The Stacks project

Proposition 10.114.2. A polynomial algebra in $n$ variables over a field is a regular ring. It has global dimension $n$. All localizations at maximal ideals are regular local rings of dimension $n$.

Proof. By Lemma 10.114.1 all localizations $k[x_1, \ldots , x_ n]_{\mathfrak m}$ at maximal ideals are regular local rings of dimension $n$. Hence we conclude by Lemma 10.110.8. $\square$

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