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  "body": "\n## Definition\n\nLet $(R, \\mathfrak m)$ be a [[Noetherian|0670T59HGE]] [[local ring|MZ02BKPHJA]] with residue field $k = R/\\mathfrak m$.\nThe *embedding dimension* of $R$ is the [[dimension|03GSZ8RTZJ]] of its [[cotangent space|Z1CGB483J6]]:\n$$\\operatorname{edim} R = \\dim_k \\mathfrak m / \\mathfrak m^2.$$\n\n## Remarks\n\nSince $\\mathfrak m$ is finitely generated, [[Nakayama's lemma|15JHBYR6KH]] shows that $\\operatorname{edim} R$ is the minimal number of generators of the ideal $\\mathfrak m$.\nThe [[dimension theorem for local rings|KW3HY77V6D]] gives $\\dim R \\le \\operatorname{edim} R$, where $\\dim R$ is the [[Krull dimension|MVGBS3J1P7]].\nWhen equality holds, $R$ is a [[regular local ring|63XXTF6B5M]].\n\nExample: for a field $k$, the ring $k[\\![x_1, \\ldots, x_n]\\!]$ has $\\mathfrak m = (x_1, \\ldots, x_n)$, and the classes of the $x_i$ form a basis of $\\mathfrak m/\\mathfrak m^2$, so $\\operatorname{edim} = n$.\nFor the cusp $R = k[x, y]_{(x,y)}/(y^2 - x^3)$, one has $\\operatorname{edim} R = 2$ but $\\dim R = 1$.\n\nNon-example: without the Noetherian hypothesis, $\\dim_k \\mathfrak m / \\mathfrak m^2$ need not count generators of $\\mathfrak m$.\nIn a valuation ring with value group $\\Q$, every element of positive value $q$ is a product of two elements of value $q/2$.\nSo $\\mathfrak m = \\mathfrak m^2$ and $\\mathfrak m / \\mathfrak m^2 = 0$, although $\\mathfrak m \\ne 0$; here $\\mathfrak m$ is not finitely generated.\n",
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