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.
$\square$

## Post a comment

Your email address will not be published. Required fields are marked.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like `$\pi$`

). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.

## Comments (0)