Show that a composition of projective morphisms is projective. Conclude that projective morphisms have the stability properties (a)–(f) of (Ex. 4.8) above.
Hint: use the Segre embedding defined in (I, Ex. 2.14), and show that it gives a closed immersion \(\PP^r \times \PP^s \injects \PP^{r + s + rs}\).
Solution
1. Let\[ X\xrightarrow{f}Y\xrightarrow{g}Z \]be projective morphisms. Choose factorizations\[ X\xhookrightarrow{i}\mathbb P^r_Y\longrightarrow Y \]and\[ Y\xhookrightarrow{j}\mathbb P^s_Z\longrightarrow Z \]with \(i\) and \(j\) closed immersions.
This is exactly the definition of a projective morphism.
2. There is a canonical identification\[ \mathbb P^r_Y \cong \mathbb P^r_Z\times_ZY. \]Under this identification, the morphism\[ \mathbb P^r_Y \longrightarrow \mathbb P^r_Z\times_Z\mathbb P^s_Z \]induced by \(j\) is a closed immersion.
Relative projective space commutes with base change: \[ \mathbb P^r_Y = \mathbb P^r_Z\times_ZY. \]
The displayed morphism is the base change of the closed immersion \[ j:Y\hookrightarrow\mathbb P^s_Z \] along the projection \[ \mathbb P^r_Z\times_Z\mathbb P^s_Z \longrightarrow \mathbb P^s_Z. \] Closed immersions are stable under base change, so it is a closed immersion.
3. The composite\[ X \xhookrightarrow{i} \mathbb P^r_Y \hookrightarrow \mathbb P^r_Z\times_Z\mathbb P^s_Z \]is a closed immersion.
Both arrows are closed immersions, and closed immersions are stable under composition.
4. The relative Segre morphism\[ \operatorname{Seg}: \mathbb P^r_Z\times_Z\mathbb P^s_Z \longrightarrow \mathbb P^{r+s+rs}_Z \]is a closed immersion.
Over \(\Spec\mathbb Z\), the ordinary Segre embedding \[ \mathbb P^r_\mathbb Z\times\mathbb P^s_\mathbb Z \hookrightarrow \mathbb P^{(r+1)(s+1)-1}_\mathbb Z \] is cut out by the \(2\times2\) minors of the matrix of homogeneous coordinates, hence is a closed immersion.
Since \[ (r+1)(s+1)-1=r+s+rs, \] base changing this closed immersion along \(Z\to\Spec\mathbb Z\) gives the displayed relative Segre closed immersion.
5. The composite\[ X\longrightarrow\mathbb P^{r+s+rs}_Z \]is a closed immersion.
4. A composition of closed immersions is a closed immersion.
6. Therefore\[ \boxed{g\circ f:X\longrightarrow Z\text{ is projective}.} \]
Step 5 gives a factorization \[ X\hookrightarrow\mathbb P^{r+s+rs}_Z\longrightarrow Z \] with first arrow a closed immersion. This is precisely the definition of projectivity.
7. Every closed immersion is projective.
If \[ i:X\hookrightarrow Y \] is a closed immersion, use \[ \mathbb P^0_Y\cong Y. \] Then \[ X\xhookrightarrow{i}\mathbb P^0_Y\longrightarrow Y \] is a projective factorization. Thus projective morphisms satisfy property (a) of Hartshorne II.4.8.
8. Projective morphisms are stable under arbitrary base extension.
Let \[ f:X\to Y \] be projective, with factorization \[ X\hookrightarrow\mathbb P^n_Y\to Y, \] and let \(Y'\to Y\) be any morphism.
After base change, \[ X\times_YY' \hookrightarrow \mathbb P^n_Y\times_YY' \cong \mathbb P^n_{Y'} \longrightarrow Y'. \] The first arrow remains a closed immersion. Hence the base-changed morphism is projective. Thus property (c) of II.4.8 holds.
9. Projective morphisms satisfy properties (a)–(c) of Hartshorne II.4.8.
Property (a) is 7, property (b) is the composition result 6, and property (c) is 8.
10. Consequently projective morphisms satisfy all six stability properties (a)–(f) of Hartshorne II.4.8.
Hartshorne II.4.8 proves formally that any class of morphisms satisfying (a)–(c) also satisfies:
- products preserve the property;
- if \(g\circ f\) has the property and \(g\) is separated, then \(f\) has the property;
- passage to reductions preserves the property.
Applying that exercise to projective morphisms using 9 proves (d)–(f) as well.
11. Q.E.D.
Step 6 proves closure under composition, and 10 gives the requested stability package.