Lemma 76.25.1. Let S be a scheme. Let f : Y \to X be a smooth morphism of algebraic spaces over S. Let \mathcal{A} be a quasi-coherent sheaf of \mathcal{O}_ X-algebras. The integral closure of \mathcal{O}_ Y in f^*\mathcal{A} is equal to f^*\mathcal{A}' where \mathcal{A}' \subset \mathcal{A} is the integral closure of \mathcal{O}_ X in \mathcal{A}.
76.25 Normalization revisited
Normalization commutes with smooth base change.
Proof. By our construction of the integral closure, see Morphisms of Spaces, Definition 67.48.2, this reduces immediately to the case where X and Y are affine. In this case the result is Algebra, Lemma 10.147.4. \square
Lemma 76.25.2 (Normalization commutes with smooth base change). Let S be a scheme. Let
\xymatrix{ Y_2 \ar[r] \ar[d] & Y_1 \ar[d]^ f \\ X_2 \ar[r]^\varphi & X_1 }
be a fibre square of algebraic spaces over S. Assume f is quasi-compact and quasi-separated and \varphi is smooth. Let Y_ i \to X_ i' \to X_ i be the normalization of X_ i in Y_ i. Then X_2' \cong X_2 \times _{X_1} X_1'.
Proof. The base change of the factorization Y_1 \to X_1' \to X_1 to X_2 is a factorization Y_2 \to X_2 \times _{X_1} X_1' \to X_1 and X_2 \times _{X_1} X_1' \to X_1 is integral (Morphisms of Spaces, Lemma 67.45.5). Hence we get a morphism h : X_2' \to X_2 \times _{X_1} X_1' by the universal property of Morphisms of Spaces, Lemma 67.48.5. Observe that X_2' is the relative spectrum of the integral closure of \mathcal{O}_{X_2} in f_{2, *}\mathcal{O}_{Y_2}. If \mathcal{A}' \subset f_{1, *}\mathcal{O}_{Y_1} denotes the integral closure of \mathcal{O}_{X_2}, then X_2 \times _{X_1} X_1' is the relative spectrum of \varphi ^*\mathcal{A}' as the construction of the relative spectrum commutes with arbitrary base change. By Cohomology of Spaces, Lemma 69.11.2 we know that f_{2, *}\mathcal{O}_{Y_2} = \varphi ^*f_{1, *}\mathcal{O}_{Y_1}. Hence the result follows from Lemma 76.25.1. \square
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