Lemma 33.15.3. Let $k$ be a field. Let $X$ be a scheme over $k$. If $X_ K$ is quasi-projective over $K$ for some field extension $K/k$, then $X$ is quasi-projective over $k$.

Proof. By definition a morphism of schemes $g : Y \to T$ is quasi-projective if it is locally of finite type, quasi-compact, and there exists a $g$-ample invertible sheaf on $Y$. Let $K/k$ be a field extension such that $X_ K$ is quasi-projective over $K$. Let $\mathop{\mathrm{Spec}}(A) \subset X$ be an affine open. Then $U_ K$ is an affine open subscheme of $X_ K$, hence $A_ K$ is a $K$-algebra of finite type. Then $A$ is a $k$-algebra of finite type by Algebra, Lemma 10.126.1. Hence $X \to \mathop{\mathrm{Spec}}(k)$ is locally of finite type. Since $X_ K \to \mathop{\mathrm{Spec}}(K)$ is quasi-compact, we see that $X_ K$ is quasi-compact, hence $X$ is quasi-compact, hence $X \to \mathop{\mathrm{Spec}}(k)$ is of finite type. By Morphisms, Lemma 29.39.4 we see that $X_ K$ has an ample invertible sheaf. Then $X$ has an ample invertible sheaf by Lemma 33.15.1. Hence $X \to \mathop{\mathrm{Spec}}(k)$ is quasi-projective by Morphisms, Lemma 29.39.4. $\square$

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