The Stacks project

Lemma 29.43.4. 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 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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