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

1. 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 $n \geq 1$.

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

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

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

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

Proof. Proof of (1). This is clear. Assume (2)(a). Define $\bar\gamma _ n(x \bmod I') = \gamma _ n(x) \bmod I'$ for $x \in I$. This is well defined since $\gamma _ n(x + y) = \gamma _ n(x) \bmod I'$ for $y \in I'$ by Definition 23.2.1 (4) and the fact that $\gamma _ j(y) \in I'$ by assumption. It is clear that $\bar\gamma$ is a divided power structure as $\gamma$ is one. Hence (2)(b) holds. Also, (2)(b) implies (2)(a) by part (1). It is clear that (2)(a) implies (2)(c). Assume (2)(c). Note that $\gamma _ n(x) = a^ n\gamma _ n(x_ i) \in I'$ for $x = ax_ i$. Hence we see that $\gamma _ n(x) \in I'$ for a set of generators of $I'$ as an abelian group. By induction on the length of an expression in terms of these, it suffices to prove $\forall n : \gamma _ n(x + y) \in I'$ if $\forall n : \gamma _ n(x), \gamma _ n(y) \in I'$. This follows immediately from the fourth axiom of a divided power structure. $\square$

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