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