Henselization

Summary

The Henselization of a local ring $R$ is the smallest Henselian local ring that receives a local map from $R$. It is characterized by a universal property among local homomorphisms from $R$ to Henselian local rings. It keeps the residue field of $R$ and, for a Noetherian $R$, it has the same completion as $R$. It is a substitute for the completion that stays algebraic over $R$, and it is used in étale cohomology and in approximation theorems.