Normalization

Summary

The normalization of an integral scheme $X$ is a normal scheme $\tilde{X}$ with a morphism $\tilde{X} \to X$. It is built by taking the integral closure of each affine coordinate ring in its fraction field. It is the best normal approximation of $X$: every dominant morphism from a normal integral scheme to $X$ factors through it. For curves it resolves singularities, which makes it a basic tool in the study of singular curves.