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: If is a scheme that is proper over a noetherian base , then there exists a projective -scheme and a surjective -morphism that induces an isomorphism for some dense open The proof here is a standard one. We can first reduce to the case where is irreducible. To start, is noetherian since it is of finite type over a noetherian base. Therefore it has finitely many irreducible components , and we claim that for each there is an irreducible proper -scheme so that has set-theoretic image and is an isomorphism on the open dense subset of . To see this, define to be the scheme-theoretic image of the open immersion Since is set-theoretically noetherian for each , the map is quasi-compact and we may compute this scheme-theoretic image affine-locally on , immediately proving the two claims. If we can produce for each a projective -scheme as in the statement of the theorem, then we can take to be the disjoint union and to be the composition : this map is projective, and an isomorphism over a dense open set of , while is a projective -scheme since it is a finite union of projective -schemes. Since each is proper over , we've completed the reduction to the case irreducible. Next, we will show that can be covered by a finite number of open subsets so that each is quasi-projective over . To do this, we may by quasi-compactness first cover by finitely many affine opens , and then cover the preimage of each in by finitely many affine opens each with a closed immersion in to since is of finite type and therefore quasi-compact. Composing this map with the open immersions and , we see that each is a closed subscheme of an open subscheme of . As is noetherian, every closed subscheme of an open subscheme is also an open subscheme of a closed subscheme, and therefore each is quasi-projective over .