Definition 29.55.8. Let $X$ be a scheme such that every quasi-compact open has finitely many irreducible components. We define the *weak normalization* of $X$ as the weak normalization

\[ X^\nu \longrightarrow X^{wn} \longrightarrow X \]

of $X$ in the normalization $X^\nu $ of $X$ (Definition 29.54.1). In a formula: $X^{wn} = X^{X^\nu /wn}$.

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