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

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

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

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