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  "body": "\n## Definition\n\nLet $R$ be a [[commutative ring|0310REJJY8]] and let $A$ be a commutative [[$R$-algebra|DPBJ04JMH8]].\nThen $A$ is *unramified* over $R$, and the structure map $R \\to A$ is an *unramified morphism*, if both conditions hold:\n\n1. $A$ is an $R$-algebra of [[finite type|AC5TFFQT9M]].\n2. The module of [[Kähler differentials|7HCFARHG7D]] $\\Omega_{A/R}$ is zero.\n\n## Remarks\n\nSome authors, following Grothendieck, require $A$ to be [[finitely presented|764GZHTGV8]] rather than only of finite type; the two agree when $R$ is Noetherian.\nCondition 2 alone is called *formally unramified*.\nAn equivalent form, for $A$ of finite type, is a condition on each [[prime ideal|00MHHZ9SNE]] $\\mathfrak q$ of $A$, with $\\mathfrak p = \\mathfrak q \\cap R$.\nOne needs $\\mathfrak p A_{\\mathfrak q} = \\mathfrak q A_{\\mathfrak q}$ in the [[localization|4C1AE84N5N]].\nOne also needs the [[residue field|NJ3APQ8BHH]] of $\\mathfrak q$ to be a finite [[separable extension|0WCX7NB4N7]] of that of $\\mathfrak p$.\n\nExample: let $R = \\Z[\\frac12]$ and $A = R[x]/(x^2 + 1) = \\Z[\\frac12][i]$.\nIt is of finite type, and $\\Omega_{A/R} = A \\, dx / (2x \\, dx) \\cong A/(2i) = 0$, since $2i$ is a unit in $A$.\n\nNon-example for clause 2: for $A = \\Z[i]$ over $\\Z$, one gets $\\Omega_{A/\\Z} \\cong \\Z[i]/(2) \\ne 0$; the prime $2$ ramifies.\nNon-example for clause 1: the algebraic closure $\\overline{\\Q}$ of $\\Q$ has $\\Omega_{\\overline{\\Q}/\\Q} = 0$, because every element is separable algebraic over $\\Q$, but it is not of finite type over $\\Q$.\n",
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