Lemma 67.48.4. Let S be a scheme. Let f : Y \to X be a quasi-compact and quasi-separated morphism of algebraic spaces over S. Let Y \to X' \to X be the normalization of X in Y.
If W \to X is an étale morphism of algebraic spaces over S, then W \times _ X X' is the normalization of W in W \times _ X Y.
If Y and X are representable, then Y' is representable and is canonically isomorphic to the normalization of the scheme X in the scheme Y as constructed in Morphisms, Section 29.54.
Proof. It is immediate from the construction that the formation of the normalization of X in Y commutes with étale base change, i.e., part (1) holds. On the other hand, if X and Y are schemes, then for U \subset X affine open, f_*\mathcal{O}_ Y(U) = \mathcal{O}_ Y(f^{-1}(U)) and hence \nu ^{-1}(U) is the spectrum of exactly the same ring as we get in the corresponding construction for schemes. \square
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