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.

We can first reduce to the case where ${\displaystyle X}$ is irreducible. To start, ${\displaystyle X}$ is noetherian since it is of finite type over a noetherian base. Therefore it has finitely many irreducible components ${\displaystyle X_{i}}$, and we claim that for each ${\displaystyle X_{i}}$ there is an irreducible proper ${\displaystyle S}$-scheme ${\displaystyle Y_{i}}$ so that ${\displaystyle Y_{i}\to X}$ has set-theoretic image ${\displaystyle X_{i}}$ and is an isomorphism on the open dense subset ${\displaystyle X_{i}\setminus \cup _{j\neq i}X_{j}}$ of ${\displaystyle X_{i}}$. To see this, define ${\displaystyle Y_{i}}$ to be the scheme-theoretic image of the open immersion

Since ${\displaystyle X\setminus \cup _{j\neq i}X_{j}}$ is set-theoretically noetherian for each ${\displaystyle i}$, the map ${\displaystyle X\setminus \cup _{j\neq i}X_{j}\to X}$ is quasi-compact and we may compute this scheme-theoretic image affine-locally on ${\displaystyle X}$, immediately proving the two claims. If we can produce for each ${\displaystyle Y_{i}}$ a projective ${\displaystyle S}$-scheme ${\displaystyle Y_{i}'}$ as in the statement of the theorem, then we can take ${\displaystyle X'}$ to be the disjoint union ${\displaystyle \coprod Y_{i}'}$ and ${\displaystyle f}$ to be the composition ${\displaystyle \coprod Y_{i}'\to \coprod Y_{i}\to X}$: this map is projective, and an isomorphism over a dense open set of ${\displaystyle X}$, while ${\displaystyle \coprod Y_{i}'}$ is a projective ${\displaystyle S}$-scheme since it is a finite union of projective ${\displaystyle S}$-schemes. Since each ${\displaystyle Y_{i}}$ is proper over ${\displaystyle S}$, we've completed the reduction to the case ${\displaystyle X}$ irreducible.

Next, we will show that ${\displaystyle X}$ can be covered by a finite number of open subsets ${\displaystyle U_{i}}$ so that each ${\displaystyle U_{i}}$ is quasi-projective over ${\displaystyle S}$. To do this, we may by quasi-compactness first cover ${\displaystyle S}$ by finitely many affine opens ${\displaystyle S_{j}}$, and then cover the preimage of each ${\displaystyle S_{j}}$ in ${\displaystyle X}$ by finitely many affine opens ${\displaystyle X_{jk}}$ each with a closed immersion in to ${\displaystyle \mathbb {A} _{S_{j}}^{n}}$ since ${\displaystyle X\to S}$ is of finite type and therefore quasi-compact. Composing this map with the open immersions ${\displaystyle \mathbb {A} _{S_{j}}^{n}\to \mathbb {P} _{S_{j}}^{n}}$ and ${\displaystyle \mathbb {P} _{S_{j}}^{n}\to \mathbb {P} _{S}^{n}}$, we see that each ${\displaystyle X_{ij}}$ is a closed subscheme of an open subscheme of ${\displaystyle \mathbb {P} _{S}^{n}}$. As ${\displaystyle \mathbb {P} _{S}^{n}}$ is noetherian, every closed subscheme of an open subscheme is also an open subscheme of a closed subscheme, and therefore each ${\displaystyle X_{ij}}$ is quasi-projective over ${\displaystyle S}$.

Now suppose ${\displaystyle \{U_{i}\}}$ is a finite open cover of ${\displaystyle X}$ by quasi-projective ${\displaystyle S}$-schemes, with ${\displaystyle \phi _{i}:U_{i}\to P_{i}}$ an open immersion in to a projective ${\displaystyle S}$-scheme. Set ${\displaystyle U=\cap _{i}U_{i}}$, which is nonempty as ${\displaystyle X}$ is irreducible. The restrictions of the ${\displaystyle \phi _{i}}$ to ${\displaystyle U}$ define a morphism

so that ${\displaystyle U\to U_{i}\to P_{i}=U{\stackrel {\phi }{\to }}P{\stackrel {p_{i}}{\to }}P_{i}}$, where ${\displaystyle U\to U_{i}}$ is the canonical injection and ${\displaystyle p_{i}:P\to P_{i}}$ is the projection. Letting ${\displaystyle j:U\to X}$ denote the canonical open immersion, we define ${\displaystyle \psi =(j,\phi )_{S}:U\to X\times _{S}P}$, which we claim is an immersion. To see this, note that this morphism can be factored as the graph morphism ${\displaystyle U\to U\times _{S}P}$ (which is a closed immersion as ${\displaystyle P\to S}$ is separated) followed by the open immersion ${\displaystyle U\times _{S}P\to X\times _{S}P}$; as ${\displaystyle X\times _{S}P}$ is noetherian, we can apply the same logic as before to see that we can swap the order of the open and closed immersions.

Now let ${\displaystyle X'}$ be the scheme-theoretic image of ${\displaystyle \psi }$, and factor ${\displaystyle \psi }$ as