Lemma 15.40.5. Let $R$ be a ring. Let $(A_ i, \varphi _{ii'})$ be a directed system of smooth $R$-algebras. Set $\Lambda = \mathop{\mathrm{colim}}\nolimits A_ i$. If the fibre rings $\Lambda \otimes _ R \kappa (\mathfrak p)$ are Noetherian for all $\mathfrak p \subset R$, then $R \to \Lambda$ is regular.

Proof. Note that $\Lambda$ is flat over $R$ by Algebra, Lemmas 10.38.3 and 10.135.10. Let $\kappa (\mathfrak p) \subset k$ be a finite purely inseparable extension. Note that

$\Lambda \otimes _ R \kappa (\mathfrak p) \otimes _{\kappa (\mathfrak p)} k = \Lambda \otimes _ R k = \mathop{\mathrm{colim}}\nolimits A_ i \otimes _ R k$

is a colimit of smooth $k$-algebras, see Algebra, Lemma 10.135.4. Since each local ring of a smooth $k$-algebra is regular by Algebra, Lemma 10.138.3 we conclude that all local rings of $\Lambda \otimes _ R k$ are regular by Algebra, Lemma 10.105.8. This proves the lemma. $\square$

## 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Your email address will not be published.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.

In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 07EP. Beware of the difference between the letter 'O' and the digit '0'.