Lemma 60.8.3. In Situation 60.7.5. Let

be a fibre square in the category of divided power thickenings of $X$ relative to $(S, \mathcal{I}, \gamma )$. If $T_2 \to T$ is flat and $U_2 = T_2 \times _ T U$, then $T_3 = T_1 \times _ T T_2$ (as schemes).

Lemma 60.8.3. In Situation 60.7.5. Let

\[ \xymatrix{ (U_3, T_3, \delta _3) \ar[d] \ar[r] & (U_2, T_2, \delta _2) \ar[d] \\ (U_1, T_1, \delta _1) \ar[r] & (U, T, \delta ) } \]

be a fibre square in the category of divided power thickenings of $X$ relative to $(S, \mathcal{I}, \gamma )$. If $T_2 \to T$ is flat and $U_2 = T_2 \times _ T U$, then $T_3 = T_1 \times _ T T_2$ (as schemes).

**Proof.**
This is true because a divided power structure extends uniquely along a flat ring map. See Divided Power Algebra, Lemma 23.4.2.
$\square$

Your email address will not be published.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.

## Comments (2)

Comment #2313 by Daxin Xu on

Comment #2389 by Johan on

There are also: