The Stacks project

28.41 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 28.41.1. 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 28.41.10. Conversely, quasi-projective morphisms are often compositions of open immersions and projective morphisms, see Lemma 28.41.12. For an overview of properties of projective morphisms over a quasi-projective base, see More on Morphisms, Section 36.45.

Example 28.41.2. 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 26.18.5.

Lemma 28.41.3. 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 26.21.5. $\square$

Lemma 28.41.4. 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 28.41.3 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 27.16.1). 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 26.11.3 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 28.39.3. Hence, by Lemma 28.41.4 above we may assume there exists a closed immersion $X \to \mathbf{P}^ n_ S$. By Lemmas 28.39.4 and 28.39.6 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 28.39.5. By Constructions, Lemma 26.8.8 the scheme $\mathbf{P}^ n_{\mathbf{Z}}$ is separated. By Constructions, Lemma 26.8.9 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 26.8.11 and the valuative criterion (see Schemes, Proposition 25.20.6). $\square$

Lemma 28.41.6. 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 28.37.4. Since $X \to S$ is proper the morphism $i$ is a closed immersion, see Lemma 28.39.7. $\square$

Lemma 28.41.7. 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 26.13.6. The diagram identifies $X$ as a closed subscheme of $\mathbf{P}^{nm + n + m}_ Z$ as desired. $\square$

Lemma 28.41.8. 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 25.18.2. $\square$

Lemma 28.41.9. 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 25.18.2. 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 28.41.5) it is quasi-compact. Hence Lemma 28.36.2 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 28.41.11. 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 28.7.7 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 28.41.12. 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 28.37.5. 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 25.24.1 (quasi-projective morphisms are quasi-compact and separated, see Lemma 28.38.4). By Properties, Lemma 27.22.6 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 28.37.1. 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 28.3.1. 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 25.21.14). By Lemma 28.7.7 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 28.41.13. 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 28.41.10 and proper by Lemma 28.41.5. 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 28.41.12. Since $X \to S$ is proper, we see that $X$ is closed in $X'$ (Lemma 28.39.7), 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 28.41.1) 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 28.41.3 and 28.41.11. $\square$

Lemma 28.41.14. 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 28.41.10 and 28.41.5 we see that $f$ and $g$ are quasi-projective and proper. By Lemmas 28.39.4 and 28.38.3 we see that $g \circ f$ is proper and quasi-projective. Thus $g \circ f$ is projective by Lemma 28.41.13. $\square$

Lemma 28.41.15. 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 an embedding $X \to \mathbf{P}(\mathcal{E})$ 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 25.21.10 (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 28.41.16. 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 27.26.1 and Lemma 27.26.11 and Constructions, Lemma 26.8.8. Hence Lemma 28.41.12 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

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

as desired. In general, let $\mathcal{L}$ be an invertible sheaf on $S$. By Properties, Proposition 27.26.13 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 26.20.1 this finishes the proof. $\square$

Lemma 28.41.17. 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 28.35.4 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 28.39.7 (the target of $r$ is separated over $S$ by Constructions, Lemma 26.16.9). Finally, the image of $r$ is dense by Properties, Lemma 27.26.11 (here we also use that it was shown in the proof of Lemma 28.35.4 that the morphism $r$ over affine opens of $S$ is given by the canonical morphism of Properties, Lemma 27.26.9). Thus we conclude that $r$ is a surjective open immersion, i.e., an isomorphism. $\square$

Lemma 28.41.18. 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 28.11.3) hence we may assume $S$ is affine. By Lemma 28.41.17 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 27.26.9) which proves the lemma by construction of $\text{Proj}(A)$, see Constructions, Section 26.8. $\square$

Comments (0)

Post a comment

Your email address will not be published. Required fields are marked.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.




In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 01W7. Beware of the difference between the letter 'O' and the digit '0'.



View Section 28.41 as pdf