The Stacks project

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. $\square$

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