Lemma 23.4.3. Let $(A, I, \gamma )$ be a divided power ring.

If $\varphi : (A, I, \gamma ) \to (B, J, \delta )$ is a homomorphism of divided power rings, then $\mathop{\mathrm{Ker}}(\varphi ) \cap I$ is preserved by $\gamma _ n$ for all $n \geq 1$.

Let $\mathfrak a \subset A$ be an ideal and set $I' = I \cap \mathfrak a$. The following are equivalent

$I'$ is preserved by $\gamma _ n$ for all $n > 0$,

$\gamma $ extends to $A/\mathfrak a$, and

there exist a set of generators $x_ i$ of $I'$ as an ideal such that $\gamma _ n(x_ i) \in I'$ for all $n > 0$.

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