Lemma 10.106.1. Let (R, \mathfrak m, \kappa ) be a regular local ring of dimension d. The graded ring \bigoplus \mathfrak m^ n / \mathfrak m^{n + 1} is isomorphic to the graded polynomial algebra \kappa [X_1, \ldots , X_ d].
Proof. Let x_1, \ldots , x_ d be a minimal set of generators for the maximal ideal \mathfrak m, see Definition 10.60.10. There is a surjection \kappa [X_1, \ldots , X_ d] \to \bigoplus \mathfrak m^ n/\mathfrak m^{n + 1}, which maps X_ i to the class of x_ i in \mathfrak m/\mathfrak m^2. Since d(R) = d by Proposition 10.60.9 we know that the numerical polynomial n \mapsto \dim _\kappa \mathfrak m^ n/\mathfrak m^{n + 1} has degree d - 1. By Lemma 10.58.10 we conclude that the surjection \kappa [X_1, \ldots , X_ d] \to \bigoplus \mathfrak m^ n/\mathfrak m^{n + 1} is an isomorphism. \square
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