The Stacks project

Remark 87.28.2. Let $A$ be a ring and let $I \subset A$ be an ideal. If $A$ is $I$-adically complete, then the $I$-adic completion $A[x_1, \ldots , x_ r]^\wedge $ of $A[x_1, \ldots , x_ r]$ is the restricted power series ring over $A$ as a ring. However, it is not clear that $A[x_1, \ldots , x_ r]^\wedge $ is $I$-adically complete. We think of the topology on $A\{ x_1, \ldots , x_ r\} $ as the limit topology (which is always complete) whereas we often think of the topology on $A[x_1, \ldots , x_ r]^\wedge $ as the $I$-adic topology (not always complete). If $I$ is finitely generated, then $A\{ x_1, \ldots , x_ r\} = A[x_1, \ldots , x_ r]^\wedge $ as topological rings, see Algebra, Lemma 10.96.3.

Comments (2)

Comment #5913 by Zhenhua Wu on

Can you clarify the definition of completeness of limit topology?

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Your email address will not be published.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 0AL0. Beware of the difference between the letter 'O' and the digit '0'.