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  "body": "\n## Summary\n\nAn unramified morphism of rings is a map $R \\to A$ of commutative rings that makes $A$ a [[finitely generated|AC5TFFQT9M]] $R$-algebra.\nMoreover its module of [[Kähler differentials|7HCFARHG7D]] $\\Omega_{A/R}$ must be zero.\nIt generalizes finite separable field extensions and unramified extensions of rings of integers in number theory.\nLocally, take a prime $\\mathfrak q$ of $A$ and the prime $\\mathfrak p$ of $R$ below it.\nThen $\\mathfrak p$ generates the maximal ideal of $A_{\\mathfrak q}$, and the residue field extension is finite and separable.\nAn unramified map that is also flat and finitely presented is [[étale|FTRW0679E2]].\n\n## Explanations\n\n### L1 human\n\nIn number theory, a prime $p$ ramifies in a larger ring of integers when it becomes a square, or a higher power, of a prime upstairs.\nIn $\\Z[i]$, the prime $2$ ramifies, since $2 = -i(1 + i)^2$.\nAn unramified map is one where nothing like this happens at any prime.\nThe test is short: the differentials of $A$ over $R$ must all vanish.\nFor $\\Z[i] = \\Z[x]/(x^2 + 1)$, the differentials are $\\Z[i]/(2i)$, which is nonzero exactly because of the prime $2$.\n",
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