The proof here is a standard one.^[2]

Reduction to the case of ${\displaystyle X}$  irreducible edit

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

${\displaystyle X\setminus \cup _{j\neq i}X_{j}\to X.}$ 

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.

${\displaystyle X}$  can be covered by finitely many quasi-projective ${\displaystyle S}$ -schemes edit

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 ${\displaystyle S}$ .

Construction of ${\displaystyle X'}$  and ${\displaystyle f:X'\to X}$  edit

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

${\displaystyle \phi :U\to P=P_{1}\times _{S}\cdots \times _{S}P_{n}}$ 

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

${\displaystyle \psi :U{\stackrel {\psi '}{\to }}X'{\stackrel {h}{\to }}X\times _{S}P}$ 

where ${\displaystyle \psi '}$  is an open immersion and ${\displaystyle h}$  is a closed immersion. Let ${\displaystyle q_{1}:X\times _{S}P\to X}$  and ${\displaystyle q_{2}:X\times _{S}P\to P}$  be the canonical projections. Set

${\displaystyle f:X'{\stackrel {h}{\to }}X\times _{S}P{\stackrel {q_{1}}{\to }}X,}$ 

${\displaystyle g:X'{\stackrel {h}{\to }}X\times _{S}P{\stackrel {q_{2}}{\to }}P.}$ 

We will show that ${\displaystyle X'}$  and ${\displaystyle f}$  satisfy the conclusion of the theorem.

Verification of the claimed properties of ${\displaystyle X'}$  and ${\displaystyle f}$  edit

To show ${\displaystyle f}$  is surjective, we first note that it is proper and therefore closed. As its image contains the dense open set ${\displaystyle U\subset X}$ , we see that ${\displaystyle f}$  must be surjective. It is also straightforward to see that ${\displaystyle f}$  induces an isomorphism on ${\displaystyle U}$ : we may just combine the facts that ${\displaystyle f^{-1}(U)=h^{-1}(U\times _{S}P)}$  and ${\displaystyle \psi }$  is an isomorphism on to its image, as ${\displaystyle \psi }$  factors as the composition of a closed immersion followed by an open immersion ${\displaystyle U\to U\times _{S}P\to X\times _{S}P}$ . It remains to show that ${\displaystyle X'}$  is projective over ${\displaystyle S}$ .

We will do this by showing that ${\displaystyle g:X'\to P}$  is an immersion. We define the following four families of open subschemes:

${\displaystyle V_{i}=\phi _{i}(U_{i})\subset P_{i}}$ 

${\displaystyle W_{i}=p_{i}^{-1}(V_{i})\subset P}$ 

${\displaystyle U_{i}'=f^{-1}(U_{i})\subset X'}$ 

${\displaystyle U_{i}''=g^{-1}(W_{i})\subset X'.}$ 

As the ${\displaystyle U_{i}}$  cover ${\displaystyle X}$ , the ${\displaystyle U_{i}'}$  cover ${\displaystyle X'}$ , and we wish to show that the ${\displaystyle U_{i}''}$  also cover ${\displaystyle X'}$ . We will do this by showing that ${\displaystyle U_{i}'\subset U_{i}''}$  for all ${\displaystyle i}$ . It suffices to show that ${\displaystyle p_{i}\circ g|_{U_{i}'}:U_{i}'\to P_{i}}$  is equal to ${\displaystyle \phi _{i}\circ f|_{U_{i}'}:U_{i}'\to P_{i}}$  as a map of topological spaces. Replacing ${\displaystyle U_{i}'}$  by its reduction, which has the same underlying topological space, we have that the two morphisms ${\displaystyle (U_{i}')_{red}\to P_{i}}$  are both extensions of the underlying map of topological space ${\displaystyle U\to U_{i}\to P_{i}}$ , so by the reduced-to-separated lemma they must be equal as ${\displaystyle U}$  is topologically dense in ${\displaystyle U_{i}}$ . Therefore ${\displaystyle U_{i}'\subset U_{i}''}$  for all ${\displaystyle i}$  and the claim is proven.

The upshot is that the ${\displaystyle W_{i}}$  cover ${\displaystyle g(X')}$ , and we can check that ${\displaystyle g}$  is an immersion by checking that ${\displaystyle g|_{U_{i}''}:U_{i}''\to W_{i}}$  is an immersion for all ${\displaystyle i}$ . For this, consider the morphism

${\displaystyle u_{i}:W_{i}{\stackrel {p_{i}}{\to }}V_{i}{\stackrel {\phi _{i}^{-1}}{\to }}U_{i}\to X.}$ 

Since ${\displaystyle X\to S}$  is separated, the graph morphism ${\displaystyle \Gamma _{u_{i}}:W_{i}\to X\times _{S}W_{i}}$  is a closed immersion and the graph ${\displaystyle T_{i}=\Gamma _{u_{i}}(W_{i})}$  is a closed subscheme of ${\displaystyle X\times _{S}W_{i}}$ ; if we show that ${\displaystyle U\to X\times _{S}W_{i}}$  factors through this graph (where we consider ${\displaystyle U\subset X'}$  via our observation that ${\displaystyle f}$  is an isomorphism over ${\displaystyle f^{-1}(U)}$  from earlier), then the map from ${\displaystyle U_{i}''}$  must also factor through this graph by construction of the scheme-theoretic image. Since the restriction of ${\displaystyle q_{2}}$  to ${\displaystyle T_{i}}$  is an isomorphism onto ${\displaystyle W_{i}}$ , the restriction of ${\displaystyle g}$  to ${\displaystyle U_{i}''}$  will be an immersion into ${\displaystyle W_{i}}$ , and our claim will be proven. Let ${\displaystyle v_{i}}$  be the canonical injection ${\displaystyle U\subset X'\to X\times _{S}W_{i}}$ ; we have to show that there is a morphism ${\displaystyle w_{i}:U\subset X'\to W_{i}}$  so that ${\displaystyle v_{i}=\Gamma _{u_{i}}\circ w_{i}}$ . By the definition of the fiber product, it suffices to prove that ${\displaystyle q_{1}\circ v_{i}=u_{i}\circ q_{2}\circ v_{i}}$ , or by identifying ${\displaystyle U\subset X}$  and ${\displaystyle U\subset X'}$ , that ${\displaystyle q_{1}\circ \psi =u_{i}\circ q_{2}\circ \psi }$ . But ${\displaystyle q_{1}\circ \psi =j}$  and ${\displaystyle q_{2}\circ \psi =\phi }$ , so the desired conclusion follows from the definition of ${\displaystyle \phi :U\to P}$  and ${\displaystyle g}$  is an immersion. Since ${\displaystyle X'\to S}$  is proper, any ${\displaystyle S}$ -morphism out of ${\displaystyle X'}$  is closed, and thus ${\displaystyle g:X'\to P}$  is a closed immersion, so ${\displaystyle X'}$  is projective. ${\displaystyle \blacksquare }$ 

In the statement of Chow's lemma, if ${\displaystyle X}$  is reduced, irreducible, or integral, we can assume that the same holds for ${\displaystyle X'}$ . If both ${\displaystyle X}$  and ${\displaystyle X'}$  are irreducible, then ${\displaystyle f:X'\to X}$  is a birational morphism.^[3]

• Grothendieck, Alexandre; Dieudonné, Jean (1961). "Éléments de géométrie algébrique: II. Étude globale élémentaire de quelques classes de morphismes". Publications Mathématiques de l'IHÉS. 8. doi:10.1007/bf02699291. MR 0217084.
• Hartshorne, Robin (1977), Algebraic Geometry, Graduate Texts in Mathematics, vol. 52, New York: Springer-Verlag, ISBN 978-0-387-90244-9, MR 0463157