Lemma 29.53.5 (035J)—The Stacks project

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Table of contents
Part 2: Schemes
Chapter 29: Morphisms of Schemes
Section 29.53: Relative normalization
Lemma 29.53.5 (cite)

numberstags

$\xymatrix{ Y_2 \ar[d]_{f_2} \ar[r] & Y_1 \ar[d]^{f_1} \\ X_2 \ar[r] & X_1 }$

be a commutative diagram of morphisms of schemes. Assume $f_1$ , $f_2$ quasi-compact and quasi-separated. Let $f_ i = \nu _ i \circ f_ i'$ , $i = 1, 2$ be the canonical factorizations, where $\nu _ i : X_ i' \to X_ i$ is the normalization of $X_ i$ in $Y_ i$ . Then there exists a unique arrow $X'_2 \to X'_1$ fitting into a commutative diagram

$\xymatrix{ Y_2 \ar[d]_{f_2'} \ar[r] & Y_1 \ar[d]^{f_1'} \\ X_2' \ar[d]_{\nu _2} \ar[r] & X_1' \ar[d]^{\nu _1} \\ X_2 \ar[r] & X_1 }$

Proof. By Lemmas 29.53.4 (1) and 29.44.6 the base change $X_2 \times _{X_1} X'_1 \to X_2$ is integral. Note that $f_2$ factors through this morphism. Hence we get a unique morphism $X'_2 \to X_2 \times _{X_1} X'_1$ from Lemma 29.53.4 (2). This gives the arrow $X'_2 \to X'_1$ fitting into the commutative diagram and uniqueness follows as well. $\square$

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