The Stacks project

Lemma 23.3.2. The category of divided power rings has all limits and they agree with limits in the category of rings.

Proof. The empty limit is the zero ring (that's weird but we need it). The product of a collection of divided power rings $(A_ t, I_ t, \gamma _ t)$, $t \in T$ is given by $(\prod A_ t, \prod I_ t, \gamma )$ where $\gamma _ n((x_ t)) = (\gamma _{t, n}(x_ t))$. The equalizer of $\alpha , \beta : (A, I, \gamma ) \to (B, J, \delta )$ is just $C = \{ a \in A \mid \alpha (a) = \beta (a)\} $ with ideal $C \cap I$ and induced divided powers. It follows that all limits exist, see Categories, Lemma 4.14.11. $\square$

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