The Stacks project

Lemma 23.8.2. Let $R$ be a regular ring. Let $\mathfrak p \subset R$ be a prime. Let $f_1, \ldots , f_ r \in \mathfrak p$ be a regular sequence. Then the completion of

\[ A = (R/(f_1, \ldots , f_ r))_\mathfrak p = R_\mathfrak p/(f_1, \ldots , f_ r)R_\mathfrak p \]

is a complete intersection in the sense defined above.

Proof. The completion of $A$ is equal to $A^\wedge = R_\mathfrak p^\wedge /(f_1, \ldots , f_ r)R_\mathfrak p^\wedge $ because completion for finite modules over the Noetherian ring $R_\mathfrak p$ is exact (Algebra, Lemma 10.96.1). The image of the sequence $f_1, \ldots , f_ r$ in $R_\mathfrak p$ is a regular sequence by Algebra, Lemmas 10.96.2 and 10.67.5. Moreover, $R_\mathfrak p^\wedge $ is a regular local ring by More on Algebra, Lemma 15.42.4. Hence the result holds by our definition of complete intersection for complete local rings. $\square$


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Your email address will not be published.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 09Q0. Beware of the difference between the letter 'O' and the digit '0'.