Chow's lemma

Chow's lemma, named after Wei-Liang Chow, is one of the foundational results in algebraic geometry.

More precisely, a version of it states the following:[1] The proof here is a standard one.

Therefore it has finitely many irreducible components

and is an isomorphism on the open dense subset

to be the scheme-theoretic image of the open immersion Since

is quasi-compact and we may compute this scheme-theoretic image affine-locally on

, immediately proving the two claims.

: this map is projective, and an isomorphism over a dense open set of

-scheme since it is a finite union of projective

, we've completed the reduction to the case

can be covered by a finite number of open subsets

by finitely many affine opens

is of finite type and therefore quasi-compact.

Composing this map with the open immersions

denote the canonical open immersion, we define

is noetherian, we can apply the same logic as before to see that we can swap the order of the open and closed immersions.

satisfy the conclusion of the theorem.

is surjective, we first note that it is proper and therefore closed.

As its image contains the dense open set

We define the following four families of open subschemes: As the

as a map of topological spaces.

by its reduction, which has the same underlying topological space, we have that the two morphisms

are both extensions of the underlying map of topological space

, so by the reduced-to-separated lemma they must be equal as

is separated, the graph morphism

must also factor through this graph by construction of the scheme-theoretic image.

By the definition of the fiber product, it suffices to prove that

, so the desired conclusion follows from the definition of

In the statement of Chow's lemma, if

is reduced, irreducible, or integral, we can assume that the same holds for