The Stacks project

Remark 23.3.5. Let

be a pushout in the category of divided power rings (pushouts exist by Lemma 23.3.4). Then

1. $B''/J'' = B/J \otimes _{A/I} B'/J'$,

2. the map $\varphi : B \otimes _ A B' \to B''$ is surjective, and

3. the map $J \otimes _ A B' + B \otimes _ A J' \to J''$ induced by $\varphi $ is surjective.

To see (1) consider maps $(B'', J'', \delta '') \to (C, (0), \emptyset )$ and apply the definition of a pushout. To see (2) consider the image $\tilde B = \varphi (B \otimes _ A B')$ and the ideal $\tilde J = \varphi (J \otimes _ A B' + B \otimes _ A J')$. Then it is clear that $\delta ''_ n(\tilde J) \subset \tilde J$ for all $n \geq 1$ because of the compatibility of $\delta ''$ with $\delta $ and $\delta '$. Hence we must have $\tilde B = B''$ and $\tilde J = J''$ by the universality of the pushout.