Chow's lemma

Chow's lemma, named after Wei-Liang Chow, is one of the foundational results in algebraic geometry. It roughly says that a proper morphism is fairly close to being a projective morphism. More precisely, a version of it states the following:^[1]

If ${\displaystyle X}$ is a scheme that is proper over a noetherian base ${\displaystyle S}$, then there exists a projective ${\displaystyle S}$-scheme ${\displaystyle X'}$ and a surjective ${\displaystyle S}$-morphism ${\displaystyle f:X'\to X}$ that induces an isomorphism ${\displaystyle f^{-1}(U)\simeq U}$ for some dense open ${\displaystyle U\subseteq X.}$