Lemma 29.53.7. Let f : Y \to X be a quasi-compact and quasi-separated morphism of schemes. Let X' be the normalization of X in Y. Then the normalization of X' in Y is X'.
Proof. If Y \to X'' \to X' is the normalization of X' in Y, then we can apply Lemma 29.53.4 to the composition X'' \to X to get a canonical morphism h : X' \to X'' over X. We omit the verification that the morphisms h and X'' \to X' are mutually inverse (using uniqueness of the factorization in the lemma). \square
Comments (0)
There are also: