Lemma 67.48.5. Let S be a scheme. Let f : Y \to X be a quasi-compact and quasi-separated morphism of algebraic spaces over S. The factorization f = \nu \circ f', where \nu : X' \to X is the normalization of X in Y is characterized by the following two properties:
the morphism \nu is integral, and
for any factorization f = \pi \circ g, with \pi : Z \to X integral, there exists a commutative diagram
\xymatrix{ Y \ar[d]_{f'} \ar[r]_ g & Z \ar[d]^\pi \\ X' \ar[ru]^ h \ar[r]^\nu & X }
for a unique morphism h : X' \to Z.
Moreover, in (2) the morphism h : X' \to Z is the normalization of Z in Y.
Proof.
Let \mathcal{O}' \subset f_*\mathcal{O}_ Y be the integral closure of \mathcal{O}_ X as in Definition 67.48.3. The morphism \nu is integral by construction, which proves (1). Assume given a factorization f = \pi \circ g with \pi : Z \to X integral as in (2). By Definition 67.45.2 \pi is affine, and hence Z is the relative spectrum of a quasi-coherent sheaf of \mathcal{O}_ X-algebras \mathcal{B}. The morphism g : X \to Z corresponds to a map of \mathcal{O}_ X-algebras \chi : \mathcal{B} \to f_*\mathcal{O}_ Y. Since \mathcal{B}(U) is integral over \mathcal{O}_ X(U) for every affine U étale over X (by Definition 67.45.2) we see from Lemma 67.48.1 that \chi (\mathcal{B}) \subset \mathcal{O}'. By the functoriality of the relative spectrum Lemma 67.20.7 this provides us with a unique morphism h : X' \to Z. We omit the verification that the diagram commutes.
It is clear that (1) and (2) characterize the factorization f = \nu \circ f' since it characterizes it as an initial object in a category. The morphism h in (2) is integral by Lemma 67.45.12. Given a factorization g = \pi ' \circ g' with \pi ' : Z' \to Z integral, we get a factorization f = (\pi \circ \pi ') \circ g' and we get a morphism h' : X' \to Z'. Uniqueness implies that \pi ' \circ h' = h. Hence the characterization (1), (2) applies to the morphism h : X' \to Z which gives the last statement of the lemma.
\square
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