V \to Y\) be an étale morphism from a scheme \(V\) and let \(v \in V\) a point mapping to \(y\). Then the scheme \(v \times_Y \Spec(k)\) is nonempty. Choose a point \(w \in v \times_Y \Spec(k)\). Consider the morphisms \[X_v \longleftarrow X_w \longrightarrow X_k\] Since \(V \to Y\) is étale and since \(w\) may be viewed as a point of \(V \times_Y \Spec(k)\), we see that \(\kappa(w)/k\) is a finite separable extension of fields (Morphisms, Lemma 02GL). Thus \(X_w \to X_k\) is a finite étale morphism as a base change of \(w \to \Spec(k)\). Hence \(X_w\) is locally Noetherian (Morphisms of Spaces, Lemma 04ZK). The morphism \(X_w \to X_v\) is a surjective, affine, flat morphism as a base change of the surjective, affine, flat morphism \(w \to v\). Then the fact that \(X_w\) is locally Noetherian implies that \(X_v\) is locally Noetherian. This can be seen by picking a surjective étale morphism \(U \to X\) and then using that \(U_w \to U_v\) is surjective, affine, and flat. Working affine locally on the scheme \(U_v\) we conclude that \(U_w\) is locally Noetherian by Algebra, Lemma 033E.

Finally, it suffices to prove that (3) implies (4) in case we have a monomorphism \(\Spec(k_0) \to Y\) in the class of \(y\). Then \(\Spec(k) \to Y\) factors as \(\Spec(k) \to \Spec(k_0) \to Y\). The argument given above then shows that \(X_k\) being locally Noetherian impies that \(X_{k_0}\) is locally Noetherian.

Definition

Let \(S\) be a scheme. Let \(f : X \to Y\) be a morphism of algebraic spaces over \(S\). Let \(y \in |Y|\). We say the fibre of \(f\) over \(y\) is locally Noetherian if the equivalent conditions (1), (2), and (3) of Lemma 0CUW are satisfied. We say the fibres of \(f\) are locally Noetherian if this holds for every \(y \in |Y|\).

Of course, the usual way to guarantee locally Noetherian fibres is to assume the morphism is locally of finite type.

Lemma

Let \(S\) be a scheme. Let \(f : X \to Y\) be a morphism of algebraic spaces over \(S\). If \(f\) is locally of finite type, then the fibres of \(f\) are locally Noetherian.

Proof

This follows from Morphisms of Spaces, Lemma 04ZK and the fact that the spectrum of a field is Noetherian.

Lemma

Let \(S\) be a scheme. Let \(f : X \to Y\) be a morphism of algebraic spaces over \(S\). Let \(x \in |X|\) and \(y = f(x) \in |Y|\). Let \(\mathcal{F}\) be a quasi-coherent \(\mathcal{O}_X\)-module. Consider commutative diagrams \[\xymatrix{ X \ar[d] & X \times_Y V \ar[d] \ar[l] & X_v \ar[d] \ar[l] \\ Y & V \ar[l] & v \ar[l] } \quad \xymatrix{ X \ar[d] & U \ar[d] \ar[l] & U_v \ar[d] \ar[l] \\ Y & V \ar[l] & v \ar[l] } \quad \xymatrix{ x \ar@{|->}[d] & x' \ar@{|->}[d] \ar@{|->}[l] & u \ar@{|->}[ld] \ar@{|->}[l] \\ y & v \ar@{|->}[l] }\] where \(V\) and \(U\) are schemes, \(V \to Y\) and \(U \to X \times_Y V\) are étale, \(v \in V\), \(x' \in |X_v|\), \(u \in U\) are points related as in the last diagram. Denote \(\mathcal{F}|_{X_v}\) and \(\mathcal{F}|_{U_v}\) the pullbacks of \(\mathcal{F}\). The following are equivalent

1. for some \(V, v, x'\) as above \(x'\) is a weakly associated point of \(\mathcal{F}|_{X_v}\),

2. for every \(V \to Y, v, x'\) as above \(x'\) is a weakly associated point of \(\mathcal{F}|_{X_v}\),

3. for some \(U, V, u, v\) as above \(u\) is a weakly associated point of \(\mathcal{F}|_{U_v}\),

4. for every \(U, V, u, v\) as above \(u\) is a weakly associated point of \(\mathcal{F}|_{U_v}\),

5. for some field \(k\) and morphism \(\Spec(k) \to Y\) representing \(y\) and some \(t \in |X_k|\) mapping to \(x\), the point \(t\) is a weakly associated point of \(\mathcal{F}|_{X_k}\).

If there exists a field \(k_0\) and a monomorphism \(\Spec(k_0) \to Y\) representing \(y\), then these are also equivalent to

1. \(x_0\) is a weakly associated point of \(\mathcal{F}|_{X_{k_0}}\) where \(x_0 \in |X_{k_0}|\) is the unique point mapping to \(x\).

If the fibre of \(f\) over \(y\) is locally Noetherian, then in conditions (1), (2), (3), (4), and (6) we may replace “weakly associated” with “associated”.

Proof

Observe that given \(V, v, x'\) as in the lemma we can find \(U \to X \times_Y V\) and \(u \in U\) mapping to \(x'\) and then the morphism \(U_v \to X_v\) is étale. Thus it is clear that (1) and (3) are equivalent as well as (2) and (4). Each of these implies (5). We will show that (5) implies (2). Suppose given \(V, v, x'\) as well as \(\Spec(k) \to X\) and \(t \in |X_k|\) such that the point \(t\) is a weakly associated point of \(\mathcal{F}|_{X_k}\). We can choose a point \(w \in v \times_Y \Spec(k)\). Then we obtain the morphisms \[X_v \longleftarrow X_w \longrightarrow X_k\] Since \(V \to Y\) is étale and since \(w\) may be viewed as a point of \(V \times_Y \Spec(k)\), we see that \(\kappa(w)/k\) is a finite separable extension of fields (Morphisms, Lemma 02GL). Thus \(X_w \to X_k\) is a finite étale morphism as a base change of \(w \to \Spec(k)\). Thus any point \(x''\) of \(X_w\) lying over \(t\) is a weakly associated point of \(\mathcal{F}|_{X_w}\) by Lemma 0CUU. We may pick \(x''\) mapping to \(x'\) (Properties of Spaces, Lemma 03H4). Then Lemma 0CUS implies that \(x'\) is a weakly associated point of \(\mathcal{F}|_{X_v}\).

To finish the proof it suffices to show that the equivalent conditions (1) – (5) imply (6) if we are given \(\Spec(k_0) \to Y\) as in (6). In this case the morphism \(\Spec(k) \to Y\) of (5) factors uniquely as \(\Spec(k) \to \Spec(k_0) \to Y\). Then \(x_0\) is the image of \(t\) under the morphism \(X_k \to X_{k_0}\). Hence the same lemma as above shows that (6) is true.

Definition

Let \(S\) be a scheme. Let \(f : X \to Y\) be a morphism of algebraic spaces over \(S\). Let \(\mathcal{F}\) be a quasi-coherent \(\mathcal{O}_X\)-module. The relative weak assassin of \(\mathcal{F}\) in \(X\) over \(Y\) is the set \(\text{WeakAss}_{X/Y}(\mathcal{F}) \subset |X|\) consisting of those \(x \in |X|\) such that the equivalent conditions of Lemma 0CUZ are satisfied. If the fibres of \(f\) are locally Noetherian (Definition 0CUX) then we use the notation \(\text{Ass}_{X/Y}(\mathcal{F})\).

With this notation we can formulate some of the results already proven for schemes.

Lemma

Let \(S\) be a scheme. Let \(f : X \to Y\) be a morphism of algebraic spaces over \(S\). Let \(\mathcal{F}\) be a quasi-coherent \(\mathcal{O}_X\)-module. Let \(\mathcal{G}\) be a quasi-coherent \(\mathcal{O}_Y\)-module. Assume

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

2. \(X\) and \(Y\) are locally Noetherian, and

3. the fibres of \(f\) are locally Noetherian.

Then \[\text{Ass}_X(\mathcal{F} \otimes_{\mathcal{O}_X} f^*\mathcal{G}) = \{x \in \text{Ass}_{X/Y}(\mathcal{F})\text{ such that } f(x) \in \text{Ass}_Y(\mathcal{G}) \}\]

Proof

Via étale localization, this is an immediate consequence of the result for schemes, see Divisors, Lemma 05DB. The result for schemes is more general only because we haven’t defined associated points for non-Noetherian algebraic spaces (hence we need to assume \(X\) and the fibres of \(X \to Y\) are locally Noetherian to even be able to formulate this result).

Lemma

Let \(S\) be a scheme. Let \[\xymatrix{ X' \ar[d]_{f'} \ar[r]_{g'} & X \ar[d]^f \\ Y' \ar[r]^g & Y }\] be a cartesian diagram of algebraic spaces over \(S\). Let \(\mathcal{F}\) be a quasi-coherent \(\mathcal{O}_X\)-module and set \(\mathcal{F}' = (g')^*\mathcal{F}\). If \(f\) is locally of finite type, then

1. \(x' \in \text{Ass}_{X'/Y'}(\mathcal{F}') \Rightarrow g'(x') \in \text{Ass}_{X/Y}(\mathcal{F})\)

2. if \(x \in \text{Ass}_{X/Y}(\mathcal{F})\), then given \(y' \in |Y'|\) with \(f(x) = g(y')\), there exists an \(x' \in \text{Ass}_{X'/Y'}(\mathcal{F}')\) with \(g'(x') = x\) and \(f'(x') = y'\).

Proof

This follows from the case of schemes by étale localization. We write out the details completely. Choose a scheme \(V\) and a surjective étale morphism \(V \to Y\). Choose a scheme \(U\) and a surjective étale morphism \(U \to V \times_Y X\). Choose a scheme \(V'\) and a surjective étale morphism \(V' \to V \times_Y Y'\). Then \(U' = V' \times_V U\) is a scheme and the morphism \(U' \to X'\) is surjective and étale.

Proof of (1). Choose \(u' \in U'\) mapping to \(x'\). Denote \(v' \in V'\) the image of \(u'\). Then \(x' \in \text{Ass}_{X'/Y'}(\mathcal{F}')\) is equivalent to \(u' \in \text{Ass}(\mathcal{F}|_{U'_{v'}})\) by definition (writing \(\text{Ass}\) instead of \(\text{WeakAss}\) makes sense as \(U'_{v'}\) is locally Noetherian). Applying Divisors, Lemma 05DC we see that the image \(u \in U\) of \(u'\) is in \(\text{Ass}(\mathcal{F}|_{U_v})\) where \(v \in V\) is the image of \(u\). This in turn means \(g'(x') \in \text{Ass}_{X/Y}(\mathcal{F})\).

Proof of (2). Choose \(u \in U\) mapping to \(x\). Denote \(v \in V\) the image of \(u\). Then \(x \in \text{Ass}_{X/Y}(\mathcal{F})\) is equivalent to \(u \in \text{Ass}(\mathcal{F}|_{U_v})\) by definition. Choose a point \(v' \in V'\) mapping to \(y' \in |Y'|\) and to \(v \in V\) (possible by Properties of Spaces, Lemma 03H4). Let \(t \in \Spec(\kappa(v') \otimes_{\kappa(v)} \kappa(u))\) be a generic point of an irreducible component. Let \(u' \in U'\) be the image of \(t\). Applying Divisors, Lemma 05DC we see that \(u' \in \text{Ass}(\mathcal{F}'|_{U'_{v'}})\). This in turn means \(x' \in \text{Ass}_{X'/Y'}(\mathcal{F}')\) where \(x' \in |X'|\) is the image of \(u'\).

Lemma

With notation and assumptions as in Lemma 0CV2. Assume \(g\) is locally quasi-finite, or more generally that for every \(y' \in |Y'|\) the transcendence degree of \(y'/g(y')\) is \(0\). Then \(\text{Ass}_{X'/Y'}(\mathcal{F}')\) is the inverse image of \(\text{Ass}_{X/Y}(\mathcal{F})\).

Proof

The transcendence degree of a point over its image is defined in Morphisms of Spaces, Definition 04NM. Let \(x' \in |X'|\) with image \(x \in |X|\). Choose a scheme \(V\) and a surjective étale morphism \(V \to Y\). Choose a scheme \(U\) and a surjective étale morphism \(U \to V \times_Y X\). Choose a scheme \(V'\) and a surjective étale morphism \(V' \to V \times_Y Y'\). Then \(U' = V' \times_V U\) is a scheme and the morphism \(U' \to X'\) is surjective and étale. Choose \(u \in U\) mapping to \(x\). Denote \(v \in V\) the image of \(u\). Then \(x \in \text{Ass}_{X/Y}(\mathcal{F})\) is equivalent to \(u \in \text{Ass}(\mathcal{F}|_{U_v})\) by definition. Choose a point \(u' \in U'\) mapping to \(x' \in |X'|\) and to \(u \in U\) (possible by Properties of Spaces, Lemma 03H4). Let \(v' \in V'\) be the image of \(u'\). Then \(x' \in \text{Ass}_{X'/Y'}(\mathcal{F}')\) is equivalent to \(u' \in \text{Ass}(\mathcal{F}'|_{U'_{v'}})\) by definition. Now the lemma follows from the discussion in Divisors, Remark 05KL applied to \(u' \in \Spec(\kappa(v') \otimes_{\kappa(v)} \kappa(u))\).

Lemma

Let \(S\) be a scheme. Let \(f : X \to Y\) be a morphism of algebraic spaces over \(S\). Let \(i : Z \to X\) be a finite morphism. Let \(\mathcal{G}\) be a quasi-coherent \(\mathcal{O}_Z\)-module. Then \(\text{WeakAss}_{X/Y}(i_*\mathcal{G}) = i(\text{WeakAss}_{Z/Y}(\mathcal{G}))\).

Proof

Follows from the case of schemes (Divisors, Lemma 0CUD) by étale localization. Details omitted.

Lemma

Let \(Y\) be a scheme. Let \(X\) be an algebraic space of finite presentation over \(Y\). Let \(\mathcal{F}\) be a quasi-coherent \(\mathcal{O}_X\)-module of finite presentation. Let \(U \subset X\) be an open subspace such that \(U \to Y\) is quasi-compact. Then the set \[E = \{y \in Y \mid \text{Ass}_{X_y}(\mathcal{F}_y) \subset |U_y|\}\] is locally constructible in \(Y\).

Proof

Note that since \(Y\) is a scheme, it makes sense to take the fibres \(X_y = \Spec(\kappa(y)) \times_Y X\). (Also, by our definitions, the set \(\text{Ass}_{X_y}(\mathcal{F}_y)\) is exactly the fibre of \(\text{Ass}_{X/Y}(\mathcal{F}) \to Y\) over \(y\), but we won’t need this.) The question is local on \(Y\), indeed, we have to show that \(E\) is constructible if \(Y\) is affine. In this case \(X\) is quasi-compact. Choose an affine scheme \(W\) and a surjective étale morphism \(\varphi : W \to X\). Then \(\text{Ass}_{X_y}(\mathcal{F}_y)\) is the image of \(\text{Ass}_{W_y}(\varphi^*\mathcal{F}_y)\) for all \(y \in Y\). Hence the lemma follows from the case of schemes for the open \(\varphi^{-1}(U) \subset W\) and the morphism \(W \to Y\). The case of schemes is More on Morphisms, Lemma 05KR.

Fitting ideals

This section is the continuation of the discussion in Divisors, Section 0C3C. Let \(S\) be a scheme. Let \(X\) be an algebraic space over \(S\). Let \(\mathcal{F}\) be a finite type, quasi-coherent \(\mathcal{O}_X\)-module. In this situation we can construct the Fitting ideals \[0 = \text{Fit}_{-1}(\mathcal{F}) \subset \text{Fit}_0(\mathcal{F}) \subset \text{Fit}_1(\mathcal{F}) \subset \ldots \subset \mathcal{O}_X\] as the sequence of quasi-coherent sheaves ideals characterized by the following property: for every affine \(U = \Spec(A)\) étale over \(X\) if \(\mathcal{F}|_U\) corresponds to the \(A\)-module \(M\), then \(\text{Fit}_i(\mathcal{F})|_U\) corresponds to the ideal \(\text{Fit}_i(M) \subset A\). This is well defined and a quasi-coherent sheaf of ideals because if \(A \to B\) is an étale ring map, then the \(i\)th Fitting ideal of \(M \otimes_A B\) over \(B\) is equal to \(\text{Fit}_i(M) B\) by More on Algebra, Lemma 07ZA part (3). More precisely (perhaps), the existence of the quasi-coherent sheaves of ideals \(\text{Fit}_0(\mathcal{O}_X)\) follows (for example) from the description of quasi-coherent sheaves in Properties of Spaces, Lemma 03LZ and the pullback property given in Divisors, Lemma 0C3D.

The advantage of constructing the Fitting ideals in this way is that we see immediately that formation of Fitting ideals commutes with étale localization hence many properties of the Fitting ideals immediately reduce to the corresponding properties in the case of schemes. Often we will use the discussion in Properties of Spaces, Section 05VR to do the translation between properties of quasi-coherent sheaves on schemes and on algebraic spaces.

Lemma

Let \(S\) be a scheme. Let \(f : X \to Y\) be a morphism of algebraic spaces over \(S\). Let \(\mathcal{F}\) be a finite type, quasi-coherent \(\mathcal{O}_Y\)-module. Then \(f^{-1}\text{Fit}_i(\mathcal{F}) \cdot \mathcal{O}_X = \text{Fit}_i(f^*\mathcal{F})\).

Proof

Reduces to Divisors, Lemma 0C3D by étale localization.

Lemma

Let \(S\) be a scheme. Let \(X\) be an algebraic space over \(S\). Let \(\mathcal{F}\) be a finitely presented \(\mathcal{O}_X\)-module. Then \(\text{Fit}_r(\mathcal{F})\) is a quasi-coherent ideal of finite type.

Proof

Reduces to Divisors, Lemma 0C3E by étale localization.

Lemma

Let \(S\) be a scheme. Let \(X\) be an algebraic space over \(S\). Let \(\mathcal{F}\) be a finite type, quasi-coherent \(\mathcal{O}_X\)-module. Let \(Z_0 \subset X\) be the closed subspace cut out by \(\text{Fit}_0(\mathcal{F})\). Let \(Z \subset X\) be the scheme theoretic support of \(\mathcal{F}\). Then

1. \(Z \subset Z_0 \subset X\) as closed subspaces,

2. \(|Z| = |Z_0| = \text{Supp}(\mathcal{F})\) as closed subsets of \(|X|\),

3. there exists a finite type, quasi-coherent \(\mathcal{O}_{Z_0}\)-module \(\mathcal{G}_0\) with \[(Z_0 \to X)_*\mathcal{G}_0 = \mathcal{F}.\]

Proof

Recall that formation of \(Z\) commutes with étale localization, see Morphisms of Spaces, Definition 07U1 (which uses Morphisms of Spaces, Lemma 07U0 to define \(Z\)). Hence (1) and (2) follow from the case of schemes, see Divisors, Lemma 0CYX. To get \(\mathcal{G}_0\) as in part (3) we can use that we have \(\mathcal{G}\) on \(Z\) as in Morphisms of Spaces, Lemma 07U0 and set \(\mathcal{G}_0 = (Z \to Z_0)_*\mathcal{G}\).

Lemma

Let \(S\) be a scheme. Let \(X\) be an algebraic space over \(S\). Let \(\mathcal{F}\) be a finite type, quasi-coherent \(\mathcal{O}_X\)-module. Let \(x \in |X|\). Then \(\mathcal{F}\) can be generated by \(r\) elements in an étale neighbourhood of \(x\) if and only if \(\text{Fit}_r(\mathcal{F})_{\overline{x}} = \mathcal{O}_{X, \overline{x}}\).

Proof

Reduces to Divisors, Lemma 0C3F by étale localization (as well as the description of the local ring in Properties of Spaces, Section 04KE and the fact that the strict henselization of a local ring is faithfully flat to see that the equality over the strict henselization is equivalent to the equality over the local ring).

Lemma

Let \(S\) be a scheme. Let \(X\) be an algebraic space over \(S\). Let \(\mathcal{F}\) be a finite type, quasi-coherent \(\mathcal{O}_X\)-module. Let \(r \geq 0\). The following are equivalent

1. \(\mathcal{F}\) is finite locally free of rank \(r\)

2. \(\text{Fit}_{r - 1}(\mathcal{F}) = 0\) and \(\text{Fit}_r(\mathcal{F}) = \mathcal{O}_X\), and

3. \(\text{Fit}_k(\mathcal{F}) = 0\) for \(k < r\) and \(\text{Fit}_k(\mathcal{F}) = \mathcal{O}_X\) for \(k \geq r\).

Proof

Reduces to Divisors, Lemma 0C3G by étale localization.

Lemma

Let \(S\) be a scheme. Let \(X\) be an algebraic space over \(S\). Let \(\mathcal{F}\) be a finite type, quasi-coherent \(\mathcal{O}_X\)-module. The closed subspaces \[X = Z_{-1} \supset Z_0 \supset Z_1 \supset Z_2 \ldots\] defined by the Fitting ideals of \(\mathcal{F}\) have the following properties

1. The intersection \(\bigcap Z_r\) is empty.

2. The functor \((\Sch/X)^{opp} \to \textit{Sets}\) defined by the rule \[T \longmapsto \left\{ \begin{matrix} \{*\} & \text{if }\mathcal{F}_T\text{ is locally generated by } \leq r\text{ sections} \\ \emptyset & \text{otherwise} \end{matrix} \right.\] is representable by the open subspace \(X \setminus Z_r\).

3. The functor \(F_r : (\Sch/X)^{opp} \to \textit{Sets}\) defined by the rule \[T \longmapsto \left\{ \begin{matrix} \{*\} & \text{if }\mathcal{F}_T\text{ locally free rank }r\\ \emptyset & \text{otherwise} \end{matrix} \right.\] is representable by the locally closed subspace \(Z_{r - 1} \setminus Z_r\) of \(X\).

If \(\mathcal{F}\) is of finite presentation, then \(Z_r \to X\), \(X \setminus Z_r \to X\), and \(Z_{r - 1} \setminus Z_r \to X\) are of finite presentation.

Proof

Reduces to Divisors, Lemma 05P8 by étale localization.

Lemma

Let \(S\) be a scheme. Let \(X\) be an algebraic space over \(S\). Let \(\mathcal{F}\) be an \(\mathcal{O}_X\)-module of finite presentation. Let \(X = Z_{-1} \subset Z_0 \subset Z_1 \subset \ldots\) be as in Lemma 0CZ9. Set \(X_r = Z_{r - 1} \setminus Z_r\). Then \(X' = \coprod_{r \geq 0} X_r\) represents the functor \[F_{flat} : \Sch/X \longrightarrow \textit{Sets},\quad\quad T \longmapsto \left\{ \begin{matrix} \{*\} & \text{if }\mathcal{F}_T\text{ flat over }T\\ \emptyset & \text{otherwise} \end{matrix} \right.\] Moreover, \(\mathcal{F}|_{X_r}\) is locally free of rank \(r\) and the morphisms \(X_r \to X\) and \(X' \to X\) are of finite presentation.

Proof

Reduces to Divisors, Lemma 05P9 by étale localization.

Effective Cartier divisors

For some reason it seem convenient to define the notion of an effective Cartier divisor before anything else. Note that in Morphisms of Spaces, Section 03MA we discussed the correspondence between closed subspaces and quasi-coherent sheaves of ideals. Moreover, in Properties of Spaces, Section 05VR, we discussed properties of quasi-coherent modules, in particular “locally generated by \(1\) element”. These references show that the following definition is compatible with the definition for schemes.

Definition

Let \(S\) be a scheme. Let \(X\) be an algebraic space over \(S\).

1. A locally principal closed subspace of \(X\) is a closed subspace whose sheaf of ideals is locally generated by \(1\) element.

2. An effective Cartier divisor on \(X\) is a closed subspace \(D \subset X\) such that the ideal sheaf \(\mathcal{I}_D \subset \mathcal{O}_X\) is an invertible \(\mathcal{O}_X\)-module.

Thus an effective Cartier divisor is a locally principal closed subspace, but the converse is not always true. Effective Cartier divisors are closed subspaces of pure codimension \(1\) in the strongest possible sense. Namely they are locally cut out by a single element which is not a zerodivisor. In particular they are nowhere dense.

Lemma

Let \(S\) be a scheme. Let \(X\) be an algebraic space over \(S\). Let \(D \subset X\) be a closed subspace. The following are equivalent:

1. The subspace \(D\) is an effective Cartier divisor on \(X\).

2. For some scheme \(U\) and surjective étale morphism \(U \to X\) the inverse image \(D \times_X U\) is an effective Cartier divisor on \(U\).

3. For every scheme \(U\) and every étale morphism \(U \to X\) the inverse image \(D \times_X U\) is an effective Cartier divisor on \(U\).

4. For every \(x \in |D|\) there exists an étale morphism \((U, u) \to (X, x)\) of pointed algebraic spaces such that \(U = \Spec(A)\) and \(D \times_X U = \Spec(A/(f))\) with \(f \in A\) not a zerodivisor.

Proof

The equivalence of (1) – (3) follows from Definition 083B and the references preceding it. Assume (1) and let \(x \in |D|\). Choose a scheme \(W\) and a surjective étale morphism \(W \to X\). Choose \(w \in D \times_X W\) mapping to \(x\). By (3) \(D \times_X W\) is an effective Cartier divisor on \(W\). Hence we can find affine étale neighbourhood \(U\) by choosing an affine open neighbourhood of \(w\) in \(W\) as in Divisors, Lemma 01WS.

Assume (4). Then we see that \(\mathcal{I}_D|_U\) is invertible by Divisors, Lemma 01WS. Since we can find an étale covering of \(X\) by the collection of all such \(U\) and \(X \setminus D\), we conclude that \(\mathcal{I}_D\) is an invertible \(\mathcal{O}_X\)-module.

Lemma

Let \(S\) be a scheme. Let \(X\) be an algebraic space over \(S\). Let \(Z \subset X\) be a locally principal closed subspace. Let \(U = X \setminus Z\). Then \(U \to X\) is an affine morphism.

Proof

The question is étale local on \(X\), see Morphisms of Spaces, Lemmas 03WG and Lemma 083C. Thus this follows from the case of schemes which is Divisors, Lemma 07ZT.

Lemma

Let \(S\) be a scheme. Let \(X\) be an algebraic space over \(S\). Let \(D \subset X\) be an effective Cartier divisor. Let \(U = X \setminus D\). Then \(U \to X\) is an affine morphism and \(U\) is scheme theoretically dense in \(X\).

Proof

Affineness is Lemma 083D. The density question is étale local on \(X\) by Morphisms of Spaces, Definition 0834. Thus this follows from the case of schemes which is Divisors, Lemma 07ZU.

Lemma

Let \(S\) be a scheme. Let \(X\) be an algebraic space over \(S\). Let \(D \subset X\) be an effective Cartier divisor. Let \(x \in |D|\). If \(\dim_x(X) < \infty\), then \(\dim_x(D) < \dim_x(X)\).

Proof

Both the definition of an effective Cartier divisor and of the dimension of an algebraic space at a point (Properties of Spaces, Definition 04N5) are étale local. Hence this lemma follows from the case of schemes which is Divisors, Lemma 056N.

Definition

Let \(S\) be a scheme. Let \(X\) be an algebraic space over \(S\). Given effective Cartier divisors \(D_1\), \(D_2\) on \(X\) we set \(D = D_1 + D_2\) equal to the closed subspace of \(X\) corresponding to the quasi-coherent sheaf of ideals \(\mathcal{I}_{D_1}\mathcal{I}_{D_2} \subset \mathcal{O}_S\). We call this the sum of the effective Cartier divisors \(D_1\) and \(D_2\).

It is clear that we may define the sum \(\sum n_iD_i\) given finitely many effective Cartier divisors \(D_i\) on \(X\) and nonnegative integers \(n_i\).

Lemma

The sum of two effective Cartier divisors is an effective Cartier divisor.

Proof

Omitted. Étale locally this reduces to the following simple algebra fact: if \(f_1, f_2 \in A\) are nonzerodivisors of a ring \(A\), then \(f_1f_2 \in A\) is a nonzerodivisor.

Lemma

Let \(S\) be a scheme. Let \(X\) be an algebraic space over \(S\). Let \(Z, Y\) be two closed subspaces of \(X\) with ideal sheaves \(\mathcal{I}\) and \(\mathcal{J}\). If \(\mathcal{I}\mathcal{J}\) defines an effective Cartier divisor \(D \subset X\), then \(Z\) and \(Y\) are effective Cartier divisors and \(D = Z + Y\).

Proof

By Lemma 083C this reduces to the case of schemes which is Divisors, Lemma 07ZV.

Recall that we have defined the inverse image of a closed subspace under any morphism of algebraic spaces in Morphisms of Spaces, Definition 083Q.

Lemma

Let \(S\) be a scheme. Let \(f : X' \to X\) be a morphism of algebraic spaces over \(S\). Let \(Z \subset X\) be a locally principal closed subspace. Then the inverse image \(f^{-1}(Z)\) is a locally principal closed subspace of \(X'\).

Proof

Omitted.

Definition

Let \(S\) be a scheme. Let \(f : X' \to X\) be a morphism of algebraic spaces over \(S\). Let \(D \subset X\) be an effective Cartier divisor. We say the pullback of \(D\) by \(f\) is defined if the closed subspace \(f^{-1}(D) \subset X'\) is an effective Cartier divisor. In this case we denote it either \(f^*D\) or \(f^{-1}(D)\) and we call it the pullback of the effective Cartier divisor.

The condition that \(f^{-1}(D)\) is an effective Cartier divisor is often satisfied in practice.

Lemma

Let \(S\) be a scheme. Let \(f : X \to Y\) be a morphism of algebraic spaces over \(S\). Let \(D \subset Y\) be an effective Cartier divisor. The pullback of \(D\) by \(f\) is defined in each of the following cases:

1. \(f(x) \not \in |D|\) for any weakly associated point \(x\) of \(X\),

2. \(f\) is flat, and

3. add more here as needed.

Proof

Working étale locally this lemma reduces to the case of schemes, see Divisors, Lemma 02OO.

Lemma

Let \(S\) be a scheme. Let \(f : X' \to X\) be a morphism of algebraic spaces over \(S\). Let \(D_1\), \(D_2\) be effective Cartier divisors on \(X\). If the pullbacks of \(D_1\) and \(D_2\) are defined then the pullback of \(D = D_1 + D_2\) is defined and \(f^*D = f^*D_1 + f^*D_2\).

Proof

Omitted.

Effective Cartier divisors and invertible sheaves

Since an effective Cartier divisor has an invertible ideal sheaf (Definition 083B) the following definition makes sense.

Definition

Let \(S\) be a scheme. Let \(X\) be an algebraic space over \(S\) and let \(D \subset X\) be an effective Cartier divisor with ideal sheaf \(\mathcal{I}_D\).

1. The invertible sheaf \(\mathcal{O}_X(D)\) associated to \(D\) is defined by \[\mathcal{O}_X(D) = \SheafHom_{\mathcal{O}_X}(\mathcal{I}_D, \mathcal{O}_X) = \mathcal{I}_D^{\otimes -1}.\]

2. The canonical section, usually denoted \(1\) or \(1_D\), is the global section of \(\mathcal{O}_X(D)\) corresponding to the inclusion mapping \(\mathcal{I}_D \to \mathcal{O}_X\).

3. We write \(\mathcal{O}_X(-D) = \mathcal{O}_X(D)^{\otimes -1} = \mathcal{I}_D\).

4. Given a second effective Cartier divisor \(D' \subset X\) we define \(\mathcal{O}_X(D - D') = \mathcal{O}_X(D) \otimes_{\mathcal{O}_X} \mathcal{O}_X(-D')\).

Some comments. We will see below that the assignment \(D \mapsto \mathcal{O}_X(D)\) turns addition of effective Cartier divisors (Definition 083U) into addition in the Picard group of \(X\) (Lemma 0842). However, the expression \(D - D'\) in the definition above does not have any geometric meaning. More precisely, we can think of the set of effective Cartier divisors on \(X\) as a commutative monoid \(\text{EffCart}(X)\) whose zero element is the empty effective Cartier divisor. Then the assignment \((D, D') \mapsto \mathcal{O}_X(D - D')\) defines a group homomorphism \[\text{EffCart}(X)^{gp} \longrightarrow \Pic(X)\] where the left hand side is the group completion of \(\text{EffCart}(X)\). In other words, when we write \(\mathcal{O}_X(D - D')\) we may think of \(D - D'\) as an element of \(\text{EffCart}(X)^{gp}\).

Lemma

Let \(S\) be a scheme. Let \(X\) be an algebraic space over \(S\). Let \(D \subset X\) be an effective Cartier divisor. Then for the conormal sheaf we have \(\mathcal{C}_{D/X} = \mathcal{I}_D|D = \mathcal{O}_X(D)^{\otimes -1}|_D\).

Proof

Omitted.

Lemma

Let \(S\) be a scheme. Let \(X\) be an algebraic space over \(S\). Let \(D_1\), \(D_2\) be effective Cartier divisors on \(X\). Let \(D = D_1 + D_2\). Then there is a unique isomorphism \[\mathcal{O}_X(D_1) \otimes_{\mathcal{O}_X} \mathcal{O}_X(D_2) \longrightarrow \mathcal{O}_X(D)\] which maps \(1_{D_1} \otimes 1_{D_2}\) to \(1_D\).

Proof

Omitted.

Definition

Let \(S\) be a scheme. Let \(X\) be an algebraic space over \(S\). Let \(\mathcal{L}\) be an invertible sheaf on \(X\). A global section \(s \in \Gamma(X, \mathcal{L})\) is called a regular section if the map \(\mathcal{O}_X \to \mathcal{L}\), \(f \mapsto fs\) is injective.

Lemma

Let \(S\) be a scheme. Let \(X\) be an algebraic space over \(S\). Let \(f \in \Gamma(X, \mathcal{O}_X)\). The following are equivalent:

1. \(f\) is a regular section, and

2. for any \(x \in X\) the image \(f \in \mathcal{O}_{X, \overline{x}}\) is not a zerodivisor.

3. for any affine \(U = \Spec(A)\) étale over \(X\) the restriction \(f|_U\) is a nonzerodivisor of \(A\), and

4. there exists a scheme \(U\) and a surjective étale morphism \(U \to X\) such that \(f|_U\) is a regular section of \(\mathcal{O}_U\).

Proof

Omitted.

Note that a global section \(s\) of an invertible \(\mathcal{O}_X\)-module \(\mathcal{L}\) may be seen as an \(\mathcal{O}_X\)-module map \(s : \mathcal{O}_X \to \mathcal{L}\). Its dual is therefore a map \(s : \mathcal{L}^{\otimes -1} \to \mathcal{O}_X\). (See Modules on Sites, Lemma 040A for the dual invertible sheaf.)

Definition

Let \(S\) be a scheme. Let \(X\) be an algebraic space over \(S\). Let \(\mathcal{L}\) be an invertible sheaf. Let \(s \in \Gamma(X, \mathcal{L})\). The zero scheme of \(s\) is the closed subspace \(Z(s) \subset X\) defined by the quasi-coherent sheaf of ideals \(\mathcal{I} \subset \mathcal{O}_X\) which is the image of the map \(s : \mathcal{L}^{\otimes -1} \to \mathcal{O}_X\).

Lemma

Let \(S\) be a scheme. Let \(X\) be an algebraic space over \(S\). Let \(\mathcal{L}\) be an invertible \(\mathcal{O}_X\)-module. Let \(s \in \Gamma(X, \mathcal{L})\).

1. Consider closed immersions \(i : Z \to X\) such that \(i^*s \in \Gamma(Z, i^*\mathcal{L}))\) is zero ordered by inclusion. The zero scheme \(Z(s)\) is the maximal element of this ordered set.

2. For any morphism of algebraic spaces \(f : Y \to X\) over \(S\) we have \(f^*s = 0\) in \(\Gamma(Y, f^*\mathcal{L})\) if and only if \(f\) factors through \(Z(s)\).

3. The zero scheme \(Z(s)\) is a locally principal closed subspace of \(X\).

4. The zero scheme \(Z(s)\) is an effective Cartier divisor on \(X\) if and only if \(s\) is a regular section of \(\mathcal{L}\).

Proof

Omitted.

Lemma

Let \(S\) be a scheme. Let \(X\) be an algebraic space over \(S\).

1. If \(D \subset X\) is an effective Cartier divisor, then the canonical section \(1_D\) of \(\mathcal{O}_X(D)\) is regular.

2. Conversely, if \(s\) is a regular section of the invertible sheaf \(\mathcal{L}\), then there exists a unique effective Cartier divisor \(D = Z(s) \subset X\) and a unique isomorphism \(\mathcal{O}_X(D) \to \mathcal{L}\) which maps \(1_D\) to \(s\).

The constructions \(D \mapsto (\mathcal{O}_X(D), 1_D)\) and \((\mathcal{L}, s) \mapsto Z(s)\) give mutually inverse maps \[\left\{ \begin{matrix} \text{effective Cartier divisors on }X \end{matrix} \right\} \leftrightarrow \left\{ \begin{matrix} \text{pairs }(\mathcal{L}, s)\text{ consisting of an invertible}\\ \mathcal{O}_X\text{-module and a regular global section} \end{matrix} \right\}\]

Proof

Omitted.

Effective Cartier divisors on Noetherian spaces

In the locally Noetherian setting most of the discussion of effective Cartier divisors and regular sections simplifies somewhat.

Lemma

Let \(S\) be a scheme and let \(X\) be a locally Noetherian algebraic space over \(S\). Let \(D \subset X\) be an effective Cartier divisor. If \(X\) is \((S_k)\), then \(D\) is \((S_{k - 1})\).

Proof

By our definition of the property \((S_k)\) for algebraic spaces (Properties of Spaces, Section 03E5) and Lemma 083C this follows from the case of schemes (Divisors, Lemma 0B3R).

Lemma

Let \(S\) be a scheme and let \(X\) be a locally Noetherian normal algebraic space over \(S\). Let \(D \subset X\) be an effective Cartier divisor. Then \(D\) is \((S_1)\).

Proof

By our definition of normality for algebraic spaces (Properties of Spaces, Section 03E5) and Lemma 083C this follows from the case of schemes (Divisors, Lemma 0B3S).

The following lemma can sometimes be used to produce effective Cartier divisors.

Lemma

Let \(S\) be a scheme. Let \(X\) be a regular Noetherian separated algebraic space over \(S\). Let \(U \subset X\) be a dense affine open. Then there exists an effective Cartier divisor \(D \subset X\) with \(U = X \setminus D\).

Proof

We claim that the reduced induced algebraic space structure \(D\) on \(X \setminus U\) (Properties of Spaces, Definition 047X) is the desired effective Cartier divisor. The construction of \(D\) commutes with étale localization, see proof of Properties of Spaces, Lemma 03IQ. Let \(X' \to X\) be a surjective étale morphism with \(X'\) affine. Since \(X\) is separated, we see that \(U' = X' \times_X U\) is affine. Since \(|X'| \to |X|\) is open, we see that \(U'\) is dense in \(X'\). Since \(D' = X' \times_X D\) is the reduced induced scheme structure on \(X' \setminus U'\), we conclude that \(D'\) is an effective Cartier divisor by Divisors, Lemma 0BCW and its proof. This is what we had to show.

Lemma

Let \(S\) be a scheme. Let \(X\) be a regular Noetherian separated algebraic space over \(S\). Then every invertible \(\mathcal{O}_X\)-module is isomorphic to \[\mathcal{O}_X(D - D') = \mathcal{O}_X(D) \otimes_{\mathcal{O}_X} \mathcal{O}_X(D')^{\otimes -1}\] for some effective Cartier divisors \(D, D'\) in \(X\).

Proof

Let \(\mathcal{L}\) be an invertible \(\mathcal{O}_X\)-module. Choose a dense affine open \(U \subset X\) such that \(\mathcal{L}|_U\) is trivial. This is possible because \(X\) has a dense open subspace which is a scheme, see Properties of Spaces, Proposition 06NH. Denote \(s : \mathcal{O}_U \to \mathcal{L}|_U\) the trivialization. The complement of \(U\) is an effective Cartier divisor \(D\). We claim that for some \(n > 0\) the map \(s\) extends uniquely to a map \[s : \mathcal{O}_X(-nD) \longrightarrow \mathcal{L}\] The claim implies the lemma because it shows that \(\mathcal{L} \otimes_{\mathcal{O}_X} \mathcal{O}_X(nD)\) has a regular global section hence is isomorphic to \(\mathcal{O}_X(D')\) for some effective Cartier divisor \(D'\) by Lemma 0847. To prove the claim we may work étale locally. Thus we may assume \(X\) is an affine Noetherian scheme. Since \(\mathcal{O}_X(-nD) = \mathcal{I}^n\) where \(\mathcal{I} = \mathcal{O}_X(-D)\) is the ideal sheaf of \(D\) in \(X\), this case follows from Cohomology of Schemes, Lemma 01YB.

The following lemma really belongs to a different section.

Lemma

Let \(R\) be a valuation ring with fraction field \(K\). Let \(X\) be an algebraic space over \(R\) such that \(X \to \Spec(R)\) is smooth. For every effective Cartier divisor \(D \subset X_K\) there exists an effective Cartier divisor \(D' \subset X\) with \(D'_K = D\).

Proof

Let \(D' \subset X\) be the scheme theoretic image of \(D \to X_K \to X\). Since this morphism is quasi-compact, formation of \(D'\) commutes with flat base change, see Morphisms of Spaces, Lemma 089E. In particular we find that \(D'_K = D\). Hence, we may assume \(X\) is affine. Say \(X = \Spec(A)\). Then \(X_K = \Spec(A \otimes_R K)\) and \(D\) corresponds to an ideal \(I \subset A \otimes_R K\). We have to show that \(J = I \cap A\) cuts out an effective Cartier divisor in \(X\). First, observe that \(A/J\) is flat over \(R\) (as a torsion free \(R\)-module, see More on Algebra, Lemma 0539), hence \(J\) is finitely generated by More on Algebra, Lemma 053E and Algebra, Lemma 0519. Thus it suffices to show that \(J_\mathfrak q \subset A_\mathfrak q\) is generated by a single element for each prime \(\mathfrak q \subset A\). Let \(\mathfrak p = R \cap \mathfrak q\). Then \(R_\mathfrak p\) is a valuation ring (Algebra, Lemma 088Y). Observe further that \(A_\mathfrak q/\mathfrak p A_\mathfrak q\) is a regular ring by Algebra, Lemma 00TT. Thus we may apply More on Algebra, Lemma 0DLQ to see that \(I(A_\mathfrak q \otimes_R K)\) is generated by a single element \(f \in A_\mathfrak p \otimes_R K\). After clearing denominators we may assume \(f \in A_\mathfrak q\). Let \(\mathfrak c \subset R_\mathfrak p\) be the content ideal of \(f\) (see More on Algebra, Definition 0ASA and More on Flatness, Lemma 0ASX). Since \(R_\mathfrak p\) is a valuation ring and since \(\mathfrak c\) is finitely generated (More on Algebra, Lemma 0ASB) we see \(\mathfrak c = (\pi)\) for some \(\pi \in R_\mathfrak p\) (Algebra, Lemma 090Q). After relacing \(f\) by \(\pi^{-1}f\) we see that \(f \in A_\mathfrak q\) and \(f \not \in \mathfrak pA_\mathfrak q\). Claim: \(I_\mathfrak q = (f)\) which finishes the proof. To see the claim, observe that \(f \in I_\mathfrak q\). Hence we have a surjection \(A_\mathfrak q/(f) \to A_\mathfrak q/I_\mathfrak q\) which is an isomorphism after tensoring over \(R\) with \(K\). Thus we are done if \(A_\mathfrak q/(f)\) is \(R_\mathfrak p\)-flat. This follows from Algebra, Lemma 046Z and our choice of \(f\).

Relative effective Cartier divisors

The following lemma shows that an effective Cartier divisor which is flat over the base is really a “family of effective Cartier divisors” over the base. For example the restriction to any fibre is an effective Cartier divisor.

Lemma

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

1. \(D\) is an effective Cartier divisor, and

2. \(D \to Y\) is a flat morphism.

Then for every morphism of schemes \(g : Y' \to Y\) the pullback \((g')^{-1}D\) is an effective Cartier divisor on \(X' = Y' \times_Y X\) where \(g' : X' \to X\) is the projection.

Proof

Using Lemma 083C the property of being an effective Cartier divisor is étale local. Thus this lemmma immediately reduces to the case of schemes which is Divisors, Lemma 056Q.

This lemma is the motivation for the following definition.

Definition

Let \(S\) be a scheme. Let \(f : X \to Y\) be a morphism of algebraic spaces over \(S\). A relative effective Cartier divisor on \(X/Y\) is an effective Cartier divisor \(D \subset X\) such that \(D \to Y\) is a flat morphism of algebraic spaces.

Meromorphic functions and sections

This section is the analogue of Divisors, Section 01X1. Beware: it is even easier to make mistakes with this material in the case of algebraic space, than it is in the case of schemes!

Let \(S\) be a scheme. Let \(X\) be an algebraic space over \(S\). For any scheme \(U\) étale over \(X\) we have defined the set \(\mathcal{S}(U) \subset \mathcal{O}_X(U)\) of regular sections of \(\mathcal{O}_X\) over \(U\), see Definition 0843. The restriction of a regular section to \(V/U\) étale is regular. Hence \(\mathcal{S} : U \mapsto \mathcal{S}(U)\) is a subsheaf (of sets) of \(\mathcal{O}_X\). We sometimes denote \(\mathcal{S} = \mathcal{S}_X\) if we want to indicate the dependence on \(X\). Moreover, \(\mathcal{S}(U)\) is a multiplicative subset of the ring \(\mathcal{O}_X(U)\) for each \(U\). Hence we may consider the presheaf of rings \[U \longmapsto \mathcal{S}(U)^{-1} \mathcal{O}_X(U),\] on \(X_\etale\) and its sheafification, see Modules on Sites, Section 0EMB.

Definition

Let \(S\) be a scheme. Let \(X\) be an algebraic space over \(S\). The sheaf of meromorphic functions on \(X\) is the sheaf \(\mathcal{K}_X\) on \(X_\etale\) associated to the presheaf displayed above. A meromorphic function on \(X\) is a global section of \(\mathcal{K}_X\).

Since each element of each \(\mathcal{S}(U)\) is a nonzerodivisor on \(\mathcal{O}_X(U)\) we see that the natural map of sheaves of rings \(\mathcal{O}_X \to \mathcal{K}_X\) is injective. Moreover, by the compatibility of sheafification and taking stalks we see that \[\mathcal{K}_{X, \overline{x}} = \mathcal{S}_{\overline{x}}^{-1}\mathcal{O}_{X, \overline{x}}\] for any geometric point \(\overline{x}\) of \(X\). The set \(\mathcal{S}_{\overline{x}}\) is a subset of the set of nonzerodivisors of \(\mathcal{O}_{X, \overline{x}}\), but in general not equal to this.

Lemma

Let \(S\) be a scheme. Let \(X\) be an algebraic space over \(S\). For \(U\) affine and étale over \(X\) the set \(\mathcal{S}_X(U)\) is the set of nonzerodivisors in \(\mathcal{O}_X(U)\).

Proof

Follows from Lemma 0844.

Next, let \(\mathcal{F}\) be a sheaf of \(\mathcal{O}_X\)-modules on \(X_\etale\). Consider the presheaf \(U \mapsto \mathcal{S}(U)^{-1}\mathcal{F}(U)\). Its sheafification is the sheaf \(\mathcal{F} \otimes_{\mathcal{O}_X} \mathcal{K}_X\), see Modules on Sites, Lemma 0EMD.

Definition

Let \(S\) be a scheme. Let \(X\) be an algebraic space over \(S\). Let \(\mathcal{F}\) be a sheaf of \(\mathcal{O}_X\)-modules on \(X_\etale\).

1. We denote \(\mathcal{K}_X(\mathcal{F})\) the sheaf of \(\mathcal{K}_X\)-modules which is the sheafification of the presheaf \(U \mapsto \mathcal{S}(U)^{-1}\mathcal{F}(U)\). Equivalently \(\mathcal{K}_X(\mathcal{F}) = \mathcal{F} \otimes_{\mathcal{O}_X} \mathcal{K}_X\) (see above).

2. A meromorphic section of \(\mathcal{F}\) is a global section of \(\mathcal{K}_X(\mathcal{F})\).

In particular we have \[\mathcal{K}_X(\mathcal{F})_{\overline{x}} = \mathcal{F}_{\overline{x}} \otimes_{\mathcal{O}_{X, \overline{x}}} \mathcal{K}_{X, \overline{x}} = \mathcal{S}_{\overline{x}}^{-1}\mathcal{F}_{\overline{x}}\] for any geometric point \(\overline{x}\) of \(X\). However, one has to be careful since it may not be the case that \(\mathcal{S}_{\overline{x}}\) is the set of nonzerodivisors in the étale local ring \(\mathcal{O}_{X, \overline{x}}\) as we pointed out above. The sheaves of meromorphic sections aren’t quasi-coherent modules in general, but they do have some properties in common with quasi-coherent modules.

Lemma

Let \(S\) be a scheme. Let \(X\) be an algebraic space over \(S\). Assume

1. every weakly associated point of \(X\) is a point of codimension \(0\), and

2. \(X\) satisfies the equivalent conditions of Morphisms of Spaces, Lemma 0BB1.

Then

1. \(\mathcal{K}_X\) is a quasi-coherent sheaf of \(\mathcal{O}_X\)-algebras,

2. for \(U \in X_\etale\) affine \(\mathcal{K}_X(U)\) is the total ring of fractions of \(\mathcal{O}_X(U)\),

3. for a geometric point \(\overline{x}\) the set \(\mathcal{S}_{\overline{x}}\) the set of nonzerodivisors of \(\mathcal{O}_{X, \overline{x}}\), and

4. for a geometric point \(\overline{x}\) the ring \(\mathcal{K}_{X, \overline{x}}\) is the total ring of fractions of \(\mathcal{O}_{X, \overline{x}}\).

Proof

By Lemma 0844 we see that \(U \in X_\etale\) affine \(\mathcal{S}_X(U) \subset \mathcal{O}_X(U)\) is the set of nonzerodivisors in \(\mathcal{O}_X(U)\). Thus the presheaf \(\mathcal{S}^{-1}\mathcal{O}_X\) is equal to \[U \longmapsto Q(\mathcal{O}_X(U))\] on \(X_{affine, \etale}\), with notation as in Algebra, Example 02C5. Observe that the codimension \(0\) points of \(X\) correspond to the generic points of \(U\), see Properties of Spaces, Lemma 0BAQ. Hence if \(U = \Spec(A)\), then \(A\) is a ring with finitely many minimal primes such that any weakly associated prime of \(A\) is minimal. The same is true for any étale extension of \(A\) (because the spectrum of such is an affine scheme étale over \(X\) hence can play the role of \(A\) in the previous sentence). In order to show that our presheaf is a sheaf and quasi-coherent it suffices to show that \[Q(A) \otimes_A B \longrightarrow Q(B)\] is an isomorphism when \(A \to B\) is an étale ring map, see Properties of Spaces, Lemma 03LZ. (To define the displayed arrow, observe that since \(A \to B\) is flat it maps nonzerodivisors to nonzerodivisors.) By Algebra, Lemmas 02LX and 05C3. we have \[Q(A) = \prod\nolimits_{\mathfrak p \subset A\text{ minimal}} A_\mathfrak p \quad\text{and}\quad Q(B) = \prod\nolimits_{\mathfrak q \subset B\text{ minimal}} B_\mathfrak q\] Since \(A \to B\) is étale, the minimal primes of \(B\) are exactly the primes of \(B\) lying over the minimal primes of \(A\) (for example by More on Algebra, Lemma 07QP). By Algebra, Lemmas 06RS, 04GG (13), and 04GJ we see that \(A_\mathfrak p \otimes_A B\) is a finite product of local rings finite étale over \(A_\mathfrak p\). This clearly implies that \(A_\mathfrak p \otimes_A B = \prod_{\mathfrak q\text{ lies over }\mathfrak p} B_\mathfrak q\) as desired.

At this point we know that (1) and (2) hold. Proof of (3). Let \(s \in \mathcal{O}_{X, \overline{x}}\) be a nonzerodivisor. Then we can find an étale neighbourhood \((U, \overline{u}) \to (X, \overline{x})\) and \(f \in \mathcal{O}_X(U)\) mapping to \(s\). Let \(u \in U\) be the point determined by \(\overline{u}\). Since \(\mathcal{O}_{U, u} \to \mathcal{O}_{X, \overline{x}}\) is faithfully flat (as a strict henselization), we see that \(f\) maps to a nonzerodivisor in \(\mathcal{O}_{U, u}\). By Divisors, Lemma 0EMF after shrinking \(U\) we find that \(f\) is a nonzerodivisor and hence a section of \(\mathcal{S}_X(U)\). Part (4) follows from (3) by computing stalks.

Lemma

Let \(S\) be a scheme. Let \(X\) be an algebraic space over \(S\). Assume

1. every weakly associated point of \(X\) is a point of codimension \(0\), and

2. \(X\) satisfies the equivalent conditions of Morphisms of Spaces, Lemma 0BB1.

3. \(X\) is representable by a scheme \(X_0\) (awkward but temporary notation).

Then the sheaf of meromorphic functions \(\mathcal{K}_X\) is the quasi-coherent sheaf of \(\mathcal{O}_X\)-algebras associated to the quasi-coherent sheaf of meromorphic functions \(\mathcal{K}_{X_0}\).

Proof

For the equivalence between \(\QCoh(\mathcal{O}_X)\) and \(\QCoh(\mathcal{O}_{X_0})\), please see Properties of Spaces, Section 03G5. The lemma is true because \(\mathcal{K}_X\) and \(\mathcal{K}_{X_0}\) are quasi-coherent and have the same value on corresponding affine opens of \(X\) and \(X_0\) by Lemma 0EN6 and Divisors, Lemma 0EMF.

Definition

Let \(S\) be a scheme. Let \(f : X \to Y\) be a morphism of algebraic spaces over \(S\). We say that pullbacks of meromorphic functions are defined for \(f\) if for every commutative diagram \[\xymatrix{ U \ar[r] \ar[d] & X \ar[d] \\ V \ar[r] & Y }\] with \(U \in X_\etale\) and \(V \in Y_\etale\) and any section \(s \in \mathcal{S}_Y(V)\) the pullback \(f^\sharp(s) \in \mathcal{O}_X(U)\) is an element of \(\mathcal{S}_X(U)\).

In this case there is an induced map \(f^\sharp : f_{small}^{-1}\mathcal{K}_Y \to \mathcal{K}_X\), in other words we obtain a commutative diagram of morphisms of ringed topoi \[\xymatrix{ (\Sh(X_\etale), \mathcal{K}_X) \ar[r] \ar[d]^{f_{small}} & (\Sh(X_\etale), \mathcal{O}_X) \ar[d]^{f_{small}} \\ (\Sh(Y_\etale), \mathcal{K}_Y) \ar[r] & (\Sh(Y_\etale), \mathcal{O}_Y) }\] We sometimes denote \(f^*(s) = f^\sharp(s)\) for a section \(s \in \Gamma(Y, \mathcal{K}_Y)\).

Lemma

Let \(S\) be a scheme. Let \(f : X \to Y\) be a morphism of algebraic spaces over \(S\). Pullbacks of meromorphic sections are defined in each of the following cases

1. weakly associated points of \(X\) are mapped to points of codimension \(0\) on \(Y\),

2. \(f\) is flat,

3. add more here as needed.

Proof

Working étale locally, this translates into the case of schemes, see Divisors, Lemma 02OU. To do the translation use Lemma 0844 (description of regular sections), Definition 0CTX (definition of weakly associated points), and Properties of Spaces, Lemma 0BAQ (description of codimension \(0\) points).

Lemma

Let \(S\) be a scheme. Let \(X\) be an algebraic space over \(S\). Assume

1. every weakly associated point of \(X\) is a point of codimension \(0\), and

2. \(X\) satisfies the equivalent conditions of Morphisms of Spaces, Lemma 0BB1,

3. every codimension \(0\) point of \(X\) can be represented by a monomorphism \(\Spec(k) \to X\).

Let \(X^0 \subset |X|\) be the set of codimension \(0\) points of \(X\). Then we have \[\mathcal{K}_X = \bigoplus\nolimits_{\eta \in X^0} j_{\eta, *}\mathcal{O}_{X, \eta} = \prod\nolimits_{\eta \in X^0} j_{\eta, *}\mathcal{O}_{X, \eta}\] where \(j_\eta : \Spec(\mathcal{O}_{X, \eta}) \to X\) is the canonical map of Schemes, Section 01J5; this makes sense because \(X^0\) is contained in the schematic locus of \(X\). Similarly, for every quasi-coherent \(\mathcal{O}_X\)-module \(\mathcal{F}\) we obtain the formula \[\mathcal{K}_X(\mathcal{F}) = \bigoplus\nolimits_{\eta \in X^0} j_{\eta, *}\mathcal{F}_\eta = \prod\nolimits_{\eta \in X^0} j_{\eta, *}\mathcal{F}_\eta\] for the sheaf of meromorphic sections of \(\mathcal{F}\). Finally, the ring of rational functions of \(X\) is the ring of meromorphic functions on \(X\), in a formula: \(R(X) = \Gamma(X, \mathcal{K}_X)\).

Proof

By Decent Spaces, Lemma 0BB8 and Section 03I7 we see that \(X\) is decent¹. Thus \(X^0 \subset |X|\) is the set of generic points of irreducible components (Decent Spaces, Lemma 0ABV) and \(X^0\) is locally finite in \(|X|\) by (b). It follows that \(X^0\) is contained in every dense open subset of \(|X|\). In particular, \(X^0\) is contained in the schematic locus (Decent Spaces, Theorem 086U). Thus the local rings \(\mathcal{O}_{X, \eta}\) and the morphisms \(j_\eta\) are defined.

Observe that a locally finite direct sum of sheaves of modules is equal to the product. This and the fact that \(X^0\) is locally finite in \(|X|\) explains the equalities between direct sums and products in the statement. Then since \(\mathcal{K}_X(\mathcal{F}) = \mathcal{F} \otimes_{\mathcal{O}_X} \mathcal{K}_X\) we see that the second equality follows from the first.

Let \(j : Y = \coprod\nolimits_{\eta \in X^0} \Spec(\mathcal{O}_{X, \eta}) \to X\) be the product of the morphisms \(j_\eta\). We have to show that \(\mathcal{K}_X = j_*\mathcal{O}_Y\). Observe that \(\mathcal{K}_Y = \mathcal{O}_Y\) as \(Y\) is a disjoint union of spectra of local rings of dimension \(0\): in a local ring of dimension zero any nonzerodivisor is a unit. Next, note that pullbacks of meromorphic functions are defined for \(j\) by Lemma 0EN9. This gives a map \[\mathcal{K}_X \longrightarrow j_*\mathcal{O}_Y.\] Let \(U \in X_\etale\) be affine. By Lemma 0EN6 the left hand side evaluates to total ring of fractions of \(\mathcal{O}_X(U)\). On the other hand, the right hand side is equal to the product of the local rings of \(U\) at the codimension \(0\) points, i.e., the generic points of \(U\). These two rings are equal (as we already saw in the proof of Lemma 0EN6) by Algebra, Lemmas 02LX and 05C3. Thus our map is an isomorphism.

Finally, we have to show that \(R(X) = \Gamma(X, \mathcal{K}_X)\). This follows from the case of schemes (Divisors, Lemma 0EMF) applied to the schematic locus \(X' \subset X\). Namely, the ring of rational functions of \(X\) is by definition the same as the ring of rational functions on \(X'\) as it is a dense open subspace of \(X\) (see above). Certainly, \(R(X')\) agrees with the ring of rational functions when \(X'\) is viewed as a scheme. On the other hand, by our description of \(\mathcal{K}_X\) above, and the fact, seen above, that \(X^0 \subset |X'|\) is contained in any dense open, we see that \(\Gamma(X, \mathcal{K}_X) = \Gamma(X', \mathcal{K}_{X'})\). Finally, use the compatibility recorded in Lemma 0EN7.

Definition

Let \(S\) be a scheme. Let \(X\) be an algebraic space over \(S\). Let \(\mathcal{L}\) be an invertible \(\mathcal{O}_X\)-module. A meromorphic section \(s\) of \(\mathcal{L}\) is said to be regular if the induced map \(\mathcal{K}_X \to \mathcal{K}_X(\mathcal{L})\) is injective.

Let us spell out when (regular) meromorphic sections can be pulled back.

Lemma

Let \(S\) be a scheme. Let \(f : X \to Y\) be a morphism of algebraic spaces over \(S\). Assume that pullbacks of meromorphic functions are defined for \(f\) (see Definition 0EN8).

1. Let \(\mathcal{F}\) be a sheaf of \(\mathcal{O}_Y\)-modules. There is a canonical pullback map \(f^* : \Gamma(Y, \mathcal{K}_Y(\mathcal{F})) \to \Gamma(X, \mathcal{K}_X(f^*\mathcal{F}))\) for meromorphic sections of \(\mathcal{F}\).

2. Let \(\mathcal{L}\) be an invertible \(\mathcal{O}_X\)-module. A regular meromorphic section \(s\) of \(\mathcal{L}\) pulls back to a regular meromorphic section \(f^*s\) of \(f^*\mathcal{L}\).

Proof

Omitted.

Lemma

Let \(S\) be a scheme. Let \(X\) be an algebraic space over \(S\) satisfying (a), (b), and (c) of Lemma 0ENA. Then every invertible \(\mathcal{O}_X\)-module \(\mathcal{L}\) has a regular meromorphic section.

Proof

With notation as in Lemma 0ENA the stalk \(\mathcal{L}_\eta\) of \(\mathcal{L}\) at is defined for all \(\eta \in X^0\) and it is a rank \(1\) free \(\mathcal{O}_{X, \eta}\)-module. Pick a generator \(s_\eta \in \mathcal{L}_\eta\) for all \(\eta \in X^0\). It follows immediately from the description of \(\mathcal{K}_X\) and \(\mathcal{K}_X(\mathcal{L})\) in Lemma 0ENA that \(s = \prod s_\eta\) is a regular meromorphic section of \(\mathcal{L}\).

Relative Proj

This section revisits the construction of the relative proj in the setting of algebraic spaces. The material in this section corresponds to the material in Constructions, Section 01NS and Divisors, Section 07ZW in the case of schemes.

Situation

Here \(S\) is a scheme, \(X\) is an algebraic space over \(S\), and \(\mathcal{A}\) is a quasi-coherent graded \(\mathcal{O}_X\)-algebra.

In Situation 0849 we are going to define a functor \(F : (\Sch/S)_{fppf}^{opp} \to \textit{Sets}\) which will turn out to be an algebraic space. We will follow (mutatis mutandis) the procedure of Constructions, Section 01NS. First, given a scheme \(T\) over \(S\) we define a quadruple over \(T\) to be a system \((d, f : T \to X, \mathcal{L}, \psi)\)

1. \(d \geq 1\) is an integer,

2. \(f : T \to X\) is a morphism over \(S\),

3. \(\mathcal{L}\) is an invertible \(\mathcal{O}_T\)-module, and

4. \(\psi : f^*\mathcal{A}^{(d)} \to \bigoplus_{n \geq 0}\mathcal{L}^{\otimes n}\) is a homomorphism of graded \(\mathcal{O}_T\)-algebras such that \(f^*\mathcal{A}_d \to \mathcal{L}\) is surjective.

We say two quadruples \((d, f, \mathcal{L}, \psi)\) and \((d', f', \mathcal{L}', \psi')\) are equivalent² if and only if we have \(f = f'\) and for some positive integer \(m = ad = a'd'\) there exists an isomorphism \(\beta : \mathcal{L}^{\otimes a} \to (\mathcal{L}')^{\otimes a'}\) with the property that \(\beta \circ \psi|_{f^*\mathcal{A}^{(m)}}\) and \(\psi'|_{f^*\mathcal{A}^{(m)}}\) agree as graded ring maps \(f^*\mathcal{A}^{(m)} \to \bigoplus_{n \geq 0} (\mathcal{L}')^{\otimes mn}\). Given a quadruple \((d, f, \mathcal{L}, \psi)\) and a morphism \(h : T' \to T\) we have the pullback \((d, f \circ h, h^*\mathcal{L}, h^*\psi)\). Pullback preserves the equivalence relation. Finally, for a quasi-compact scheme \(T\) over \(S\) we set \[F(T) = \text{the set of equivalence classes of quadruples over }T\] and for an arbitrary scheme \(T\) over \(S\) we set \[F(T) = \lim_{V \subset T\text{ quasi-compact open}} F(V).\] In other words, an element \(\xi\) of \(F(T)\) corresponds to a compatible system of choices of elements \(\xi_V \in F(V)\) where \(V\) ranges over the quasi-compact opens of \(T\). Thus we have defined our functor [084A]\[\begin{equation} F : \Sch^{opp} \longrightarrow \textit{Sets} \end{equation}\] There is a morphism \(F \to X\) of functors sending the quadruple \((d, f, \mathcal{L}, \psi)\) to \(f\).

Lemma

In Situation 0849. The functor \(F\) above is an algebraic space. For any morphism \(g : Z \to X\) where \(Z\) is a scheme there is a canonical isomorphism \(\underline{\text{Proj}}_Z(g^*\mathcal{A}) = Z \times_X F\) compatible with further base change.

Proof

It suffices to prove the second assertion, see Spaces, Lemma 02WY. Let \(g : Z \to X\) be a morphism where \(Z\) is a scheme. Let \(F'\) be the functor of quadruples associated to the graded quasi-coherent \(\mathcal{O}_Z\)-algebra \(g^*\mathcal{A}\). Then there is a canonical isomorphism \(F' = Z \times_X F\), sending a quadruple \((d, f : T \to Z, \mathcal{L}, \psi)\) for \(F'\) to \((d, g \circ f, \mathcal{L}, \psi)\) (details omitted, see proof of Constructions, Lemma 01NT). By Constructions, Lemmas 01NW, 01NY, and 01NZ and Definition 01O0 we see that \(F'\) is representable by \(\underline{\text{Proj}}_Z(g^*\mathcal{A})\).

The lemma above tells us the following definition makes sense.

Definition

Let \(S\) be a scheme. Let \(X\) be an algebraic space over \(S\). Let \(\mathcal{A}\) be a quasi-coherent sheaf of graded \(\mathcal{O}_X\)-algebras. The relative homogeneous spectrum of \(\mathcal{A}\) over \(X\), or the homogeneous spectrum of \(\mathcal{A}\) over \(X\), or the relative Proj of \(\mathcal{A}\) over \(X\) is the algebraic space \(F\) over \(X\) of Lemma 084B. We denote it \(\pi : \underline{\text{Proj}}_X(\mathcal{A}) \to X\).

In particular the structure morphism of the relative Proj is representable by construction. We can also think about the relative Proj via glueing. Let \(\varphi : U \to X\) be a surjective étale morphism, where \(U\) is a scheme. Set \(R = U \times_X U\) with projection morphisms \(s, t : R \to U\). By Lemma 084B there exists a canonical isomorphism \[\gamma : \underline{\text{Proj}}_U(\varphi^*\mathcal{A}) \longrightarrow \underline{\text{Proj}}_X(\mathcal{A}) \times_X U\] over \(U\). Let \(\alpha : t^*\varphi^*\mathcal{A} \to s^*\varphi^*\mathcal{A}\) be the canonical isomorphism of Properties of Spaces, Proposition 03M3. Then the diagram \[\xymatrix{ & \underline{\text{Proj}}_U(\varphi^*\mathcal{A}) \times_{U, s} R \ar@{=}[r] & \underline{\text{Proj}}_R(s^*\varphi^*\mathcal{A}) \ar[dd]_{\text{induced by }\alpha} \\ \underline{\text{Proj}}_X(\mathcal{A}) \times_X R \ar[ru]_{s^*\gamma} \ar[rd]^{t^*\gamma} \\ & \underline{\text{Proj}}_U(\varphi^*\mathcal{A}) \times_{U, t} R \ar@{=}[r] & \underline{\text{Proj}}_R(t^*\varphi^*\mathcal{A}) }\] is commutative (the equal signs come from Constructions, Lemma 01O3). Thus, if we denote \(\mathcal{A}_U\), \(\mathcal{A}_R\) the pullback of \(\mathcal{A}\) to \(U\), \(R\), then \(P = \underline{\text{Proj}}_X(\mathcal{A})\) has an étale covering by the scheme \(P_U = \underline{\text{Proj}}_U(\mathcal{A}_U)\) and \(P_U \times_P P_U\) is equal to \(P_R = \underline{\text{Proj}}_R(\mathcal{A}_R)\). Using these remarks we can argue in the usual fashion using étale localization to transfer results on the relative proj from the case of schemes to the case of algebraic spaces.

Lemma

In Situation 0849. The relative Proj comes equipped with a quasi-coherent sheaf of \(\mathbf{Z}\)-graded algebras \(\bigoplus_{n \in \mathbf{Z}} \mathcal{O}_{\underline{\text{Proj}}_X(\mathcal{A})}(n)\) and a canonical homomorphism of graded algebras \[\psi : \pi^*\mathcal{A} \longrightarrow \bigoplus\nolimits_{n \geq 0} \mathcal{O}_{\underline{\text{Proj}}_X(\mathcal{A})}(n)\] whose base change to any scheme over \(X\) agrees with Constructions, Lemma 01NR.

Proof

As in the discussion following Definition 084C choose a scheme \(U\) and a surjective étale morphism \(U \to X\), set \(R = U \times_X U\) with projections \(s, t : R \to U\), \(\mathcal{A}_U = \mathcal{A}|_U\), \(\mathcal{A}_R = \mathcal{A}|_R\), and \(\pi : P = \underline{\text{Proj}}_X(\mathcal{A}) \to X\), \(\pi_U : P_U = \underline{\text{Proj}}_U(\mathcal{A}_U)\) and \(\pi_R : P_R = \underline{\text{Proj}}_U(\mathcal{A}_R)\). By the Constructions, Lemma 01NR we have a quasi-coherent sheaf of \(\mathbf{Z}\)-graded \(\mathcal{O}_{P_U}\)-algebras \(\bigoplus_{n \in \mathbf{Z}} \mathcal{O}_{P_U}(n)\) and a canonical map \(\psi_U : \pi_U^*\mathcal{A}_U \to \bigoplus_{n \geq 0} \mathcal{O}_{P_U}(n)\) and similarly for \(P_R\). By Constructions, Lemma 01O3 the pullback of \(\mathcal{O}_{P_U}(n)\) and \(\psi_U\) by either projection \(P_R \to P_U\) is equal to \(\mathcal{O}_{P_R}(n)\) and \(\psi_R\). By Properties of Spaces, Proposition 03M3 we obtain \(\mathcal{O}_{P}(n)\) and \(\psi\). We omit the verification of compatibility with pullback to arbitrary schemes over \(X\).

Having constructed the relative Proj we turn to some basic properties.

Lemma

Let \(S\) be a scheme. Let \(g : X' \to X\) be a morphism of algebraic spaces over \(S\) and let \(\mathcal{A}\) be a quasi-coherent sheaf of graded \(\mathcal{O}_X\)-algebras. Then there is a canonical isomorphism \[r : \underline{\text{Proj}}_{X'}(g^*\mathcal{A}) \longrightarrow X' \times_X \underline{\text{Proj}}_X(\mathcal{A})\] as well as a corresponding isomorphism \[\theta : r^*\text{pr}_2^*\left(\bigoplus\nolimits_{d \in \mathbf{Z}} \mathcal{O}_{\underline{\text{Proj}}_X(\mathcal{A})}(d)\right) \longrightarrow \bigoplus\nolimits_{d \in \mathbf{Z}} \mathcal{O}_{\underline{\text{Proj}}_{X'}(g^*\mathcal{A})}(d)\] of \(\mathbf{Z}\)-graded \(\mathcal{O}_{\underline{\text{Proj}}_{X'}(g^*\mathcal{A})}\)-algebras.

Proof

Let \(F\) be the functor (084A) and let \(F'\) be the corresponding functor defined using \(g^*\mathcal{A}\) on \(X'\). We claim there is a canonical isomorphism \(r : F' \to X' \times_X F\) of functors (and of course \(r\) is the isomorphism of the lemma). It suffices to construct the bijection \(r : F'(T) \to X'(T) \times_{X(T)} F(T)\) for quasi-compact schemes \(T\) over \(S\). First, if \(\xi = (d', f', \mathcal{L}', \psi')\) is a quadruple over \(T\) for \(F'\), then we can set \(r(\xi) = (f', (d', g \circ f', \mathcal{L}', \psi'))\). This makes sense as \((g \circ f')^*\mathcal{A}^{(d)} = (f')^*(g^*\mathcal{A})^{(d)}\). The inverse map sends the pair \((f', (d, f, \mathcal{L}, \psi))\) to the quadruple \((d, f', \mathcal{L}, \psi)\). We omit the proof of the final assertion (hint: reduce to the case of schemes by étale localization and apply Constructions, Lemma 01O3).

Lemma

In Situation 0849 the morphism \(\pi : \underline{\text{Proj}}_X(\mathcal{A}) \to X\) is separated.

Proof

By Morphisms of Spaces, Lemma 03KM and the construction of the relative Proj this follows from the case of schemes which is Constructions, Lemma 01O2.

Lemma

In Situation 0849. If one of the following holds

1. \(\mathcal{A}\) is of finite type as a sheaf of \(\mathcal{A}_0\)-algebras,

2. \(\mathcal{A}\) is generated by \(\mathcal{A}_1\) as an \(\mathcal{A}_0\)-algebra and \(\mathcal{A}_1\) is a finite type \(\mathcal{A}_0\)-module,

3. there exists a finite type quasi-coherent \(\mathcal{A}_0\)-submodule \(\mathcal{F} \subset \mathcal{A}_{+}\) such that \(\mathcal{A}_{+}/\mathcal{F}\mathcal{A}\) is a locally nilpotent sheaf of ideals of \(\mathcal{A}/\mathcal{F}\mathcal{A}\),

then \(\pi : \underline{\text{Proj}}_X(\mathcal{A}) \to X\) is quasi-compact.

Proof

By Morphisms of Spaces, Lemma 03KG and the construction of the relative Proj this follows from the case of schemes which is Divisors, Lemma 07ZX.

Lemma

In Situation 0849. If \(\mathcal{A}\) is of finite type as a sheaf of \(\mathcal{O}_X\)-algebras, then \(\pi : \underline{\text{Proj}}_X(\mathcal{A}) \to X\) is of finite type.

Proof

By Morphisms of Spaces, Lemma 040Y and the construction of the relative Proj this follows from the case of schemes which is Divisors, Lemma 07ZY.

Lemma

In Situation 0849. If \(\mathcal{O}_X \to \mathcal{A}_0\) is an integral algebra map³ and \(\mathcal{A}\) is of finite type as an \(\mathcal{A}_0\)-algebra, then \(\pi : \underline{\text{Proj}}_X(\mathcal{A}) \to X\) is universally closed.

Proof

By Morphisms of Spaces, Lemma 03IT and the construction of the relative Proj this follows from the case of schemes which is Divisors, Lemma 07ZZ.

Lemma

In Situation 0849. The following conditions are equivalent

1. \(\mathcal{A}_0\) is a finite type \(\mathcal{O}_X\)-module and \(\mathcal{A}\) is of finite type as an \(\mathcal{A}_0\)-algebra,

2. \(\mathcal{A}_0\) is a finite type \(\mathcal{O}_X\)-module and \(\mathcal{A}\) is of finite type as an \(\mathcal{O}_X\)-algebra.

If these conditions hold, then \(\pi : \underline{\text{Proj}}_X(\mathcal{A}) \to X\) is proper.

Proof

By Morphisms of Spaces, Lemma 083R and the construction of the relative Proj this follows from the case of schemes which is Divisors, Lemma 07ZZ.

Lemma

Let \(S\) be a scheme. Let \(X\) be an algebraic space over \(S\). Let \(\mathcal{A}\) be a quasi-coherent sheaf of graded \(\mathcal{O}_X\)-modules generated as an \(\mathcal{A}_0\)-algebra by \(\mathcal{A}_1\). With \(P = \underline{\text{Proj}}_X(\mathcal{A})\) we have

1. \(P\) represents the functor \(F_1\) which associates to \(T\) over \(S\) the set of isomorphism classes of triples \((f, \mathcal{L}, \psi)\), where \(f : T \to X\) is a morphism over \(S\), \(\mathcal{L}\) is an invertible \(\mathcal{O}_T\)-module, and \(\psi : f^*\mathcal{A} \to \bigoplus_{n \geq 0} \mathcal{L}^{\otimes n}\) is a map of graded \(\mathcal{O}_T\)-algebras inducing a surjection \(f^*\mathcal{A}_1 \to \mathcal{L}\),

2. the canonical map \(\pi^*\mathcal{A}_1 \to \mathcal{O}_P(1)\) is surjective, and

3. each \(\mathcal{O}_P(n)\) is invertible and the multiplication maps induce isomorphisms \(\mathcal{O}_P(n) \otimes_{\mathcal{O}_P} \mathcal{O}_P(m) = \mathcal{O}_P(n + m)\).

Proof

Omitted. See Constructions, Lemma 01O4 for the case of schemes.

Functoriality of relative proj

This section is the analogue of Constructions, Section 07ZF.

Lemma

Let \(S\) be a scheme. Let \(X\) be an algebraic space over \(S\). Let \(\psi : \mathcal{A} \to \mathcal{B}\) be a map of quasi-coherent graded \(\mathcal{O}_X\)-algebras. Set \(P = \underline{\text{Proj}}_X(\mathcal{A}) \to X\) and \(Q = \underline{\text{Proj}}_X(\mathcal{B}) \to X\). There is a canonical open subspace \(U(\psi) \subset Q\) and a canonical morphism of algebraic spaces \[r_\psi : U(\psi) \longrightarrow P\] over \(X\) and a map of \(\mathbf{Z}\)-graded \(\mathcal{O}_{U(\psi)}\)-algebras \[\theta = \theta_\psi : r_\psi^*\left( \bigoplus\nolimits_{d \in \mathbf{Z}} \mathcal{O}_P(d) \right) \longrightarrow \bigoplus\nolimits_{d \in \mathbf{Z}} \mathcal{O}_{U(\psi)}(d).\] The triple \((U(\psi), r_\psi, \theta)\) is characterized by the property that for any scheme \(W\) étale over \(X\) the triple \[(U(\psi) \times_X W,\quad r_\psi|_{U(\psi) \times_X W} : U(\psi) \times_X W \to P \times_X W,\quad \theta|_{U(\psi) \times_X W})\] is equal to the triple associated to \(\psi : \mathcal{A}|_W \to \mathcal{B}|_W\) of Constructions, Lemma 07ZG.

Proof

This lemma follows from étale localization and the case of schemes, see discussion following Definition 084C. Details omitted.

Lemma

Let \(S\) be a scheme. Let \(X\) be an algebraic space over \(S\). Let \(\mathcal{A}\), \(\mathcal{B}\), and \(\mathcal{C}\) be quasi-coherent graded \(\mathcal{O}_X\)-algebras. Set \(P = \underline{\text{Proj}}_X(\mathcal{A})\), \(Q = \underline{\text{Proj}}_X(\mathcal{B})\) and \(R = \underline{\text{Proj}}_X(\mathcal{C})\). Let \(\varphi : \mathcal{A} \to \mathcal{B}\), \(\psi : \mathcal{B} \to \mathcal{C}\) be graded \(\mathcal{O}_X\)-algebra maps. Then we have \[U(\psi \circ \varphi) = r_\varphi^{-1}(U(\psi)) \quad \text{and} \quad r_{\psi \circ \varphi} = r_\varphi \circ r_\psi|_{U(\psi \circ \varphi)}.\] In addition we have \[\theta_\psi \circ r_\psi^*\theta_\varphi = \theta_{\psi \circ \varphi}\] with obvious notation.

Proof

Omitted.

Lemma

With hypotheses and notation as in Lemma 085F above. Assume \(\mathcal{A}_d \to \mathcal{B}_d\) is surjective for \(d \gg 0\). Then

1. \(U(\psi) = Q\),

2. \(r_\psi : Q \to R\) is a closed immersion, and

3. the maps \(\theta : r_\psi^*\mathcal{O}_P(n) \to \mathcal{O}_Q(n)\) are surjective but not isomorphisms in general (even if \(\mathcal{A} \to \mathcal{B}\) is surjective).

Proof

Follows from the case of schemes (Constructions, Lemma 07ZI) by étale localization.

Lemma

With hypotheses and notation as in Lemma 085F above. Assume \(\mathcal{A}_d \to \mathcal{B}_d\) is an isomorphism for all \(d \gg 0\). Then

1. \(U(\psi) = Q\),

2. \(r_\psi : Q \to P\) is an isomorphism, and

3. the maps \(\theta : r_\psi^*\mathcal{O}_P(n) \to \mathcal{O}_Q(n)\) are isomorphisms.

Proof

Follows from the case of schemes (Constructions, Lemma 07ZJ) by étale localization.

Lemma

With hypotheses and notation as in Lemma 085F above. Assume \(\mathcal{A}_d \to \mathcal{B}_d\) is surjective for \(d \gg 0\) and that \(\mathcal{A}\) is generated by \(\mathcal{A}_1\) over \(\mathcal{A}_0\). Then

1. \(U(\psi) = Q\),

2. \(r_\psi : Q \to P\) is a closed immersion, and

3. the maps \(\theta : r_\psi^*\mathcal{O}_P(n) \to \mathcal{O}_Q(n)\) are isomorphisms.

Proof

Follows from the case of schemes (Constructions, Lemma 07ZK) by étale localization.

Invertible sheaves and morphisms into relative Proj

It seems that we may need the following lemma somewhere. The situation is the following:

1. Let \(S\) be a scheme and \(Y\) an algebraic space over \(S\).

2. Let \(\mathcal{A}\) be a quasi-coherent graded \(\mathcal{O}_Y\)-algebra.

3. Denote \(\pi : \underline{\text{Proj}}_Y(\mathcal{A}) \to Y\) the relative Proj of \(\mathcal{A}\) over \(Y\).

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

5. Let \(\mathcal{L}\) be an invertible \(\mathcal{O}_X\)-module.

6. Let \(\psi : f^*\mathcal{A} \to \bigoplus_{d \geq 0} \mathcal{L}^{\otimes d}\) be a homomorphism of graded \(\mathcal{O}_X\)-algebras.

Given this data let \(U(\psi) \subset X\) be the open subspace with \[|U(\psi)| = \bigcup\nolimits_{d \geq 1} \{\text{locus where }f^*\mathcal{A}_d \to \mathcal{L}^{\otimes d} \text{ is surjective}\}\] Formation of \(U(\psi) \subset X\) commutes with pullback by any morphism \(X' \to X\).

Lemma

With assumptions and notation as above. The morphism \(\psi\) induces a canonical morphism of algebraic spaces over \(Y\) \[r_{\mathcal{L}, \psi} : U(\psi) \longrightarrow \underline{\text{Proj}}_Y(\mathcal{A})\] together with a map of graded \(\mathcal{O}_{U(\psi)}\)-algebras \[\theta : r_{\mathcal{L}, \psi}^*\left( \bigoplus\nolimits_{d \geq 0} \mathcal{O}_{\underline{\text{Proj}}_Y(\mathcal{A})}(d) \right) \longrightarrow \bigoplus\nolimits_{d \geq 0} \mathcal{L}^{\otimes d}|_{U(\psi)}\] characterized by the following properties:

1. For \(V \to Y\) étale and \(d \geq 0\) the diagram \[\xymatrix{ \mathcal{A}_d(V) \ar[d]_{\psi} \ar[r]_{\psi} & \Gamma(V \times_Y X, \mathcal{L}^{\otimes d}) \ar[d]^{restrict} \\ \Gamma(V \times_Y \underline{\text{Proj}}_Y(\mathcal{A}), \mathcal{O}_{\underline{\text{Proj}}_Y(\mathcal{A})}(d)) \ar[r]^-\theta & \Gamma(V \times_Y U(\psi), \mathcal{L}^{\otimes d}) }\] is commutative.

2. For any \(d \geq 1\) and any morphism \(W \to X\) where \(W\) is a scheme such that \(\psi|_W : f^*\mathcal{A}_d|_W \to \mathcal{L}^{\otimes d}|_W\) is surjective we have (a) \(W \to X\) factors through \(U(\psi)\) and (b) composition of \(W \to U(\psi)\) with \(r_{\mathcal{L}, \psi}\) agrees with the morphism \(W \to \underline{\text{Proj}}_Y(\mathcal{A})\) which exists by the construction of \(\underline{\text{Proj}}_Y(\mathcal{A})\), see Definition 084C.

3. Consider a commutative diagram \[\xymatrix{ X' \ar[r]_{g'} \ar[d]_{f'} & X \ar[d]^f \\ Y' \ar[r]^g & Y }\] where \(X'\) and \(Y'\) are schemes, set \(\mathcal{A}' = g^*\mathcal{A}\) and \(\mathcal{L}' = (g')^*\mathcal{L}\) and denote \(\psi' : (f')^*\mathcal{A} \to \bigoplus_{d \geq 0} (\mathcal{L}')^{\otimes d}\) the pullback of \(\psi\). Let \(U(\psi')\), \(r_{\psi', \mathcal{L}'}\), and \(\theta'\) be the open, morphism, and homomorphism constructed in Constructions, Lemma 0D2Z. Then \(U(\psi') = (g')^{-1}(U(\psi))\) and \(r_{\psi', \mathcal{L}'}\) agrees with the base change of \(r_{\psi, \mathcal{L}}\) via the isomorphism \(\underline{\text{Proj}}_{Y'}(\mathcal{A}') = Y' \times_Y \underline{\text{Proj}}_Y(\mathcal{A})\) of Lemma 085C. Moreover, \(\theta'\) is the pullback of \(\theta\).

Proof

Omitted. Hints: First we observe that for a quasi-compact scheme \(W\) over \(X\) the following are equivalent

1. \(W \to X\) factors through \(U(\psi)\), and

2. there exists a \(d\) such that \(\psi|_W : f^*\mathcal{A}_d|_W \to \mathcal{L}^{\otimes d}|_W\) is surjective.

This gives a description of \(U(\psi)\) as a subfunctor of \(X\) on our base category \((\Sch/S)_{fppf}\). For such a \(W\) and \(d\) we consider the quadruple \((d, W \to Y, \mathcal{L}|_W, \psi^{(d)}|_W)\). By definition of \(\underline{\text{Proj}}_Y(\mathcal{A})\) we obtain a morphism \(W \to \underline{\text{Proj}}_Y(\mathcal{A})\). By our notion of equivalence of quadruples one sees that this morphism is independent of the choice of \(d\). This clearly defines a transformation of functors \(r_{\psi, \mathcal{L}} : U(\psi) \to \underline{\text{Proj}}_Y(\mathcal{A})\), i.e., a morphism of algebraic spaces. By construction this morphism satisfies (2). Since the morphism constructed in Constructions, Lemma 01O9 satisfies the same property, we see that (3) is true.

To construct \(\theta\) and check the compatibility (1) of the lemma, work étale locally on \(Y\) and \(X\), arguing as in the discussion following Definition 084C.

Relatively ample sheaves

This section is the analogue of Morphisms, Section 01VG for algebraic spaces. Our definition of a relatively ample invertible sheaf is as follows.

Definition

Let \(S\) be a scheme. Let \(f : X \to Y\) be a morphism of algebraic spaces over \(S\). Let \(\mathcal{L}\) be an invertible \(\mathcal{O}_X\)-module. We say \(\mathcal{L}\) is relatively ample, or \(f\)-relatively ample, or ample on \(X/Y\), or \(f\)-ample if \(f : X \to Y\) is representable and for every morphism \(Z \to Y\) where \(Z\) is a scheme, the pullback \(\mathcal{L}_Z\) of \(\mathcal{L}\) to \(X_Z = Z \times_Y X\) is ample on \(X_Z/Z\) as in Morphisms, Definition 01VH.

We will almost always reduce questions about relatively ample invertible sheaves to the case of schemes. Thus in this section we have mainly sanity checks.

Lemma

Let \(S\) be a scheme. Let \(f : X \to Y\) be a morphism of algebraic spaces over \(S\). Let \(\mathcal{L}\) be an invertible \(\mathcal{O}_X\)-module. Assume \(Y\) is a scheme. The following are equivalent

1. \(\mathcal{L}\) is ample on \(X/Y\) in the sense of Definition 0D31, and

2. \(X\) is a scheme and \(\mathcal{L}\) is ample on \(X/Y\) in the sense of Morphisms, Definition 01VH.

Proof

This follows from the definitions and Morphisms, Lemma 0893 (which says that being relatively ample for schemes is preserved under base change).

Lemma

Let \(S\) be a scheme. Let \(f : X \to Y\) be a morphism of algebraic spaces over \(S\). Let \(\mathcal{L}\) be an invertible \(\mathcal{O}_X\)-module. Let \(Y' \to Y\) be a morphism of algebraic spaces over \(S\). Let \(f' : X' \to Y'\) be the base change of \(f\) and denote \(\mathcal{L}'\) the pullback of \(\mathcal{L}\) to \(X'\). If \(\mathcal{L}\) is \(f\)-ample, then \(\mathcal{L}'\) is \(f'\)-ample.

Proof

This follows immediately from the definition! (Hint: transitivity of base change.)

Lemma

Let \(S\) be a scheme. Let \(f : X \to Y\) be a morphism of algebraic spaces over \(S\). If there exists an \(f\)-ample invertible sheaf, then \(f\) is representable, quasi-compact, and separated.

Proof

This is clear from the definitions and Morphisms, Lemma 01VI. (If in doubt, take a look at the principle of Algebraic Spaces, Lemma 02YO.)

Lemma

Let \(V \to U\) be a surjective étale morphism of affine schemes. Let \(X\) be an algebraic space over \(U\). Let \(\mathcal{L}\) be an invertible \(\mathcal{O}_X\)-module. Let \(Y = V \times_U X\) and let \(\mathcal{N}\) be the pullback of \(\mathcal{L}\) to \(Y\). The following are equivalent

1. \(\mathcal{L}\) is ample on \(X/U\), and

2. \(\mathcal{N}\) is ample on \(Y/V\).

Proof

The implication (1) \(\Rightarrow\) (2) follows from Lemma 0D33. Assume (2). This implies that \(Y \to V\) is quasi-compact and separated (Lemma 0D34) and \(Y\) is a scheme. It follows that the morphism \(f : X \to U\) is quasi-compact and separated (Morphisms of Spaces, Lemmas 03KG and 03KM). Set \(\mathcal{A} = \bigoplus_{d \geq 0} f_*\mathcal{L}^{\otimes d}\). This is a quasi-coherent sheaf of graded \(\mathcal{O}_U\)-algebras (Morphisms of Spaces, Lemma 03M9). By adjunction we have a map \(\psi : f^*\mathcal{A} \to \bigoplus_{d \geq 0} \mathcal{L}^{\otimes d}\). Applying Lemma 0D2Z we obtain an open subspace \(U(\psi) \subset X\) and a morphism \[r_{\mathcal{L}, \psi} : U(\psi) \to \underline{\text{Proj}}_U(\mathcal{A})\] Since \(h : V \to U\) is étale we have \(\mathcal{A}|_V = (Y \to V)_*(\bigoplus_{d \geq 0} \mathcal{N}^{\otimes d})\), see Properties of Spaces, Lemma 03LX. It follows that the pullback \(\psi'\) of \(\psi\) to \(Y\) is the adjunction map for the situation \((Y \to V, \mathcal{N})\) as in Morphisms, Lemma 01VJ part (5). Since \(\mathcal{N}\) is ample on \(Y/V\) we conclude from the lemma just cited that \(U(\psi') = Y\) and that \(r_{\mathcal{N}, \psi'}\) is an open immersion. Since Lemma 0D2Z tells us that the formation of \(r_{\mathcal{L}, \psi}\) commutes with base change, we conclude that \(U(\psi) = X\) and that we have a commutative diagram \[\xymatrix{ Y \ar[r]_-{r'} \ar[d] & \underline{\text{Proj}}_V(\mathcal{A}|_V) \ar[d] \ar[r] & V \ar[d] \\ X \ar[r]^-r & \underline{\text{Proj}}_U(\mathcal{A}) \ar[r] & U }\] whose squares are fibre products. We conclude that \(r\) is an open immersion by Morphisms of Spaces, Lemma 03M4. Thus \(X\) is a scheme. Then we can apply Morphisms, Lemma 01VJ part (5) to conclude that \(\mathcal{L}\) is ample on \(X/U\).

Lemma

Let \(S\) be a scheme. Let \(f : X \to Y\) be a morphism of algebraic spaces over \(S\). Let \(\mathcal{L}\) be an invertible \(\mathcal{O}_X\)-module. The following are equivalent

1. \(\mathcal{L}\) is ample on \(X/Y\),

2. for every scheme \(Z\) and every morphism \(Z \to Y\) the algebraic space \(X_Z = Z \times_Y X\) is a scheme and the pullback \(\mathcal{L}_Z\) is ample on \(X_Z/Z\),

3. for every affine scheme \(Z\) and every morphism \(Z \to Y\) the algebraic space \(X_Z = Z \times_Y X\) is a scheme and the pullback \(\mathcal{L}_Z\) is ample on \(X_Z/Z\),

4. there exists a scheme \(V\) and a surjective étale morphism \(V \to Y\) such that the algebraic space \(X_V = V \times_Y X\) is a scheme and the pullback \(\mathcal{L}_V\) is ample on \(X_V/V\).

Proof

Parts (1) and (2) are equivalent by definition. The implication (2) \(\Rightarrow\) (3) is immediate. If (3) holds and \(Z \to Y\) is as in (2), then we see that \(X_Z \to Z\) is affine locally on \(Z\) representable. Hence \(X_Z\) is a scheme for example by Properties of Spaces, Lemma 03JH. Then it follows that \(\mathcal{L}_Z\) is ample on \(X_Z/Z\) because it holds locally on \(Z\) and we can use Morphisms, Lemma 01VJ. Thus (1), (2), and (3) are equivalent. Clearly these conditions imply (4).

Assume (4). Let \(Z \to Y\) be a morphism with \(Z\) affine. Then \(U = V \times_Y Z \to Z\) is a surjective étale morphism such that the pullback of \(\mathcal{L}_Z\) by \(X_U \to X_Z\) is relatively ample on \(X_U/U\). Of course we may replace \(U\) by an affine open. It follows that \(\mathcal{L}_Z\) is ample on \(X_Z/Z\) by Lemma 0D35. Thus (4) \(\Rightarrow\) (3) and the proof is complete.

Lemma

Let \(S\) be a scheme. Let \(f : X \to Y\) be a morphism of algebraic spaces over \(S\). Then \(f\) is quasi-affine if and only if \(\mathcal{O}_X\) is \(f\)-relatively ample.

Proof

Follows from the case of schemes, see Morphisms, Lemma 0891.

Relative ampleness and cohomology

This section contains some results related to the results in Cohomology of Schemes, Sections 02OF and 01XO.

The following lemma is just an example of what we can do.

Lemma

Let \(R\) be a Noetherian ring. Let \(X\) be an algebraic space over \(R\) such that the structure morphism \(f : X \to \Spec(R)\) is proper. Let \(\mathcal{L}\) be an invertible \(\mathcal{O}_X\)-module. The following are equivalent

1. \(\mathcal{L}\) is ample on \(X/R\) (Definition 0D31),

2. for every coherent \(\mathcal{O}_X\)-module \(\mathcal{F}\) there exists an \(n_0 \geq 0\) such that \(H^p(X, \mathcal{F} \otimes \mathcal{L}^{\otimes n}) = 0\) for all \(n \geq n_0\) and \(p > 0\).

Proof

The implication (1) \(\Rightarrow\) (2) follows from Cohomology of Schemes, Lemma 0B5T because assumption (1) implies that \(X\) is a scheme. The implication (2) \(\Rightarrow\) (1) is Cohomology of Spaces, Lemma 0D2W.

Lemma

Let \(Y\) be a Noetherian scheme. Let \(X\) be an algebraic space over \(Y\) such that the structure morphism \(f : X \to Y\) is proper. Let \(\mathcal{L}\) be an invertible \(\mathcal{O}_X\)-module. Let \(\mathcal{F}\) be a coherent \(\mathcal{O}_X\)-module. Let \(y \in Y\) be a point such that \(X_y\) is a scheme and \(\mathcal{L}_y\) is ample on \(X_y\). Then there exists a \(d_0\) such that for all \(d \geq d_0\) we have \[R^pf_*(\mathcal{F} \otimes_{\mathcal{O}_X} \mathcal{L}^{\otimes d})_y = 0 \text{ for }p > 0\] and the map \[f_*(\mathcal{F} \otimes_{\mathcal{O}_X} \mathcal{L}^{\otimes d})_y \longrightarrow H^0(X_y, \mathcal{F}_y \otimes_{\mathcal{O}_{X_y}} \mathcal{L}_y^{\otimes d})\] is surjective.

Proof

Note that \(\mathcal{O}_{Y, y}\) is a Noetherian local ring. Consider the canonical morphism \(c : \Spec(\mathcal{O}_{Y, y}) \to Y\), see Schemes, Equation (02NA). This is a flat morphism as it identifies local rings. Denote momentarily \(f' : X' \to \Spec(\mathcal{O}_{Y, y})\) the base change of \(f\) to this local ring. We see that \(c^*R^pf_*\mathcal{F} = R^pf'_*\mathcal{F}'\) by Cohomology of Spaces, Lemma 073K. Moreover, the fibres \(X_y\) and \(X'_y\) are identified. Hence we may assume that \(Y = \Spec(A)\) is the spectrum of a Noetherian local ring \((A, \mathfrak m, \kappa)\) and \(y \in Y\) corresponds to \(\mathfrak m\). In this case \(R^pf_*(\mathcal{F} \otimes_{\mathcal{O}_X} \mathcal{L}^{\otimes d})_y = H^p(X, \mathcal{F} \otimes_{\mathcal{O}_X} \mathcal{L}^{\otimes d})\) for all \(p \geq 0\). Denote \(f_y : X_y \to \Spec(\kappa)\) the projection.

Let \(B = \text{Gr}_\mathfrak m(A) = \bigoplus_{n \geq 0} \mathfrak m^n/\mathfrak m^{n + 1}\). Consider the sheaf \(\mathcal{B} = f_y^*\widetilde{B}\) of quasi-coherent graded \(\mathcal{O}_{X_y}\)-algebras. We will use notation as in Cohomology of Spaces, Section 08AU with \(I\) replaced by \(\mathfrak m\). Since \(X_y\) is the closed subspace of \(X\) cut out by \(\mathfrak m\mathcal{O}_X\) we may think of \(\mathfrak m^n\mathcal{F}/\mathfrak m^{n + 1}\mathcal{F}\) as a coherent \(\mathcal{O}_{X_y}\)-module, see Cohomology of Spaces, Lemma 08AM. Then \(\bigoplus_{n \geq 0} \mathfrak m^n\mathcal{F}/\mathfrak m^{n + 1}\mathcal{F}\) is a quasi-coherent graded \(\mathcal{B}\)-module of finite type because it is generated in degree zero over \(\mathcal{B}\) abd because the degree zero part is \(\mathcal{F}_y = \mathcal{F}/\mathfrak m \mathcal{F}\) which is a coherent \(\mathcal{O}_{X_y}\)-module. Hence by Cohomology of Schemes, Lemma 0897 part (2) there exists a \(d_0\) such that \[H^p(X_y, \mathfrak m^n \mathcal{F}/ \mathfrak m^{n + 1}\mathcal{F} \otimes_{\mathcal{O}_{X_y}} \mathcal{L}_y^{\otimes d}) = 0\] for all \(p > 0\), \(d \geq d_0\), and \(n \geq 0\). By Cohomology of Spaces, Lemma 0D2U this is the same as the statement that \(H^p(X, \mathfrak m^n \mathcal{F}/ \mathfrak m^{n + 1}\mathcal{F} \otimes_{\mathcal{O}_X} \mathcal{L}^{\otimes d}) = 0\) for all \(p > 0\), \(d \geq d_0\), and \(n \geq 0\).

Consider the short exact sequences \[0 \to \mathfrak m^n\mathcal{F}/\mathfrak m^{n + 1} \mathcal{F} \to \mathcal{F}/\mathfrak m^{n + 1} \mathcal{F} \to \mathcal{F}/\mathfrak m^n \mathcal{F} \to 0\] of coherent \(\mathcal{O}_X\)-modules. Tensoring with \(\mathcal{L}^{\otimes d}\) is an exact functor and we obtain short exact sequences \[0 \to \mathfrak m^n\mathcal{F}/\mathfrak m^{n + 1} \mathcal{F} \otimes_{\mathcal{O}_X} \mathcal{L}^{\otimes d} \to \mathcal{F}/\mathfrak m^{n + 1} \mathcal{F} \otimes_{\mathcal{O}_X} \mathcal{L}^{\otimes d} \to \mathcal{F}/\mathfrak m^n \mathcal{F} \otimes_{\mathcal{O}_X} \mathcal{L}^{\otimes d} \to 0\] Using the long exact cohomology sequence and the vanishing above we conclude (using induction) that

1. \(H^p(X, \mathcal{F}/\mathfrak m^n \mathcal{F} \otimes_{\mathcal{O}_X} \mathcal{L}^{\otimes d}) = 0\) for all \(p > 0\), \(d \geq d_0\), and \(n \geq 0\), and

2. \(H^0(X, \mathcal{F}/\mathfrak m^n \mathcal{F} \otimes_{\mathcal{O}_X} \mathcal{L}^{\otimes d}) \to H^0(X_y, \mathcal{F}_y \otimes_{\mathcal{O}_{X_y}} \mathcal{L}_y^{\otimes d})\) is surjective for all \(d \geq d_0\) and \(n \geq 1\).

By the theorem on formal functions (Cohomology of Spaces, Theorem 08AZ) we find that the \(\mathfrak m\)-adic completion of \(H^p(X, \mathcal{F} \otimes_{\mathcal{O}_X} \mathcal{L}^{\otimes d})\) is zero for all \(d \geq d_0\) and \(p > 0\). Since \(H^p(X, \mathcal{F} \otimes_{\mathcal{O}_X} \mathcal{L}^{\otimes d})\) is a finite \(A\)-module by Cohomology of Spaces, Lemma 08AS it follows from Nakayama’s lemma (Algebra, Lemma 00DV) that \(H^p(X, \mathcal{F} \otimes_{\mathcal{O}_X} \mathcal{L}^{\otimes d})\) is zero for all \(d \geq d_0\) and \(p > 0\). For \(p = 0\) we deduce from Cohomology of Spaces, Lemma 08AY part (3) that \(H^0(X, \mathcal{F} \otimes_{\mathcal{O}_X} \mathcal{L}^{\otimes d}) \to H^0(X_y, \mathcal{F}_y \otimes_{\mathcal{O}_{X_y}} \mathcal{L}_y^{\otimes d})\) is surjective, which gives the final statement of the lemma.

Lemma

(For a more general version see Descent on Spaces, Lemma 0D3D). Let \(Y\) be a Noetherian scheme. Let \(X\) be an algebraic space over \(Y\) such that the structure morphism \(f : X \to Y\) is proper. Let \(\mathcal{L}\) be an invertible \(\mathcal{O}_X\)-module. Let \(y \in Y\) be a point such that \(X_y\) is a scheme and \(\mathcal{L}_y\) is ample on \(X_y\). Then there is an open neighbourhood \(V \subset Y\) of \(y\) such that \(\mathcal{L}|_{f^{-1}(V)}\) is ample on \(f^{-1}(V)/V\) (as in Definition 0D31).

Proof

Pick \(d_0\) as in Lemma 0D39 for \(\mathcal{F} = \mathcal{O}_X\). Pick \(d \geq d_0\) so that we can find \(r \geq 0\) and sections \(s_{y, 0}, \ldots, s_{y, r} \in H^0(X_y, \mathcal{L}_y^{\otimes d})\) which define a closed immersion \[\varphi_y = \varphi_{\mathcal{L}_y^{\otimes d}, (s_{y, 0}, \ldots, s_{y, r})} : X_y \to \mathbf{P}^r_{\kappa(y)}.\] This is possible by Morphisms, Lemma 01VT but we also use Morphisms, Lemma 01W6 to see that \(\varphi_y\) is a closed immersion and Constructions, Section 01ND for the description of morphisms into projective space in terms of invertible sheaves and sections. By our choice of \(d_0\), after replacing \(Y\) by an open neighbourhood of \(y\), we can choose \(s_0, \ldots, s_r \in H^0(X, \mathcal{L}^{\otimes d})\) mapping to \(s_{y, 0}, \ldots, s_{y, r}\). Let \(X_{s_i} \subset X\) be the open subspace where \(s_i\) is a generator of \(\mathcal{L}^{\otimes d}\). Since the \(s_{y, i}\) generate \(\mathcal{L}_y^{\otimes d}\) we see that \(|X_y| \subset U = \bigcup |X_{s_i}|\). Since \(X \to Y\) is closed, we see that there is an open neighbourhood \(y \in V \subset Y\) such that \(|f|^{-1}(V) \subset U\). After replacing \(Y\) by \(V\) we may assume that the \(s_i\) generate \(\mathcal{L}^{\otimes d}\). Thus we obtain a morphism \[\varphi = \varphi_{\mathcal{L}^{\otimes d}, (s_0, \ldots, s_r)} : X \longrightarrow \mathbf{P}^r_Y\] with \(\mathcal{L}^{\otimes d} \cong \varphi^*\mathcal{O}_{\mathbf{P}^r_Y}(1)\) whose base change to \(y\) gives \(\varphi_y\) (strictly speaking we need to write out a proof that the construction of morphisms into projective space given in Constructions, Section 01ND also works to describe morphisms of algebraic spaces into projective space; we omit the details).

We will finish the proof by a sleight of hand; the “correct” proof proceeds by directly showing that \(\varphi\) is a closed immersion after base changing to an open neighbourhood of \(y\). Namely, by Cohomology of Spaces, Lemma 0A4W we see that \(\varphi\) is a finite over an open neighbourhood of the fibre \(\mathbf{P}^r_{\kappa(y)}\) of \(\mathbf{P}^r_Y \to Y\) above \(y\). Using that \(\mathbf{P}^r_Y \to Y\) is closed, after shrinking \(Y\) we may assume that \(\varphi\) is finite. In particular \(X\) is a scheme. Then \(\mathcal{L}^{\otimes d} \cong \varphi^*\mathcal{O}_{\mathbf{P}^r_Y}(1)\) is ample by the very general Morphisms, Lemma 0892.

Closed subspaces of relative proj

Some auxiliary lemmas about closed subspaces of relative proj. This section is the analogue of Divisors, Section 084M.

Lemma

Let \(S\) be a scheme. Let \(X\) be an algebraic space over \(S\). Let \(\mathcal{A}\) be a quasi-coherent graded \(\mathcal{O}_X\)-algebra. Let \(\pi : P = \underline{\text{Proj}}_X(\mathcal{A}) \to X\) be the relative Proj of \(\mathcal{A}\). Let \(i : Z \to P\) be a closed subspace. Denote \(\mathcal{I} \subset \mathcal{A}\) the kernel of the canonical map \[\mathcal{A} \longrightarrow \bigoplus\nolimits_{d \geq 0} \pi_*\left((i_*\mathcal{O}_Z)(d)\right)\] If \(\pi\) is quasi-compact, then there is an isomorphism \(Z = \underline{\text{Proj}}_X(\mathcal{A}/\mathcal{I})\).

Proof

The morphism \(\pi\) is separated by Lemma 084E. As \(\pi\) is quasi-compact, \(\pi_*\) transforms quasi-coherent modules into quasi-coherent modules, see Morphisms of Spaces, Lemma 03M9. Hence \(\mathcal{I}\) is a quasi-coherent \(\mathcal{O}_X\)-module. In particular, \(\mathcal{B} = \mathcal{A}/\mathcal{I}\) is a quasi-coherent graded \(\mathcal{O}_X\)-algebra. The functoriality morphism \(Z' = \underline{\text{Proj}}_X(\mathcal{B}) \to \underline{\text{Proj}}_X(\mathcal{A})\) is everywhere defined and a closed immersion, see Lemma 085H. Hence it suffices to prove \(Z = Z'\) as closed subspaces of \(P\).

Having said this, the question is étale local on the base and we reduce to the case of schemes (Divisors, Lemma 0801) by étale localization.

In case the closed subspace is locally cut out by finitely many equations we can define it by a finite type ideal sheaf of \(\mathcal{A}\).

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 graded \(\mathcal{O}_X\)-algebra. Let \(\pi : P = \underline{\text{Proj}}_X(\mathcal{A}) \to X\) be the relative Proj of \(\mathcal{A}\). Let \(i : Z \to P\) be a closed subscheme. If \(\pi\) is quasi-compact and \(i\) of finite presentation, then there exists a \(d > 0\) and a quasi-coherent finite type \(\mathcal{O}_X\)-submodule \(\mathcal{F} \subset \mathcal{A}_d\) such that \(Z = \underline{\text{Proj}}_X(\mathcal{A}/\mathcal{F}\mathcal{A})\).

Proof

The reader can redo the arguments used in the case of schemes. However, we will show the lemma follows from the case of schemes by a trick. Let \(\mathcal{I} \subset \mathcal{A}\) be the quasi-coherent graded ideal cutting out \(Z\) of Lemma 085L. Choose an affine scheme \(U\) and a surjective étale morphism \(U \to X\), see Properties of Spaces, Lemma 03H6. By the case of schemes (Divisors, Lemma 0802) there exists a \(d > 0\) and a quasi-coherent finite type \(\mathcal{O}_U\)-submodule \(\mathcal{F}' \subset \mathcal{I}_d|_U \subset \mathcal{A}_d|_U\) such that \(Z \times_X U\) is equal to \(\underline{\text{Proj}}_U(\mathcal{A}|_U/\mathcal{F}'\mathcal{A}|_U)\). By Limits of Spaces, Lemma 0829 we can find a finite type quasi-coherent submodule \(\mathcal{F} \subset \mathcal{I}_d\) such that \(\mathcal{F}' \subset \mathcal{F}|_U\). Let \(Z' = \underline{\text{Proj}}_X(\mathcal{A}/\mathcal{F}\mathcal{A})\). Then \(Z' \to P\) is a closed immersion (Lemma 085J) and \(Z \subset Z'\) as \(\mathcal{F}\mathcal{A} \subset \mathcal{I}\). On the other hand, \(Z' \times_X U \subset Z \times_X U\) by our choice of \(\mathcal{F}\). Thus \(Z = Z'\) as desired.

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 graded \(\mathcal{O}_X\)-algebra. Let \(\pi : P = \underline{\text{Proj}}_X(\mathcal{A}) \to X\) be the relative Proj of \(\mathcal{A}\). Let \(i : Z \to X\) be a closed subspace. Let \(U \subset X\) be an open. Assume that

1. \(\pi\) is quasi-compact,

2. \(i\) of finite presentation,

3. \(|U| \cap |\pi|(|i|(|Z|)) = \emptyset\),

4. \(U\) is quasi-compact,

5. \(\mathcal{A}_n\) is a finite type \(\mathcal{O}_X\)-module for all \(n\).

Then there exists a \(d > 0\) and a quasi-coherent finite type \(\mathcal{O}_X\)-submodule \(\mathcal{F} \subset \mathcal{A}_d\) with (a) \(Z = \underline{\text{Proj}}_X(\mathcal{A}/\mathcal{F}\mathcal{A})\) and (b) the support of \(\mathcal{A}_d/\mathcal{F}\) is disjoint from \(U\).

Proof

We use the same trick as in the proof of Lemma 085M to reduce to the case of schemes. Let \(\mathcal{I} \subset \mathcal{A}\) be the quasi-coherent graded ideal cutting out \(Z\) of Lemma 085L. Choose an affine scheme \(W\) and a surjective étale morphism \(W \to X\), see Properties of Spaces, Lemma 03H6. By the case of schemes (Divisors, Lemma 0803) there exists a \(d > 0\) and a quasi-coherent finite type \(\mathcal{O}_W\)-submodule \(\mathcal{F}' \subset \mathcal{I}_d|_W \subset \mathcal{A}_d|_W\) such that (a) \(Z \times_X W\) is equal to \(\underline{\text{Proj}}_W(\mathcal{A}|_W/\mathcal{F}'\mathcal{A}|_W)\) and (b) the support of \(\mathcal{A}_d|_W/\mathcal{F}'\) is disjoint from \(U \times_X W\). By Limits of Spaces, Lemma 0829 we can find a finite type quasi-coherent submodule \(\mathcal{F} \subset \mathcal{I}_d\) such that \(\mathcal{F}' \subset \mathcal{F}|_W\). Let \(Z' = \underline{\text{Proj}}_X(\mathcal{A}/\mathcal{F}\mathcal{A})\). Then \(Z' \to P\) is a closed immersion (Lemma 085J) and \(Z \subset Z'\) as \(\mathcal{F}\mathcal{A} \subset \mathcal{I}\). On the other hand, \(Z' \times_X W \subset Z \times_X W\) by our choice of \(\mathcal{F}\). Thus \(Z = Z'\). Finally, we see that \(\mathcal{A}_d/\mathcal{F}\) is supported on \(X \setminus U\) as \(\mathcal{A}_d|_W/\mathcal{F}|_W\) is a quotient of \(\mathcal{A}_d|_W/\mathcal{F}'\) which is supported on \(W \setminus U \times_X W\). Thus the lemma follows.

Lemma

Let \(S\) be a scheme and let \(X\) be an algebraic space over \(S\). Let \(\mathcal{E}\) be a quasi-coherent \(\mathcal{O}_X\)-module. There is a bijection \[\left\{ \begin{matrix} \text{sections }\sigma\text{ of the } \\ \text{morphism } \mathbf{P}(\mathcal{E}) \to X \end{matrix} \right\} \leftrightarrow \left\{ \begin{matrix} \text{surjections }\mathcal{E} \to \mathcal{L}\text{ where} \\ \mathcal{L}\text{ is an invertible }\mathcal{O}_X\text{-module} \end{matrix} \right\}\] In this case \(\sigma\) is a closed immersion and there is a canonical isomorphism \[\Ker(\mathcal{E} \to \mathcal{L}) \otimes_{\mathcal{O}_X} \mathcal{L}^{\otimes -1} \longrightarrow \mathcal{C}_{\sigma(X)/\mathbf{P}(\mathcal{E})}\] Both the bijection and isomorphism are compatible with base change.

Proof

Because the constructions are compatible with base change, it suffices to check the statement étale locally on \(X\). Thus we may assume \(X\) is a scheme and the result is Divisors, Lemma 0B3V.

Blowing up

Blowing up is an important tool in algebraic geometry.

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, and let \(Z \subset X\) be the closed subspace corresponding to \(\mathcal{I}\) (Morphisms of Spaces, Lemma 03MB). The blowing up of \(X\) along \(Z\), or the blowing up of \(X\) in the ideal sheaf \(\mathcal{I}\) is the morphism \[b : \underline{\text{Proj}}_X \left(\bigoplus\nolimits_{n \geq 0} \mathcal{I}^n\right) \longrightarrow X\] The exceptional divisor of the blowup is the inverse image \(b^{-1}(Z)\). Sometimes \(Z\) is called the center of the blowup.

We will see later that the exceptional divisor is an effective Cartier divisor. Moreover, the blowing up is characterized as the “smallest” algebraic space over \(X\) such that the inverse image of \(Z\) is an effective Cartier divisor.

If \(b : X' \to X\) is the blowup of \(X\) in \(Z\), then we often denote \(\mathcal{O}_{X'}(n)\) the twists of the structure sheaf. Note that these are invertible \(\mathcal{O}_{X'}\)-modules and that \(\mathcal{O}_{X'}(n) = \mathcal{O}_{X'}(1)^{\otimes n}\) because \(X'\) is the relative Proj of a quasi-coherent graded \(\mathcal{O}_X\)-algebra which is generated in degree \(1\), see Lemma 085D.

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 \(U = \Spec(A)\) be an affine scheme étale over \(X\) and let \(I \subset A\) be the ideal corresponding to \(\mathcal{I}|_U\). If \(X' \to X\) is the blowup of \(X\) in \(\mathcal{I}\), then there is a canonical isomorphism \[U \times_X X' = \text{Proj}(\bigoplus\nolimits_{d \geq 0} I^d)\] of schemes over \(U\), where the right hand side is the homogeneous spectrum of the Rees algebra of \(I\) in \(A\). Moreover, \(U \times_X X'\) has an affine open covering by spectra of the affine blowup algebras \(A[\frac{I}{a}]\).

Proof

Note that the restriction \(\mathcal{I}|_U\) is equal to the pullback of \(\mathcal{I}\) via the morphism \(U \to X\), see Properties of Spaces, Section 03LT. Thus the lemma follows on combining Lemma 084B with Divisors, Lemma 0804.

Lemma

Let \(S\) be a scheme. Let \(X_1 \to X_2\) be a flat morphism of algebraic spaces over \(S\). Let \(Z_2 \subset X_2\) be a closed subspace. Let \(Z_1\) be the inverse image of \(Z_2\) in \(X_1\). Let \(X'_i\) be the blowup of \(Z_i\) in \(X_i\). Then there exists a cartesian diagram \[\xymatrix{ X_1' \ar[r] \ar[d] & X_2' \ar[d] \\ X_1 \ar[r] & X_2 }\] of algebraic spaces over \(S\).

Proof

Let \(\mathcal{I}_2\) be the ideal sheaf of \(Z_2\) in \(X_2\). Denote \(g : X_1 \to X_2\) the given morphism. Then the ideal sheaf \(\mathcal{I}_1\) of \(Z_1\) is the image of \(g^*\mathcal{I}_2 \to \mathcal{O}_{X_1}\) (see Morphisms of Spaces, Definition 083Q and discussion following the definition). By Lemma 085C we see that \(X_1 \times_{X_2} X_2'\) is the relative Proj of \(\bigoplus_{n \geq 0} g^*\mathcal{I}_2^n\). Because \(g\) is flat the map \(g^*\mathcal{I}_2^n \to \mathcal{O}_{X_1}\) is injective with image \(\mathcal{I}_1^n\). Thus we see that \(X_1 \times_{X_2} X_2' = X_1'\).

Lemma

Let \(S\) be a scheme. Let \(X\) be an algebraic space over \(S\). Let \(Z \subset X\) be a closed subspace. The blowing up \(b : X' \to X\) of \(Z\) in \(X\) has the following properties:

1. \(b|_{b^{-1}(X \setminus Z)} : b^{-1}(X \setminus Z) \to X \setminus Z\) is an isomorphism,

2. the exceptional divisor \(E = b^{-1}(Z)\) is an effective Cartier divisor on \(X'\),

3. there is a canonical isomorphism \(\mathcal{O}_{X'}(-1) = \mathcal{O}_{X'}(E)\)

Proof

Let \(U\) be a scheme and let \(U \to X\) be a surjective étale morphism. As blowing up commutes with flat base change (Lemma 085S) we can prove each of these statements after base change to \(U\). This reduces us to the case of schemes. In this case the result is Divisors, Lemma 02OS.

Lemma

Let \(S\) be a scheme. Let \(X\) be an algebraic space over \(S\). Let \(Z \subset X\) be a closed subspace. Let \(\mathcal{C}\) be the full subcategory of \((\textit{Spaces}/X)\) consisting of \(Y \to X\) such that the inverse image of \(Z\) is an effective Cartier divisor on \(Y\). Then the blowing up \(b : X' \to X\) of \(Z\) in \(X\) is a final object of \(\mathcal{C}\).

Proof

We see that \(b : X' \to X\) is an object of \(\mathcal{C}\) according to Lemma 085T. Let \(f : Y \to X\) be an object of \(\mathcal{C}\). We have to show there exists a unique morphism \(Y \to X'\) over \(X\). Let \(D = f^{-1}(Z)\). Let \(\mathcal{I} \subset \mathcal{O}_X\) be the ideal sheaf of \(Z\) and let \(\mathcal{I}_D\) be the ideal sheaf of \(D\). Then \(f^*\mathcal{I} \to \mathcal{I}_D\) is a surjection to an invertible \(\mathcal{O}_Y\)-module. This extends to a map \(\psi : \bigoplus f^*\mathcal{I}^d \to \bigoplus \mathcal{I}_D^d\) of graded \(\mathcal{O}_Y\)-algebras. (We observe that \(\mathcal{I}_D^d = \mathcal{I}_D^{\otimes d}\) as \(D\) is an effective Cartier divisor.) By Lemma 085D. the triple \((f : Y \to X, \mathcal{I}_D, \psi)\) defines a morphism \(Y \to X'\) over \(X\). The restriction \[Y \setminus D \longrightarrow X' \setminus b^{-1}(Z) = X \setminus Z\] is unique. The open \(Y \setminus D\) is scheme theoretically dense in \(Y\) according to Lemma 083S. Thus the morphism \(Y \to X'\) is unique by Morphisms of Spaces, Lemma 084N (also \(b\) is separated by Lemma 084E).

Lemma

Let \(S\) be a scheme. Let \(X\) be an algebraic space over \(S\). Let \(Z \subset X\) be an effective Cartier divisor. The blowup of \(X\) in \(Z\) is the identity morphism of \(X\).

Proof

Immediate from the universal property of blowups (Lemma 085U).

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. If \(X\) is reduced, then the blowup \(X'\) of \(X\) in \(\mathcal{I}\) is reduced.

Proof

Let \(U\) be a scheme and let \(U \to X\) be a surjective étale morphism. As blowing up commutes with flat base change (Lemma 085S) we can prove each of these statements after base change to \(U\). This reduces us to the case of schemes. In this case the result is Divisors, Lemma 0808.

Lemma

Let \(S\) be a scheme. Let \(X\) be an algebraic space over \(S\). Let \(b : X' \to X\) be the blowup of \(X\) in a closed subspace. If \(X\) satisfies the equivalent conditions of Morphisms of Spaces, Lemma 0BB1 then so does \(X'\).

Proof

Follows immediately from the lemma cited in the statement, the étale local description of blowing ups in Lemma 085R, and Divisors, Lemma 0BFM.

Lemma

Let \(S\) be a scheme. Let \(X\) be an algebraic space over \(S\). Let \(b : X' \to X\) be a blowup of \(X\) in a closed subspace. For any effective Cartier divisor \(D\) on \(X\) the pullback \(b^{-1}D\) is defined (see Definition 083Y).

Proof

By Lemmas 085R and 083C this reduces to the following algebra fact: Let \(A\) be a ring, \(I \subset A\) an ideal, \(a \in I\), and \(x \in A\) a nonzerodivisor. Then the image of \(x\) in \(A[\frac{I}{a}]\) is a nonzerodivisor. Namely, suppose that \(x (y/a^n) = 0\) in \(A[\frac{I}{a}]\). Then \(a^mxy = 0\) in \(A\) for some \(m\). Hence \(a^my = 0\) as \(x\) is a nonzerodivisor. Whence \(y/a^n\) is zero in \(A[\frac{I}{a}]\) as desired.

Lemma

Let \(S\) be a scheme. Let \(X\) be an algebraic space over \(S\). Let \(\mathcal{I} \subset \mathcal{O}_X\) and \(\mathcal{J}\) be quasi-coherent sheaves of ideals. Let \(b : X' \to X\) be the blowing up of \(X\) in \(\mathcal{I}\). Let \(b' : X'' \to X'\) be the blowing up of \(X'\) in \(b^{-1}\mathcal{J} \mathcal{O}_{X'}\). Then \(X'' \to X\) is canonically isomorphic to the blowing up of \(X\) in \(\mathcal{I}\mathcal{J}\).

Proof

Let \(E \subset X'\) be the exceptional divisor of \(b\) which is an effective Cartier divisor by Lemma 085T. Then \((b')^{-1}E\) is an effective Cartier divisor on \(X''\) by Lemma 085X. Let \(E' \subset X''\) be the exceptional divisor of \(b'\) (also an effective Cartier divisor). Consider the effective Cartier divisor \(E'' = E' + (b')^{-1}E\). By construction the ideal of \(E''\) is \((b \circ b')^{-1}\mathcal{I} (b \circ b')^{-1}\mathcal{J} \mathcal{O}_{X''}\). Hence according to Lemma 085U there is a canonical morphism from \(X''\) to the blowup \(c : Y \to X\) of \(X\) in \(\mathcal{I}\mathcal{J}\). Conversely, as \(\mathcal{I}\mathcal{J}\) pulls back to an invertible ideal we see that \(c^{-1}\mathcal{I}\mathcal{O}_Y\) defines an effective Cartier divisor, see Lemma 083W. Thus a morphism \(c' : Y \to X'\) over \(X\) by Lemma 085U. Then \((c')^{-1}b^{-1}\mathcal{J}\mathcal{O}_Y = c^{-1}\mathcal{J}\mathcal{O}_Y\) which also defines an effective Cartier divisor. Thus a morphism \(c'' : Y \to X''\) over \(X'\). We omit the verification that this morphism is inverse to the morphism \(X'' \to Y\) constructed earlier.

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 \(b : X' \to X\) be the blowing up of \(X\) in the ideal sheaf \(\mathcal{I}\). If \(\mathcal{I}\) is of finite type, then \(b : X' \to X\) is a proper morphism.

Proof

Let \(U\) be a scheme and let \(U \to X\) be a surjective étale morphism. As blowing up commutes with flat base change (Lemma 085S) we can prove each of these statements after base change to \(U\) (see Morphisms of Spaces, Lemma 083R). This reduces us to the case of schemes. In this case the morphism \(b\) is projective by Divisors, Lemma 02NS hence proper by Morphisms, Lemma 01WC.

Lemma

Let \(S\) be a scheme and let \(X\) be an algebraic space over \(S\). Assume \(X\) is quasi-compact and quasi-separated. Let \(Z \subset X\) be a closed subspace of finite presentation. Let \(b : X' \to X\) be the blowing up with center \(Z\). Let \(Z' \subset X'\) be a closed subspace of finite presentation. Let \(X'' \to X'\) be the blowing up with center \(Z'\). There exists a closed subspace \(Y \subset X\) of finite presentation, such that

1. \(|Y| = |Z| \cup |b|(|Z'|)\), and

2. the composition \(X'' \to X\) is isomorphic to the blowing up of \(X\) in \(Y\).

Proof

The condition that \(Z \to X\) is of finite presentation means that \(Z\) is cut out by a finite type quasi-coherent sheaf of ideals \(\mathcal{I} \subset \mathcal{O}_X\), see Morphisms of Spaces, Lemma 084Q. Write \(\mathcal{A} = \bigoplus_{n \geq 0} \mathcal{I}^n\) so that \(X' = \underline{\text{Proj}}(\mathcal{A})\). Note that \(X \setminus Z\) is a quasi-compact open subspace of \(X\) by Limits of Spaces, Lemma 0855. Since \(b^{-1}(X \setminus Z) \to X \setminus Z\) is an isomorphism (Lemma 085T) the same result shows that \(b^{-1}(X \setminus Z) \setminus Z'\) is quasi-compact open subspace in \(X'\). Hence \(U = X \setminus (Z \cup b(Z'))\) is quasi-compact open subspace in \(X\). By Lemma 085N there exist a \(d > 0\) and a finite type \(\mathcal{O}_X\)-submodule \(\mathcal{F} \subset \mathcal{I}^d\) such that \(Z' = \underline{\text{Proj}}(\mathcal{A}/\mathcal{F}\mathcal{A})\) and such that the support of \(\mathcal{I}^d/\mathcal{F}\) is contained in \(X \setminus U\).

Since \(\mathcal{F} \subset \mathcal{I}^d\) is an \(\mathcal{O}_X\)-submodule we may think of \(\mathcal{F} \subset \mathcal{I}^d \subset \mathcal{O}_X\) as a finite type quasi-coherent sheaf of ideals on \(X\). Let’s denote this \(\mathcal{J} \subset \mathcal{O}_X\) to prevent confusion. Since \(\mathcal{I}^d / \mathcal{J}\) and \(\mathcal{O}/\mathcal{I}^d\) are supported on \(|X| \setminus |U|\) we see that \(|V(\mathcal{J})|\) is contained in \(|X| \setminus |U|\). Conversely, as \(\mathcal{J} \subset \mathcal{I}^d\) we see that \(|Z| \subset |V(\mathcal{J})|\). Over \(X \setminus Z \cong X' \setminus b^{-1}(Z)\) the sheaf of ideals \(\mathcal{J}\) cuts out \(Z'\) (see displayed formula below). Hence \(|V(\mathcal{J})|\) equals \(|Z| \cup |b|(|Z'|)\). It follows that also \(|V(\mathcal{I}\mathcal{J})| = |Z| \cup |b|(|Z'|)\). Moreover, \(\mathcal{I}\mathcal{J}\) is an ideal of finite type as a product of two such. We claim that \(X'' \to X\) is isomorphic to the blowing up of \(X\) in \(\mathcal{I}\mathcal{J}\) which finishes the proof of the lemma by setting \(Y = V(\mathcal{I}\mathcal{J})\).

First, recall that the blowup of \(X\) in \(\mathcal{I}\mathcal{J}\) is the same as the blowup of \(X'\) in \(b^{-1}\mathcal{J} \mathcal{O}_{X'}\), see Lemma 085Y. Hence it suffices to show that the blowup of \(X'\) in \(b^{-1}\mathcal{J} \mathcal{O}_{X'}\) agrees with the blowup of \(X'\) in \(Z'\). We will show that \[b^{-1}\mathcal{J} \mathcal{O}_{X'} = \mathcal{I}_E^d \mathcal{I}_{Z'}\] as ideal sheaves on \(X''\). This will prove what we want as \(\mathcal{I}_E^d\) cuts out the effective Cartier divisor \(dE\) and we can use Lemmas 085V and 085Y.

To see the displayed equality of the ideals we may work locally. With notation \(A\), \(I\), \(a \in I\) as in Lemma 085R we see that \(\mathcal{F}\) corresponds to an \(R\)-submodule \(M \subset I^d\) mapping isomorphically to an ideal \(J \subset R\). The condition \(Z' = \underline{\text{Proj}}(\mathcal{A}/\mathcal{F}\mathcal{A})\) means that \(Z' \cap \Spec(A[\frac{I}{a}])\) is cut out by the ideal generated by the elements \(m/a^d\), \(m \in M\). Say the element \(m \in M\) corresponds to the function \(f \in J\). Then in the affine blowup algebra \(A' = A[\frac{I}{a}]\) we see that \(f = (a^dm)/a^d = a^d (m/a^d)\). Thus the equality holds.

Strict transform

This section is the analogue of Divisors, Section 080C. Let \(S\) be a scheme, let \(B\) be an algebraic space over \(S\), and let \(Z \subset B\) be a closed subspace. Let \(b : B' \to B\) be the blowing up of \(B\) in \(Z\) and denote \(E \subset B'\) the exceptional divisor \(E = b^{-1}Z\). In the following we will often consider an algebraic space \(X\) over \(B\) and form the cartesian diagram \[\xymatrix{ \text{pr}_{B'}^{-1}E \ar[r] \ar[d] & X \times_B B' \ar[r]_-{\text{pr}_X} \ar[d]_{\text{pr}_{B'}} & X \ar[d]^f \\ E \ar[r] & B' \ar[r] & B }\] Since \(E\) is an effective Cartier divisor (Lemma 085T) we see that \(\text{pr}_{B'}^{-1}E \subset X \times_B B'\) is locally principal (Lemma 083X). Thus the inclusion morphism of the complement of \(\text{pr}_{B'}^{-1}E\) in \(X \times_B B'\) is affine and in particular quasi-compact (Lemma 083D). Consequently, for a quasi-coherent \(\mathcal{O}_{X \times_B B'}\)-module \(\mathcal{G}\) the subsheaf of sections supported on \(|\text{pr}_{B'}^{-1}E|\) is a quasi-coherent submodule, see Limits of Spaces, Definition 085A. If \(\mathcal{G}\) is a quasi-coherent sheaf of algebras, e.g., \(\mathcal{G} = \mathcal{O}_{X \times_B B'}\), then this subsheaf is an ideal of \(\mathcal{G}\).

Definition

With \(Z \subset B\) and \(f : X \to B\) as above.

1. Given a quasi-coherent \(\mathcal{O}_X\)-module \(\mathcal{F}\) the strict transform of \(\mathcal{F}\) with respect to the blowup of \(B\) in \(Z\) is the quotient \(\mathcal{F}'\) of \(\text{pr}_X^*\mathcal{F}\) by the submodule of sections supported on \(|\text{pr}_{B'}^{-1}E|\).

2. The strict transform of \(X\) is the closed subspace \(X' \subset X \times_B B'\) cut out by the quasi-coherent ideal of sections of \(\mathcal{O}_{X \times_B B'}\) supported on \(|\text{pr}_{B'}^{-1}E|\).

Note that taking the strict transform along a blowup depends on the closed subspace used for the blowup (and not just on the morphism \(B' \to B\)).

Lemma

In the situation of Definition 0862. Let \[\xymatrix{ U \ar[r] \ar[d] & X \ar[d] \\ V \ar[r] & B }\] be a commutative diagram of morphisms with \(U\) and \(V\) schemes and étale horizontal arrows. Let \(V' \to V\) be the blowup of \(V\) in \(Z \times_B V\). Then

1. \(V' = V \times_B B'\) and the maps \(V' \to B'\) and \(U \times_V V' \to X \times_B B'\) are étale,

2. the strict transform \(U'\) of \(U\) relative to \(V' \to V\) is equal to \(X' \times_X U\) where \(X'\) is the strict transform of \(X\) relative to \(B' \to B\), and

3. for a quasi-coherent \(\mathcal{O}_X\)-module \(\mathcal{F}\) the restriction of the strict transform \(\mathcal{F}'\) to \(U \times_V V'\) is the strict transform of \(\mathcal{F}|_U\) relative to \(V' \to V\).

Proof

Part (1) follows from the fact that blowup commutes with flat base change (Lemma 085S), the fact that étale morphisms are flat, and that the base change of an étale morphism is étale. Part (3) then follows from the fact that taking the sheaf of sections supported on a closed commutes with pullback by étale morphisms, see Limits of Spaces, Lemma 0859. Part (2) follows from (3) applied to \(\mathcal{F} = \mathcal{O}_X\).

Lemma

In the situation of Definition 0862.

1. The strict transform \(X'\) of \(X\) is the blowup of \(X\) in the closed subspace \(f^{-1}Z\) of \(X\).

2. For a quasi-coherent \(\mathcal{O}_X\)-module \(\mathcal{F}\) the strict transform \(\mathcal{F}'\) is canonically isomorphic to the pushforward along \(X' \to X \times_B B'\) of the strict transform of \(\mathcal{F}\) relative to the blowing up \(X' \to X\).

Proof

Let \(X'' \to X\) be the blowup of \(X\) in \(f^{-1}Z\). By the universal property of blowing up (Lemma 085U) there exists a commutative diagram \[\xymatrix{ X'' \ar[r] \ar[d] & X \ar[d] \\ B' \ar[r] & B }\] whence a morphism \(i : X'' \to X \times_B B'\). The first assertion of the lemma is that \(i\) is a closed immersion with image \(X'\). The second assertion of the lemma is that \(\mathcal{F}' = i_*\mathcal{F}''\) where \(\mathcal{F}''\) is the strict transform of \(\mathcal{F}\) with respect to the blowing up \(X'' \to X\). We can check these assertions étale locally on \(X\), hence we reduce to the case of schemes (Divisors, Lemma 080E). Some details omitted.

Lemma

In the situation of Definition 0862.

1. If \(X\) is flat over \(B\) at all points lying over \(Z\), then the strict transform of \(X\) is equal to the base change \(X \times_B B'\).

2. Let \(\mathcal{F}\) be a quasi-coherent \(\mathcal{O}_X\)-module. If \(\mathcal{F}\) is flat over \(B\) at all points lying over \(Z\), then the strict transform \(\mathcal{F}'\) of \(\mathcal{F}\) is equal to the pullback \(\text{pr}_X^*\mathcal{F}\).

Proof

Omitted. Hint: Follows from the case of schemes (Divisors, Lemma 080F) by étale localization (Lemma 0863).

Lemma

Let \(S\) be a scheme. Let \(B\) be an algebraic space over \(S\). Let \(Z \subset B\) be a closed subspace. Let \(b : B' \to B\) be the blowing up of \(Z\) in \(B\). Let \(g : X \to Y\) be an affine morphism of spaces over \(B\). Let \(\mathcal{F}\) be a quasi-coherent sheaf on \(X\). Let \(g' : X \times_B B' \to Y \times_B B'\) be the base change of \(g\). Let \(\mathcal{F}'\) be the strict transform of \(\mathcal{F}\) relative to \(b\). Then \(g'_*\mathcal{F}'\) is the strict transform of \(g_*\mathcal{F}\).

Proof

Omitted. Hint: Follows from the case of schemes (Divisors, Lemma 080G) by étale localization (Lemma 0863).

Lemma

Let \(S\) be a scheme. Let \(B\) be an algebraic space over \(S\). Let \(Z \subset B\) be a closed subspace. Let \(D \subset B\) be an effective Cartier divisor. Let \(Z' \subset B\) be the closed subspace cut out by the product of the ideal sheaves of \(Z\) and \(D\). Let \(B' \to B\) be the blowup of \(B\) in \(Z\).

1. The blowup of \(B\) in \(Z'\) is isomorphic to \(B' \to B\).

2. Let \(f : X \to B\) be a morphism of algebraic spaces and let \(\mathcal{F}\) be a quasi-coherent \(\mathcal{O}_X\)-module. If the subsheaf of \(\mathcal{F}\) of sections supported on \(|f^{-1}D|\) is zero, then the strict transform of \(\mathcal{F}\) relative to the blowing up in \(Z\) agrees with the strict transform of \(\mathcal{F}\) relative to the blowing up of \(B\) in \(Z'\).

Proof

Omitted. Hint: Follows from the case of schemes (Divisors, Lemma 080H) by étale localization (Lemma 0863).

Lemma

Let \(S\) be a scheme. Let \(B\) be an algebraic space over \(S\). Let \(Z \subset B\) be a closed subspace. Let \(b : B' \to B\) be the blowing up with center \(Z\). Let \(Z' \subset B'\) be a closed subspace. Let \(B'' \to B'\) be the blowing up with center \(Z'\). Let \(Y \subset B\) be a closed subscheme such that \(|Y| = |Z| \cup |b|(|Z'|)\) and the composition \(B'' \to B\) is isomorphic to the blowing up of \(B\) in \(Y\). In this situation, given any scheme \(X\) over \(B\) and \(\mathcal{F} \in \QCoh(\mathcal{O}_X)\) we have

1. the strict transform of \(\mathcal{F}\) with respect to the blowing up of \(B\) in \(Y\) is equal to the strict transform with respect to the blowup \(B'' \to B'\) in \(Z'\) of the strict transform of \(\mathcal{F}\) with respect to the blowup \(B' \to B\) of \(B\) in \(Z\), and

2. the strict transform of \(X\) with respect to the blowing up of \(B\) in \(Y\) is equal to the strict transform with respect to the blowup \(B'' \to B'\) in \(Z'\) of the strict transform of \(X\) with respect to the blowup \(B' \to B\) of \(B\) in \(Z\).

Proof

Omitted. Hint: Follows from the case of schemes (Divisors, Lemma 080I) by étale localization (Lemma 0863).

Lemma

In the situation of Definition 0862. Suppose that \[0 \to \mathcal{F}_1 \to \mathcal{F}_2 \to \mathcal{F}_3 \to 0\] is an exact sequence of quasi-coherent sheaves on \(X\) which remains exact after any base change \(T \to B\). Then the strict transforms of \(\mathcal{F}_i'\) relative to any blowup \(B' \to B\) form a short exact sequence \(0 \to \mathcal{F}'_1 \to \mathcal{F}'_2 \to \mathcal{F}'_3 \to 0\) too.

Proof

Omitted. Hint: Follows from the case of schemes (Divisors, Lemma 080W) by étale localization (Lemma 0863).

Lemma

Let \(S\) be a scheme. Let \(B\) be an algebraic space over \(S\). Let \(\mathcal{F}\) be a finite type quasi-coherent \(\mathcal{O}_B\)-module. Let \(Z_k \subset S\) be the closed subscheme cut out by \(\text{Fit}_k(\mathcal{F})\), see Section 0CZ3. Let \(B' \to B\) be the blowup of \(B\) in \(Z_k\) and let \(\mathcal{F}'\) be the strict transform of \(\mathcal{F}\). Then \(\mathcal{F}'\) can locally be generated by \(\leq k\) sections.

Proof

Omitted. Follows from the case of schemes (Divisors, Lemma 0CZP) by étale localization (Lemma 0863).

Lemma

Let \(S\) be a scheme. Let \(B\) be an algebraic space over \(S\). Let \(\mathcal{F}\) be a finite type quasi-coherent \(\mathcal{O}_B\)-module. Let \(Z_k \subset S\) be the closed subscheme cut out by \(\text{Fit}_k(\mathcal{F})\), see Section 0CZ3. Assume that \(\mathcal{F}\) is locally free of rank \(k\) on \(B \setminus Z_k\). Let \(B' \to B\) be the blowup of \(B\) in \(Z_k\) and let \(\mathcal{F}'\) be the strict transform of \(\mathcal{F}\). Then \(\mathcal{F}'\) is locally free of rank \(k\).

Proof

Omitted. Follows from the case of schemes (Divisors, Lemma 0CZQ) by étale localization (Lemma 0863).

Admissible blowups

To have a bit more control over our blowups we introduce the following standard terminology.

Definition

Let \(S\) be a scheme. Let \(X\) be an algebraic space over \(S\). Let \(U \subset X\) be an open subspace. A morphism \(X' \to X\) is called a \(U\)-admissible blowup if there exists a closed immersion \(Z \to X\) of finite presentation with \(Z\) disjoint from \(U\) such that \(X'\) is isomorphic to the blowup of \(X\) in \(Z\).

We recall that \(Z \to X\) is of finite presentation if and only if the ideal sheaf \(\mathcal{I}_Z \subset \mathcal{O}_X\) is of finite type, see Morphisms of Spaces, Lemma 084Q. In particular, a \(U\)-admissible blowup is a proper morphism, see Lemma 085Z. Note that there can be multiple centers which give rise to the same morphism. Hence the requirement is just the existence of some center disjoint from \(U\) which produces \(X'\). Finally, as the morphism \(b : X' \to X\) is an isomorphism over \(U\) (see Lemma 085T) we will often abuse notation and think of \(U\) as an open subspace of \(X'\) as well.

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 subspace. Let \(b : X' \to X\) be a \(U\)-admissible blowup. Let \(X'' \to X'\) be a \(U\)-admissible blowup. Then the composition \(X'' \to X\) is a \(U\)-admissible blowup.

Proof

Immediate from the more precise Lemma 0860.

Lemma

Let \(S\) be a scheme. Let \(X\) be a quasi-compact and quasi-separated algebraic space. Let \(U, V \subset X\) be quasi-compact open subspaces. Let \(b : V' \to V\) be a \(U \cap V\)-admissible blowup. Then there exists a \(U\)-admissible blowup \(X' \to X\) whose restriction to \(V\) is \(V'\).

Proof

Let \(\mathcal{I} \subset \mathcal{O}_V\) be the finite type quasi-coherent sheaf of ideals such that \(V(\mathcal{I})\) is disjoint from \(U \cap V\) and such that \(V'\) is isomorphic to the blowup of \(V\) in \(\mathcal{I}\). Let \(\mathcal{I}' \subset \mathcal{O}_{U \cup V}\) be the quasi-coherent sheaf of ideals whose restriction to \(U\) is \(\mathcal{O}_U\) and whose restriction to \(V\) is \(\mathcal{I}\). By Limits of Spaces, Lemma 0853 there exists a finite type quasi-coherent sheaf of ideals \(\mathcal{J} \subset \mathcal{O}_X\) whose restriction to \(U \cup V\) is \(\mathcal{I}'\). The lemma follows.

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 subspace. Let \(b_i : X_i \to X\), \(i = 1, \ldots, n\) be \(U\)-admissible blowups. There exists a \(U\)-admissible blowup \(b : X' \to X\) such that (a) \(b\) factors as \(X' \to X_i \to X\) for \(i = 1, \ldots, n\) and (b) each of the morphisms \(X' \to X_i\) is a \(U\)-admissible blowup.

Proof

Let \(\mathcal{I}_i \subset \mathcal{O}_X\) be the finite type quasi-coherent sheaf of ideals such that \(V(\mathcal{I}_i)\) is disjoint from \(U\) and such that \(X_i\) is isomorphic to the blowup of \(X\) in \(\mathcal{I}_i\). Set \(\mathcal{I} = \mathcal{I}_1 \cdot \ldots \cdot \mathcal{I}_n\) and let \(X'\) be the blowup of \(X\) in \(\mathcal{I}\). Then \(X' \to X\) factors through \(b_i\) by Lemma 085Y.

Lemma

Let \(S\) be a scheme. Let \(X\) be a quasi-compact and quasi-separated algebraic space over \(S\). Let \(U, V\) be quasi-compact disjoint open subspaces of \(X\). Then there exist a \(U \cup V\)-admissible blowup \(b : X' \to X\) such that \(X'\) is a disjoint union of open subspaces \(X' = X'_1 \amalg X'_2\) with \(b^{-1}(U) \subset X'_1\) and \(b^{-1}(V) \subset X'_2\).

Proof

Choose a finite type quasi-coherent sheaf of ideals \(\mathcal{I}\), resp. \(\mathcal{J}\) such that \(X \setminus U = V(\mathcal{I})\), resp. \(X \setminus V = V(\mathcal{J})\), see Limits of Spaces, Lemma 0855. Then \(|V(\mathcal{I}\mathcal{J})| = |X|\). Hence \(\mathcal{I}\mathcal{J}\) is a locally nilpotent sheaf of ideals. Since \(\mathcal{I}\) and \(\mathcal{J}\) are of finite type and \(X\) is quasi-compact there exists an \(n > 0\) such that \(\mathcal{I}^n \mathcal{J}^n = 0\). We may and do replace \(\mathcal{I}\) by \(\mathcal{I}^n\) and \(\mathcal{J}\) by \(\mathcal{J}^n\). Whence \(\mathcal{I} \mathcal{J} = 0\). Let \(b : X' \to X\) be the blowing up in \(\mathcal{I} + \mathcal{J}\). This is \(U \cup V\)-admissible as \(|V(\mathcal{I} + \mathcal{J})| = |X| \setminus |U| \cup |V|\). We will show that \(X'\) is a disjoint union of open subspaces \(X' = X'_1 \amalg X'_2\) as in the statement of the lemma.

Since \(|V(\mathcal{I} + \mathcal{J})|\) is the complement of \(|U \cup V|\) we conclude that \(V \cup U\) is scheme theoretically dense in \(X'\), see Lemmas 085T and 083S. Thus if such a decomposition \(X' = X'_1 \amalg X'_2\) into open and closed subspaces exists, then \(X'_1\) is the scheme theoretic closure of \(U\) in \(X'\) and similarly \(X'_2\) is the scheme theoretic closure of \(V\) in \(X'\). Since \(U \to X'\) and \(V \to X'\) are quasi-compact taking scheme theoretic closures commutes with étale localization (Morphisms of Spaces, Lemma 082Z). Hence to verify the existence of \(X'_1\) and \(X'_2\) we may work étale locally on \(X\). This reduces us to the case of schemes which is treated in the proof of Divisors, Lemma 080P.