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  "body": "\n## Summary\n\nAn étale morphism of rings is a map $R \\to A$ of commutative rings that is both [[smooth|286G0P1B69]] and [[unramified|GY3QA6NH0E]].\nEquivalently, $A$ is finitely presented, flat and unramified over $R$.\nIt is the algebraic analogue of a local diffeomorphism, or of a covering map.\nOver a field $k$, the étale algebras are the finite products of finite separable field extensions of $k$.\nÉtale maps are the basis of étale cohomology and of the Henselization of a local ring.\n\n## Explanations\n\n### L1 human\n\nAn étale map looks, near each point, like an isomorphism to first order, even though it may not be an isomorphism.\nA typical example is adjoining a simple root of a polynomial: $R[x]/(f)$ where the derivative $f'$ is a unit.\nOver $\\Z[\\frac12]$, adjoining a square root of $-1$ is étale, while over $\\Z$ it is not, because of the prime $2$.\nOver a field, an étale algebra is a finite product of separable field extensions, such as $\\Q(\\sqrt 2) \\times \\Q$.\nÉtale maps let one pass to finer \"neighbourhoods\" than the Zariski topology allows.\n",
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