Lemma 33.15.5. Let $k$ be a field. Let $X$ be a scheme over $k$. If $X_ K$ is projective over $K$ for some field extension $K/k$, then $X$ is projective over $k$.

Proof. A scheme over $k$ is projective over $k$ if and only if it is quasi-projective and proper over $k$. See Morphisms, Lemma 29.43.13. Thus the lemma follows from Lemmas 33.15.3 and 33.15.4. $\square$

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