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  "body": "\n## Summary\n\nThe strict Henselization of a [[local ring|MZ02BKPHJA]] $R$ is a [[Henselian local ring|RNXPV3KC5J]] over $R$.\nIts residue field is a separable closure of the residue field of $R$.\nIt is universal among local maps from $R$ to strictly Henselian local rings, that is, Henselian local rings with a separably closed residue field.\nIt refines the [[Henselization|917PGHVZPM]], which keeps the residue field unchanged.\nIn algebraic geometry the strict Henselizations of the local rings of a scheme are the stalks of its structure sheaf for the étale topology.\n\n## Explanations\n\n### L1 human\n\nThe Henselization of a local ring lifts every simple root that exists modulo the maximal ideal.\nThe strict Henselization goes further: it first enlarges the residue field until every separable polynomial has a root there, and then lifts these roots.\nFor a field $k$, the strict Henselization is a separable closure of $k$.\nFor the $p$-adic integers, it is the ring of integers of the maximal unramified extension of the $p$-adic numbers.\nIt is the ring that sees a point of a space together with all its étale neighbourhoods.\n",
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