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  "body": "\n## Summary\n\nThe sheaf of relative differentials $\\Omega_{X/Y}$ of a morphism of schemes $X \\to Y$ is the sheaf version of the module of [[Kähler differentials|7HCFARHG7D]].\nOn an affine piece $\\operatorname{Spec} B \\to \\operatorname{Spec} A$ it is given by the $B$-module $\\Omega_{B/A}$.\nFor a smooth variety it is a vector bundle, the cotangent bundle, whose dual is the tangent bundle.\nIts exterior powers give the sheaves of differential forms, and the top one is the [[canonical bundle|W05VYY4HBR]].\n\n## Explanations\n\n### L1 human\n\nOn a manifold, the cotangent bundle collects the differentials $df$ of functions at every point.\nThe sheaf of differentials does the same on a scheme, using only algebra: $d$ is linear and satisfies the product rule.\nOn the affine plane with coordinates $x, y$, every differential is $g \\, dx + h \\, dy$ with polynomial $g, h$.\nOn a curve defined by $f(x, y) = 0$, there is one extra relation $\\partial_x f \\, dx + \\partial_y f \\, dy = 0$.\nAt smooth points this leaves a module of rank one, the cotangent line of the curve.\n",
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