The Stacks project

29.43 Projective morphisms

We will use the definition of a projective morphism from [EGA]. The version of the definition with the “H” is the one from [H]. The resulting definitions are different. Both are useful.

Definition 29.43.1. Let $f : X \to S$ be a morphism of schemes.

  1. We say $f$ is projective if $X$ is isomorphic as an $S$-scheme to a closed subscheme of a projective bundle $\mathbf{P}(\mathcal{E})$ for some quasi-coherent, finite type $\mathcal{O}_ S$-module $\mathcal{E}$.

  2. We say $f$ is H-projective if there exists an integer $n$ and a closed immersion $X \to \mathbf{P}^ n_ S$ over $S$.

  3. We say $f$ is locally projective if there exists an open covering $S = \bigcup U_ i$ such that each $f^{-1}(U_ i) \to U_ i$ is projective.

As expected, a projective morphism is quasi-projective, see Lemma 29.43.10. Conversely, quasi-projective morphisms are often compositions of open immersions and projective morphisms, see Lemma 29.43.12. For an overview of properties of projective morphisms over a quasi-projective base, see More on Morphisms, Section 37.50.

Example 29.43.2. Let $S$ be a scheme. Let $\mathcal{A}$ be a quasi-coherent graded $\mathcal{O}_ S$-algebra generated by $\mathcal{A}_1$ over $\mathcal{A}_0$. Assume furthermore that $\mathcal{A}_1$ is of finite type over $\mathcal{O}_ S$. Set $X = \underline{\text{Proj}}_ S(\mathcal{A})$. In this case $X \to S$ is projective. Namely, the morphism associated to the graded $\mathcal{O}_ S$-algebra map

\[ \text{Sym}_{\mathcal{O}_ X}^*(\mathcal{A}_1) \longrightarrow \mathcal{A} \]

is a closed immersion, see Constructions, Lemma 27.18.5.

Lemma 29.43.3. An H-projective morphism is H-quasi-projective. An H-projective morphism is projective.

Proof. The first statement is immediate from the definitions. The second holds as $\mathbf{P}^ n_ S$ is a projective bundle over $S$, see Constructions, Lemma 27.21.5. $\square$

Lemma 29.43.4. Let $f : X \to S$ be a morphism of schemes. The following are equivalent:

  1. The morphism $f$ is locally projective.

  2. There exists an open covering $S = \bigcup U_ i$ such that each $f^{-1}(U_ i) \to U_ i$ is H-projective.

Proof. By Lemma 29.43.3 we see that (2) implies (1). Assume (1). For every point $s \in S$ we can find $\mathop{\mathrm{Spec}}(R) = U \subset S$ an affine open neighbourhood of $s$ such that $X_ U$ is isomorphic to a closed subscheme of $\mathbf{P}(\mathcal{E})$ for some finite type, quasi-coherent sheaf of $\mathcal{O}_ U$-modules $\mathcal{E}$. Write $\mathcal{E} = \widetilde{M}$ for some finite type $R$-module $M$ (see Properties, Lemma 28.16.1). Choose generators $x_0, \ldots , x_ n \in M$ of $M$ as an $R$-module. Consider the surjective graded $R$-algebra map

\[ R[X_0, \ldots , X_ n] \longrightarrow \text{Sym}_ R(M). \]

According to Constructions, Lemma 27.11.3 the corresponding morphism

\[ \mathbf{P}(\mathcal{E}) \to \mathbf{P}^ n_ R \]

is a closed immersion. Hence we conclude that $f^{-1}(U)$ is isomorphic to a closed subscheme of $\mathbf{P}^ n_ U$ (as a scheme over $U$). In other words: (2) holds. $\square$

Proof. Let $f : X \to S$ be locally projective. In order to show that $f$ is proper we may work locally on the base, see Lemma 29.41.3. Hence, by Lemma 29.43.4 above we may assume there exists a closed immersion $X \to \mathbf{P}^ n_ S$. By Lemmas 29.41.4 and 29.41.6 it suffices to prove that $\mathbf{P}^ n_ S \to S$ is proper. Since $\mathbf{P}^ n_ S \to S$ is the base change of $\mathbf{P}^ n_{\mathbf{Z}} \to \mathop{\mathrm{Spec}}(\mathbf{Z})$ it suffices to show that $\mathbf{P}^ n_{\mathbf{Z}} \to \mathop{\mathrm{Spec}}(\mathbf{Z})$ is proper, see Lemma 29.41.5. By Constructions, Lemma 27.8.8 the scheme $\mathbf{P}^ n_{\mathbf{Z}}$ is separated. By Constructions, Lemma 27.8.9 the scheme $\mathbf{P}^ n_{\mathbf{Z}}$ is quasi-compact. It is clear that $\mathbf{P}^ n_{\mathbf{Z}} \to \mathop{\mathrm{Spec}}(\mathbf{Z})$ is locally of finite type since $\mathbf{P}^ n_{\mathbf{Z}}$ is covered by the affine opens $D_{+}(X_ i)$ each of which is the spectrum of the finite type $\mathbf{Z}$-algebra

\[ \mathbf{Z}[X_0/X_ i, \ldots , X_ n/X_ i]. \]

Finally, we have to show that $\mathbf{P}^ n_{\mathbf{Z}} \to \mathop{\mathrm{Spec}}(\mathbf{Z})$ is universally closed. This follows from Constructions, Lemma 27.8.11 and the valuative criterion (see Schemes, Proposition 26.20.6). $\square$

Lemma 29.43.6. Let $f : X \to S$ be a proper morphism of schemes. If there exists an $f$-ample invertible sheaf on $X$, then $f$ is locally projective.

Proof. If there exists an $f$-ample invertible sheaf, then we can locally on $S$ find an immersion $i : X \to \mathbf{P}^ n_ S$, see Lemma 29.39.4. Since $X \to S$ is proper the morphism $i$ is a closed immersion, see Lemma 29.41.7. $\square$

Lemma 29.43.7. A composition of H-projective morphisms is H-projective.

Proof. Suppose $X \to Y$ and $Y \to Z$ are H-projective. Then there exist closed immersions $X \to \mathbf{P}^ n_ Y$ over $Y$, and $Y \to \mathbf{P}^ m_ Z$ over $Z$. Consider the following diagram

\[ \xymatrix{ X \ar[r] \ar[d] & \mathbf{P}^ n_ Y \ar[r] \ar[dl] & \mathbf{P}^ n_{\mathbf{P}^ m_ Z} \ar[dl] \ar@{=}[r] & \mathbf{P}^ n_ Z \times _ Z \mathbf{P}^ m_ Z \ar[r] & \mathbf{P}^{nm + n + m}_ Z \ar[ddllll] \\ Y \ar[r] \ar[d] & \mathbf{P}^ m_ Z \ar[dl] & \\ Z & & } \]

Here the rightmost top horizontal arrow is the Segre embedding, see Constructions, Lemma 27.13.6. The diagram identifies $X$ as a closed subscheme of $\mathbf{P}^{nm + n + m}_ Z$ as desired. $\square$

Lemma 29.43.8. A base change of a H-projective morphism is H-projective.

Proof. This is true because the base change of projective space over a scheme is projective space, and the fact that the base change of a closed immersion is a closed immersion, see Schemes, Lemma 26.18.2. $\square$

Lemma 29.43.9. A base change of a (locally) projective morphism is (locally) projective.

Proof. This is true because the base change of a projective bundle over a scheme is a projective bundle, the pullback of a finite type $\mathcal{O}$-module is of finite type (Modules, Lemma 17.9.2) and the fact that the base change of a closed immersion is a closed immersion, see Schemes, Lemma 26.18.2. Some details omitted. $\square$

Proof. Let $f : X \to S$ be a projective morphism. Choose a closed immersion $i : X \to \mathbf{P}(\mathcal{E})$ where $\mathcal{E}$ is a quasi-coherent, finite type $\mathcal{O}_ S$-module. Then $\mathcal{L} = i^*\mathcal{O}_{\mathbf{P}(\mathcal{E})}(1)$ is $f$-very ample. Since $f$ is proper (Lemma 29.43.5) it is quasi-compact. Hence Lemma 29.38.2 implies that $\mathcal{L}$ is $f$-ample. Since $f$ is proper it is of finite type. Thus we've checked all the defining properties of quasi-projective holds and we win. $\square$

Lemma 29.43.11. Let $f : X \to S$ be a H-quasi-projective morphism. Then $f$ factors as $X \to X' \to S$ where $X \to X'$ is an open immersion and $X' \to S$ is H-projective.

Proof. By definition we can factor $f$ as a quasi-compact immersion $i : X \to \mathbf{P}^ n_ S$ followed by the projection $\mathbf{P}^ n_ S \to S$. By Lemma 29.7.7 there exists a closed subscheme $X' \subset \mathbf{P}^ n_ S$ such that $i$ factors through an open immersion $X \to X'$. The lemma follows. $\square$

Lemma 29.43.12. Let $f : X \to S$ be a quasi-projective morphism with $S$ quasi-compact and quasi-separated. Then $f$ factors as $X \to X' \to S$ where $X \to X'$ is an open immersion and $X' \to S$ is projective.

Proof. Let $\mathcal{L}$ be $f$-ample. Since $f$ is of finite type and $S$ is quasi-compact $\mathcal{L}^{\otimes n}$ is $f$-very ample for some $n > 0$, see Lemma 29.39.5. Replace $\mathcal{L}$ by $\mathcal{L}^{\otimes n}$. Write $\mathcal{F} = f_*\mathcal{L}$. This is a quasi-coherent $\mathcal{O}_ S$-module by Schemes, Lemma 26.24.1 (quasi-projective morphisms are quasi-compact and separated, see Lemma 29.40.4). By Properties, Lemma 28.22.7 we can find a directed set $I$ and a system of finite type quasi-coherent $\mathcal{O}_ S$-modules $\mathcal{E}_ i$ over $I$ such that $\mathcal{F} = \mathop{\mathrm{colim}}\nolimits \mathcal{E}_ i$. Consider the compositions $\psi _ i : f^*\mathcal{E}_ i \to f^*\mathcal{F} \to \mathcal{L}$. Choose a finite affine open covering $S = \bigcup _{j = 1, \ldots , m} V_ j$. For each $j$ we can choose sections

\[ s_{j, 0}, \ldots , s_{j, n_ j} \in \Gamma (f^{-1}(V_ j), \mathcal{L}) = f_*\mathcal{L}(V_ j) = \mathcal{F}(V_ j) \]

which generate $\mathcal{L}$ over $f^{-1}V_ j$ and define an immersion

\[ f^{-1}V_ j \longrightarrow \mathbf{P}^{n_ j}_{V_ j}, \]

see Lemma 29.39.1. Choose $i$ such that there exist sections $e_{j, t} \in \mathcal{E}_ i(V_ j)$ mapping to $s_{j, t}$ in $\mathcal{F}$ for all $j = 1, \ldots , m$ and $t = 1, \ldots , n_ j$. Then the map $\psi _ i$ is surjective as the sections $f^*e_{j, t}$ have the same image as the sections $s_{j, t}$ which generate $\mathcal{L}|_{f^{-1}V_ j}$. Whence we obtain a morphism

\[ r_{\mathcal{L}, \psi _ i} : X \longrightarrow \mathbf{P}(\mathcal{E}_ i) \]

over $S$ such that over $V_ j$ we have a factorization

\[ f^{-1}V_ j \to \mathbf{P}(\mathcal{E}_ i)|_{V_ j} \to \mathbf{P}^{n_ j}_{V_ j} \]

of the immersion given above. It follows that $r_{\mathcal{L}, \psi _ i}|_{V_ j}$ is an immersion, see Lemma 29.3.1. Since $S = \bigcup V_ j$ we conclude that $r_{\mathcal{L}, \psi _ i}$ is an immersion. Note that $r_{\mathcal{L}, \psi _ i}$ is quasi-compact as $X \to S$ is quasi-compact and $\mathbf{P}(\mathcal{E}_ i) \to S$ is separated (see Schemes, Lemma 26.21.14). By Lemma 29.7.7 there exists a closed subscheme $X' \subset \mathbf{P}(\mathcal{E}_ i)$ such that $i$ factors through an open immersion $X \to X'$. Then $X' \to S$ is projective by definition and we win. $\square$

Lemma 29.43.13. Let $S$ be a quasi-compact and quasi-separated scheme. Let $f : X \to S$ be a morphism of schemes. Then

  1. $f$ is projective if and only if $f$ is quasi-projective and proper, and

  2. $f$ is H-projective if and only if $f$ is H-quasi-projective and proper.

Proof. If $f$ is projective, then $f$ is quasi-projective by Lemma 29.43.10 and proper by Lemma 29.43.5. Conversely, if $X \to S$ is quasi-projective and proper, then we can choose an open immersion $X \to X'$ with $X' \to S$ projective by Lemma 29.43.12. Since $X \to S$ is proper, we see that $X$ is closed in $X'$ (Lemma 29.41.7), i.e., $X \to X'$ is a (open and) closed immersion. Since $X'$ is isomorphic to a closed subscheme of a projective bundle over $S$ (Definition 29.43.1) we see that the same thing is true for $X$, i.e., $X \to S$ is a projective morphism. This proves (1). The proof of (2) is the same, except it uses Lemmas 29.43.3 and 29.43.11. $\square$

Lemma 29.43.14. Let $f : X \to Y$ and $g : Y \to S$ be morphisms of schemes. If $S$ is quasi-compact and quasi-separated and $f$ and $g$ are projective, then $g \circ f$ is projective.

Proof. By Lemmas 29.43.10 and 29.43.5 we see that $f$ and $g$ are quasi-projective and proper. By Lemmas 29.41.4 and 29.40.3 we see that $g \circ f$ is proper and quasi-projective. Thus $g \circ f$ is projective by Lemma 29.43.13. $\square$

Lemma 29.43.15. Let $g : Y \to S$ and $f : X \to Y$ be morphisms of schemes. If $g \circ f$ is projective and $g$ is separated, then $f$ is projective.

Proof. Choose a closed immersion $X \to \mathbf{P}(\mathcal{E})$ where $\mathcal{E}$ is a quasi-coherent, finite type $\mathcal{O}_ S$-module. Then we get a morphism $X \to \mathbf{P}(\mathcal{E}) \times _ S Y$. This morphism is a closed immersion because it is the composition

\[ X \to X \times _ S Y \to \mathbf{P}(\mathcal{E}) \times _ S Y \]

where the first morphism is a closed immersion by Schemes, Lemma 26.21.10 (and the fact that $g$ is separated) and the second as the base change of a closed immersion. Finally, the fibre product $\mathbf{P}(\mathcal{E}) \times _ S Y$ is isomorphic to $\mathbf{P}(g^*\mathcal{E})$ and pullback preserves quasi-coherent, finite type modules. $\square$

Lemma 29.43.16. Let $S$ be a scheme which admits an ample invertible sheaf. Then

  1. any projective morphism $X \to S$ is H-projective, and

  2. any quasi-projective morphism $X \to S$ is H-quasi-projective.

Proof. The assumptions on $S$ imply that $S$ is quasi-compact and separated, see Properties, Definition 28.26.1 and Lemma 28.26.11 and Constructions, Lemma 27.8.8. Hence Lemma 29.43.12 applies and we see that (1) implies (2). Let $\mathcal{E}$ be a finite type quasi-coherent $\mathcal{O}_ S$-module. By our definition of projective morphisms it suffices to show that $\mathbf{P}(\mathcal{E}) \to S$ is H-projective. If $\mathcal{E}$ is generated by finitely many global sections, then the corresponding surjection $\mathcal{O}_ S^{\oplus n} \to \mathcal{E}$ induces a closed immersion

\[ \mathbf{P}(\mathcal{E}) \longrightarrow \mathbf{P}(\mathcal{O}_ S^{\oplus n}) = \mathbf{P}^{n - 1}_ S \]

as desired. In general, let $\mathcal{L}$ be an ample invertible sheaf on $S$. By Properties, Proposition 28.26.13 there exists an integer $n$ such that $\mathcal{E} \otimes _{\mathcal{O}_ S} \mathcal{L}^{\otimes n}$ is globally generated by finitely many sections. Since $\mathbf{P}(\mathcal{E}) = \mathbf{P}(\mathcal{E} \otimes _{\mathcal{O}_ S} \mathcal{L}^{\otimes n})$ by Constructions, Lemma 27.20.1 this finishes the proof. $\square$

Lemma 29.43.17. Let $f : X \to S$ be a universally closed morphism. Let $\mathcal{L}$ be an $f$-ample invertible $\mathcal{O}_ X$-module. Then the canonical morphism

\[ r : X \longrightarrow \underline{\text{Proj}}_ S \left( \bigoplus \nolimits _{d \geq 0} f_*\mathcal{L}^{\otimes d} \right) \]

of Lemma 29.37.4 is an isomorphism.

Proof. Observe that $f$ is quasi-compact because the existence of an $f$-ample invertible module forces $f$ to be quasi-compact. By the lemma cited the morphism $r$ is an open immersion. On the other hand, the image of $r$ is closed by Lemma 29.41.7 (the target of $r$ is separated over $S$ by Constructions, Lemma 27.16.9). Finally, the image of $r$ is dense by Properties, Lemma 28.26.11 (here we also use that it was shown in the proof of Lemma 29.37.4 that the morphism $r$ over affine opens of $S$ is given by the canonical morphism of Properties, Lemma 28.26.9). Thus we conclude that $r$ is a surjective open immersion, i.e., an isomorphism. $\square$

Lemma 29.43.18. Let $f : X \to S$ be a universally closed morphism. Let $\mathcal{L}$ be an $f$-ample invertible $\mathcal{O}_ X$-module. Let $s \in \Gamma (X, \mathcal{L})$. Then $X_ s \to S$ is an affine morphism.

Proof. The question is local on $S$ (Lemma 29.11.3) hence we may assume $S$ is affine. By Lemma 29.43.17 we can write $X = \text{Proj}(A)$ where $A$ is a graded ring and $s$ corresponds to $f \in A_1$ and $X_ s = D_+(f)$ (Properties, Lemma 28.26.9) which proves the lemma by construction of $\text{Proj}(A)$, see Constructions, Section 27.8. $\square$

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