Definition 29.53.3. Let $f : Y \to X$ be a quasi-compact and quasi-separated morphism of schemes. Let $\mathcal{O}'$ be the integral closure of $\mathcal{O}_ X$ in $f_*\mathcal{O}_ Y$. The normalization of $X$ in $Y$ is the scheme1

$\nu : X' = \underline{\mathop{\mathrm{Spec}}}_ X(\mathcal{O}') \to X$

over $X$. It comes equipped with a natural factorization

$Y \xrightarrow {f'} X' \xrightarrow {\nu } X$

of the initial morphism $f$.

[1] The scheme $X'$ need not be normal, for example if $Y = X$ and $f = \text{id}_ X$, then $X' = X$.

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