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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