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  "body": "\n## Summary\n\nThe module of Kähler differentials $\\Omega_{A/R}$ of an algebra $A$ over a commutative ring $R$ is the algebraic version of the module of differential $1$-forms.\nIt is generated by symbols $da$ for $a \\in A$, subject only to the rules of differentiation.\nIt is the universal target of [[derivations|3J89REV9WQ]] of $A$ over $R$.\nIt is the basic tool to define [[smooth|286G0P1B69]], [[unramified|GY3QA6NH0E]] and [[étale|FTRW0679E2]] maps of rings without any analysis.\n\n## Explanations\n\n### L1 human\n\nOn a curve or surface given by polynomial equations, one can differentiate polynomial functions without limits, using only the product rule and linearity.\nKähler differentials collect all such formal differentials $df$ in one module.\nFor the polynomial ring $k[x, y]$, every differential is a combination $g \\, dx + h \\, dy$, and $d(x^2 y) = 2xy \\, dx + x^2 \\, dy$.\nOn the curve $y^2 = x^3$, one also has the relation $2y \\, dy = 3x^2 \\, dx$, which comes from differentiating the equation.\nAt the cusp $x = y = 0$ this relation becomes $0 = 0$, and this signals the singular point.\n",
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