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  "body": "\n## Summary\n\nA reduced scheme is a [[scheme|5A35TYZD02]] whose rings of functions have no nonzero [[nilpotent elements|0ZB2JG1M9C]].\nOn a reduced scheme, a function is determined by its values at all points.\nClassical varieties are reduced, while non-reduced schemes record extra infinitesimal structure, such as a double point.\n\n## Explanations\n\n### L1 human\n\nOn a scheme a function can be nonzero and still have value zero at every point.\nThis happens when the function is nilpotent, like $x$ in the ring $k[x]/(x^2)$, where $x^2 = 0$.\nThat scheme is a single point \"thickened\" in one direction, which remembers a tangent direction.\nA reduced scheme is one with no such thickening, like the line $\\operatorname{Spec} k[x]$ itself.\n",
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