Lemma 29.43.4. Let $f : X \to S$ be a morphism of schemes. The following are equivalent:
The morphism $f$ is locally projective.
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. $\square$
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Your email address will not be published.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).
All contributions are licensed under the GNU Free Documentation License.
Comments (2)
Comment #7520 by Firmaprim on
Comment #7521 by Johan on