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
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Comment #7520 by Firmaprim on
Comment #7521 by Johan on