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  "body": "\n## Summary\n\nA local homomorphism is a [[ring homomorphism|0VTH8KPCG2]] between [[local rings|MZ02BKPHJA]] that sends the maximal ideal into the maximal ideal.\nThese are the natural maps between local rings, and they induce maps of [[residue fields|NJ3APQ8BHH]].\nIn geometry, the map of local rings induced by a morphism of spaces at a point is local.\n\n## Explanations\n\n### L1 human\n\nThe maximal ideal of a local ring consists of the \"functions that vanish at the point\".\nA local homomorphism must send functions vanishing at the point to functions vanishing at the image point.\nFor example, reducing the ring $\\Z_{(p)}$ modulo $p$ to the field $\\Z/p\\Z$ is local.\nBut including $\\Z_{(p)}$ into $\\Q$ is not, since $p$ is not invertible in $\\Z_{(p)}$ but becomes invertible in $\\Q$.\nThe [[definition|RM0FMT0X8W]] states the condition.\n",
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