The Stacks project

Remark 23.5.2. Let $(A, I, \gamma )$ be a divided power ring. There is a variant of Lemma 23.5.1 for infinitely many variables. First note that if $s < t$ then there is a canonical map

\[ A\langle x_1, \ldots , x_ s \rangle \to A\langle x_1, \ldots , x_ t\rangle \]

Hence if $W$ is any set, then we set

\[ A\langle x_ w: w \in W\rangle = \mathop{\mathrm{colim}}\nolimits _{E \subset W} A\langle x_ e:e \in E\rangle \]

(colimit over $E$ finite subset of $W$) with transition maps as above. By the definition of a colimit we see that the universal mapping property of $A\langle x_ w: w \in W\rangle $ is completely analogous to the mapping property stated in Lemma 23.5.1.


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Your email address will not be published.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 07H6. Beware of the difference between the letter 'O' and the digit '0'.