Algebraic Normalization

Algebraic normalization of a reduced algebraic variety replaces its affine coordinate rings by their integral closures in the total rings of fractions. The resulting morphism is integral and birational on each irreducible component, and is a disjoint union of normal varieties (Stacks Project). The same construction applies to a reduced scheme with finitely many irreducible components, over a field or more generally. For each affine scheme open in , the inverse image is , where is the integral closure of in its total ring of fractions. These affine pieces glue together, and is an integral morphism (Stacks Project, Tag 035E).

For the cusp , its coordinate ring has normalization , with map .

Indeed, is integral over because it satisfies the monic polynomial with coefficients in . Also, belongs to the field of fractions of , so this field of fractions is . The polynomial ring is integrally closed and is an integral extension of . Any element of integral over is also integral over and therefore belongs to . This proves that is the integral closure of in their common field of fractions.

Normalization separates geometric branches of a nodal curve. It does not generally remove singularities in higher dimensions, since normal varieties can be singular.