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  "body": "\n## Summary\n\nAn étale morphism of schemes is a morphism that is flat, unramified and locally of finite presentation.\nIt is the algebraic analogue of a local diffeomorphism, or of a covering map in topology.\nÉtale morphisms are the open sets of the étale topology, which gives étale cohomology and the étale fundamental group.\n\n## Explanations\n\n### L1 human\n\nThe Zariski topology has very few open sets, so the inverse function theorem fails for it.\nÉtale morphisms replace the missing small open sets: they are maps that look locally like an isomorphism on tangent spaces.\nFor example, over a field of characteristic not $2$, the squaring map $t \\mapsto t^2$ from the punctured line to itself is étale.\nIt is a two-sheeted covering, like the map $z \\mapsto z^2$ on the punctured complex plane.\nAt the origin, the full squaring map of the line is not étale, because there the two sheets meet.\n",
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