= Normalization of an integral scheme
{title2=$X^\nu\longrightarrow X$}
For an <integral scheme>, take on each <affine chart> the <integral closure> of its coordinate <ring> in its <function field>. <Integral closure> commutes with <localization>, so these <rings> glue to a <normal scheme> with an integral birational morphism to the original <scheme>. This is the normalization. It need not be finite without additional hypotheses; for an algebraic curve of finite type over a <field> it is finite. It is an <isomorphism> exactly when the original <scheme> is normal.
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