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 n.d.). For the cusp 
, its coordinate ring ![C[t^2,t^3]](/images/equations/AlgebraicNormalization/Inline4.svg)
has normalization ![C[t]](/images/equations/AlgebraicNormalization/Inline5.svg)
, with map 
. Normalization separates branches of a nodal
curve. It does not generally remove singularities in higher dimensions,
since normal varieties can be singular.
Algebraic Normalization
See also
Conductor Ideal, Integral Closure, Normal Variety, Resolution of SingularitiesExplore with Wolfram|Alpha
References
Stacks Project. "Normalization." Tag 035E. https://stacks.math.columbia.edu/tag/035E.Cite this as:
Weisstein, Eric W. "Algebraic Normalization." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/AlgebraicNormalization.html