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  "body": "\n## Summary\n\nA map of commutative rings $R \\to A$ is formally smooth when it has a lifting property.\nEvery map from $A$ to a quotient $C/J$ by an ideal $J$ with $J^2 = 0$ must lift to a map to $C$.\nThis is the algebraic form of an idea from geometry.\nOn a smooth space, there is no obstruction to extending a point to a small thickening.\nTogether with finite presentation it gives the notion of a [[smooth morphism of rings|286G0P1B69]], and with uniqueness of the lift it gives formally étale maps.\n\n## Explanations\n\n### L1 human\n\nThink of a map $A \\to C/J$ as a point of the space of $A$ with coordinates in $C/J$.\nSince $J^2 = 0$, the ring $C$ is only a little bigger than $C/J$, like adding a first-order neighbourhood.\nFormal smoothness says that every such point extends to $C$.\nA polynomial ring has this property, because one may lift the images of the variables freely.\nThe ring $k[x]/(x^2)$ does not: the point $x \\mapsto x$ of $k[x]/(x^2)$ does not extend to $k[x]/(x^3)$, since no lift squares to $0$.\n",
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