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  "body": "\n## Definition\n\nLet $f \\colon X \\to Y$ be a [[morphism of schemes|JXRKH6P1Z4]].\nLet $\\Delta \\colon X \\to X \\times_Y X$ be the diagonal morphism into the [[fiber product|1DZMT4S9FB]].\nThe morphism $\\Delta$ is an immersion, so there is an open subscheme $W \\subseteq X \\times_Y X$ such that $\\Delta$ is a closed immersion of $X$ into $W$.\nLet $\\mathcal{I} \\subseteq \\mathcal{O}_W$ be the ideal sheaf of $\\Delta(X)$ in $W$.\nThe *sheaf of relative differentials* of $X$ over $Y$ is\n$$\\Omega_{X/Y} = \\Delta^*(\\mathcal{I}/\\mathcal{I}^2),$$\na [[quasi-coherent|KPKND0K7W6]] $\\mathcal{O}_X$-module.\n\n## Remarks\n\nIf $U = \\operatorname{Spec} B \\subseteq X$ maps into $V = \\operatorname{Spec} A \\subseteq Y$, then $\\Omega_{X/Y}|_U$ is the quasi-coherent sheaf associated to the module of [[Kähler differentials|7HCFARHG7D]] $\\Omega_{B/A}$.\nThere is a universal derivation $d \\colon \\mathcal{O}_X \\to \\Omega_{X/Y}$, and $\\Omega_{X/Y}$ is generated by the sections $dg$.\nIf $X$ is a variety over a field $k$, one writes $\\Omega_X = \\Omega_{X/k}$, the *cotangent sheaf*.\n\nExample: for $X = \\mathbb{A}^n_k = \\operatorname{Spec} k[x_1, \\dots, x_n]$, $\\Omega_{X/k}$ is free of rank $n$ with basis $dx_1, \\dots, dx_n$.\nThis matches $\\Omega_{B/k}$ for the polynomial ring $B$.\n\nNon-example of local freeness: for the cuspidal curve $X = \\operatorname{Spec} k[x, y]/(y^2 - x^3)$ with $\\operatorname{char} k \\ne 2, 3$, $\\Omega_{X/k}$ is generated by $dx, dy$ with the relation $2y \\, dy = 3x^2 \\, dx$.\nIts fiber at the origin has dimension $2$, while the generic rank is $1$, so $\\Omega_{X/k}$ is not locally free there.\nAn irreducible variety $X$ over a perfect field $k$ is smooth over $k$ if and only if $\\Omega_{X/k}$ is locally free of rank $\\dim X$.\n",
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