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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 *smooth* over $R$, and the structure map $R \\to A$ is a *smooth morphism*, if both conditions hold:\n\n1. $A$ is a [[finitely presented|764GZHTGV8]] $R$-algebra.\n2. $A$ is [[formally smooth|8KPDNVKX7B]] over $R$.\n\n## Remarks\n\nIf $A$ is smooth over $R$, then $A$ is [[flat|73RYYEVDKA]] over $R$, and the module of [[Kähler differentials|7HCFARHG7D]] $\\Omega_{A/R}$ is a finitely generated projective $A$-module.\n\nExample: the [[polynomial ring|0SDC934R52]] $A = R[x_1, \\ldots, x_n]$ is smooth over $R$.\nIt is finitely presented, with no relations, and it is formally smooth, because a lift is obtained by lifting the images of the variables.\n\nNon-example: let $k$ be a field and $A = k[x_1, x_2, x_3, \\ldots]$ in infinitely many variables.\nIt is formally smooth over $k$ by the same argument, but it is not finitely generated, so clause 1 fails.\n\nAnother non-example: $A = k[x, y]/(y^2 - x^3)$ is finitely presented over $k$ but not smooth.\nHere $\\Omega_{A/k}$ is generated by $dx, dy$ with the one relation $2y \\, dy - 3x^2 \\, dx = 0$.\nAt the maximal ideal $\\mathfrak m = (x, y)$, this relation vanishes modulo $\\mathfrak m$, so $\\Omega_{A/k} \\otimes_A A/\\mathfrak m$ has dimension $2$.\nA smooth algebra over a field has differentials that are locally free of rank equal to the local [[Krull dimension|MVGBS3J1P7]], which is $1$ here.\n",
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