The Stacks project

Remark 23.3.5. The forgetful functor $(A, I, \gamma ) \mapsto A$ does not commute with colimits. For example, let

\[ \xymatrix{ (B, J, \delta ) \ar[r] & (B'', J'', \delta '') \\ (A, I, \gamma ) \ar[r] \ar[u] & (B', J', \delta ') \ar[u] } \]

be a pushout in the category of divided power rings. Then in general the map $B \otimes _ A B' \to B''$ isn't an isomorphism. (It is always surjective.) An explicit example is given by $(A, I, \gamma ) = (\mathbf{Z}, (0), \emptyset )$, $(B, J, \delta ) = (\mathbf{Z}/4\mathbf{Z}, 2\mathbf{Z}/4\mathbf{Z}, \delta )$, and $(B', J', \delta ') = (\mathbf{Z}/4\mathbf{Z}, 2\mathbf{Z}/4\mathbf{Z}, \delta ')$ where $\delta _2(2) = 2$ and $\delta '_2(2) = 0$. More precisely, using Lemma 23.5.3 we let $\delta $, resp. $\delta '$ be the unique divided power structure on $J$, resp. $J'$ such that $\delta _2 : J \to J$, resp. $\delta '_2 : J' \to J'$ is the map $0 \mapsto 0, 2 \mapsto 2$, resp. $0 \mapsto 0, 2 \mapsto 0$. Then $(B'', J'', \delta '') = (\mathbf{F}_2, (0), \emptyset )$ which doesn't agree with the tensor product. However, note that it is always true that

\[ B''/J'' = B/J \otimes _{A/I} B'/J' \]

as can be seen from the universal property of the pushout by considering maps into divided power algebras of the form $(C, (0), \emptyset )$.

Comments (3)

Comment #7050 by nkym on

is not a divided power structure since .

Comment #7051 by on

Indeed, we should set and so on. A divided power structure does exist because we can apply Lemma 23.5.3 to the map mapping to and to . Thanks for pointing this out! I will fix this the next time I go through all the comments.

There are also:

  • 2 comment(s) on Section 23.3: Divided power rings

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Your email address will not be published.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