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:

The proof here is a standard one (cf. EGA II, 5.6.1).

It is easy to reduce to the case when ${\displaystyle X}$ is irreducible, as follows. ${\displaystyle X}$ is noetherian since it is of finite type over a noetherian base. Then it's also topologically noetherian, and consists of a finite number of irreducible components ${\displaystyle X_{i}}$, which are each proper over ${\displaystyle S}$ (because they're closed immersions in the scheme ${\displaystyle X}$ which is proper over ${\displaystyle S}$). If, within each of these irreducible components, there exists a dense open ${\displaystyle U_{i}\subset X_{i}}$, then we can take ${\displaystyle U:=\bigsqcup (U_{i}\setminus \bigcup \limits _{i\neq j}X_{j})}$. It is not hard to see that each of the disjoint pieces are dense in their respective ${\displaystyle X_{i}}$, so the full set ${\displaystyle U}$ is dense in ${\displaystyle X}$. In addition, it's clear that we can similarly find a morphism ${\displaystyle g}$ which satisfies the density condition.

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