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  "body": "\n## Summary\n\nThe cotangent space of a [[local ring|MZ02BKPHJA]] $(R, \\mathfrak m, k)$ is the [[vector space|000AYPA9AJ]] $\\mathfrak m / \\mathfrak m^2$ over the [[residue field|NJ3APQ8BHH]] $k$.\nIts dual is the Zariski tangent space at the point that $R$ describes.\nIts [[dimension|03GSZ8RTZJ]] is the [[embedding dimension|DP3KSCRKFP]] of $R$, which is used to define [[regular local rings|63XXTF6B5M]], that is, smooth points.\n\n## Explanations\n\n### L1 human\n\nNear a point of a curve or a surface, the elements of $\\mathfrak m$ are the functions that vanish at the point.\nTwo such functions have the same image in $\\mathfrak m / \\mathfrak m^2$ when they agree to first order, so the quotient records only their linear parts.\nFor the plane at the origin, the linear parts are spanned by $x$ and $y$, so the cotangent space has dimension $2$.\nThe [[definition|TCRZV021XR]] gives the construction.\n",
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