The Stacks project

Lemma 15.49.7. Let $R$ be a Noetherian ring. Then $R$ is a G-ring if and only if $R_\mathfrak m$ has geometrically regular formal fibres for every maximal ideal $\mathfrak m$ of $R$.

Proof. Assume $R_\mathfrak m \to R_\mathfrak m^\wedge $ is regular for every maximal ideal $\mathfrak m$ of $R$. Let $\mathfrak p$ be a prime of $R$ and choose a maximal ideal $\mathfrak p \subset \mathfrak m$. Since $R_\mathfrak m \to R_\mathfrak m^\wedge $ is faithfully flat we can choose a prime $\mathfrak p'$ if $R_\mathfrak m^\wedge $ lying over $\mathfrak pR_\mathfrak m$. Consider the commutative diagram

\[ \xymatrix{ R_\mathfrak m^\wedge \ar[r] & (R_\mathfrak m^\wedge )_{\mathfrak p'} \ar[r] & (R_\mathfrak m^\wedge )_{\mathfrak p'}^\wedge \\ R_\mathfrak m \ar[u] \ar[r] & R_\mathfrak p \ar[u] \ar[r] & R_\mathfrak p^\wedge \ar[u] } \]

By assumption the ring map $R_\mathfrak m \to R_\mathfrak m^\wedge $ is regular. By Proposition 15.49.6 $(R_\mathfrak m^\wedge )_{\mathfrak p'} \to (R_\mathfrak m^\wedge )_{\mathfrak p'}^\wedge $ is regular. The localization $R_\mathfrak m^\wedge \to (R_\mathfrak m^\wedge )_{\mathfrak p'}$ is regular. Hence $R_\mathfrak m \to (R_\mathfrak m^\wedge )_{\mathfrak p'}^\wedge $ is regular by Lemma 15.40.4. Since it factors through the localization $R_\mathfrak p$, also the ring map $R_\mathfrak p \to (R_\mathfrak m^\wedge )_{\mathfrak p'}^\wedge $ is regular. Thus we may apply Lemma 15.40.7 to see that $R_\mathfrak p \to R_\mathfrak p^\wedge $ is regular. $\square$


Comments (0)


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Your email address will not be published.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 07PT. Beware of the difference between the letter 'O' and the digit '0'.