The Stacks project

Remark 41.19.6. The result on “reducedness” does not hold with a weaker definition of étale local ring maps $A \to B$ where one drops the assumption that $B$ is essentially of finite type over $A$. Namely, it can happen that a Noetherian local domain $A$ has nonreduced completion $A^\wedge $, see Examples, Section 110.17. But the ring map $A \to A^\wedge $ is flat, and $\mathfrak m_ AA^\wedge $ is the maximal ideal of $A^\wedge $ and of course $A$ and $A^\wedge $ have the same residue fields. This is why it is important to consider this notion only for ring extensions which are essentially of finite type (or essentially of finite presentation if $A$ is not Noetherian).


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Your email address will not be published.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 039U. Beware of the difference between the letter 'O' and the digit '0'.