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. $\square$

Comment #2976 by Dario Weißmann on

Maybe we could clarify the statement: $d$ is the dimension of $R$.

Comment #3528 by Jonas Ehrhard on

Defining the map $\kappa[X_1,\dots,X_n] \rightarrow \bigoplus_n \mathfrak m^n / \mathfrak m^{n+1}$ it should be "which maps $X_i$ to the class of $x_i$" instead of the other way round.

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