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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 *formally smooth* over $R$, and the structure map $R \\to A$ is a *formally smooth map*, if the following holds.\nLet $C$ be a commutative $R$-algebra and let $J \\subseteq C$ be an [[ideal|096Z9MP2ME]] with $J^2 = 0$.\nLet $\\pi \\colon C \\to C/J$ be the [[quotient map|0076HWHGHQ]], and let $u \\colon A \\to C/J$ be an $R$-algebra [[homomorphism|0VTH8KPCG2]].\nThe condition is that for every such $C$, $J$ and $u$ there is an $R$-algebra homomorphism $v \\colon A \\to C$ with $\\pi \\circ v = u$.\n\n## Remarks\n\nThe lift $v$ need not be unique.\nIf the lift always exists and is unique, $A$ is *formally étale* over $R$; if it is always unique, $A$ is *formally unramified* over $R$.\n\nExample: let $A = R[x_i : i \\in I]$ be a [[polynomial ring|0SDC934R52]] in any set of variables.\nGiven $u \\colon A \\to C/J$, choose for each $i$ an element $c_i \\in C$ with $\\pi(c_i) = u(x_i)$, and let $v$ be the $R$-algebra map with $v(x_i) = c_i$.\nThen $\\pi \\circ v$ and $u$ agree on the variables, so they are equal.\n\nNon-example: let $k$ be a field and $A = k[x]/(x^2)$.\nTake $C = k[x]/(x^3)$ and $J = (x^2)/(x^3)$, so $J^2 = 0$ and $C/J = A$, and let $u$ be the identity of $A$.\nA lift $v$ would send $x$ to $x + a x^2$ for some $a \\in k$, and then $v(x)^2 = x^2 \\ne 0$ in $C$, while $v(x^2) = v(0) = 0$.\nSo no lift exists, and $A$ is not formally smooth over $k$.\n",
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