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  "body": "\n## Definition\n\nLet $(R, \\mathfrak{m}, k)$ be a [[local ring|MZ02BKPHJA]] with [[residue field|NJ3APQ8BHH]] $k$, and let $k^{s}$ be a [[separable closure|JJ91A1V8NQ]] of $k$.\nA local ring is *strictly Henselian* if it is a [[Henselian local ring|RNXPV3KC5J]] and its residue field is [[separably closed|8B57B7EGK4]].\nA *strict Henselization* of $R$ is a local ring $R^{sh}$ with a [[local homomorphism|P8YTMGEN9R]] $i \\colon R \\to R^{sh}$.\nThe residue field of $R^{sh}$ is identified with $k^{s}$, and the following hold:\n\n1. $R^{sh}$ is strictly Henselian.\n2. The map of residue fields induced by $i$ is the inclusion $k \\subseteq k^{s}$.\n3. Let $S$ be a strictly Henselian local ring with residue field $\\kappa_S$, let $f \\colon R \\to S$ be a local homomorphism, and let $\\sigma \\colon k^{s} \\to \\kappa_S$ be a $k$-embedding.\nThen there is exactly one local homomorphism $g \\colon R^{sh} \\to S$ with $g \\circ i = f$ that induces $\\sigma$ on residue fields.\n\n## Remarks\n\nA strict Henselization exists and is unique up to a unique isomorphism once $k^{s}$ is fixed.\nIt is a direct limit of localizations of [[étale|FTRW0679E2]] $R$-algebras, and the map $i$ is [[faithfully flat|C31PYYPRGC]].\nThe map $i$ factors through the [[Henselization|917PGHVZPM]] of $R$.\n\nExample: for a field $k$, the field $k^{s}$ with the inclusion $k \\to k^{s}$ is a strict Henselization.\nA field is a Henselian local ring with maximal ideal $0$, its residue field is itself, and $k^{s}$ is separably closed.\n\nNon-example: the ring $\\Z_p$ of $p$-adic integers with the identity map fails clause 1.\nIt is Henselian, but its residue field $\\F_p$ is not separably closed.\n",
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