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Tag 07KD

Lemma 23.4.5. Let $(A, I, \gamma)$ be a divided power ring. Let $p$ be a prime. If $p$ is nilpotent in $A/I$, then

1. the $p$-adic completion $A^\wedge = \mathop{\rm lim}\nolimits_e A/p^eA$ surjects onto $A/I$,
2. the kernel of this map is the $p$-adic completion $I^\wedge$ of $I$, and
3. each $\gamma_n$ is continuous for the $p$-adic topology and extends to $\gamma_n^\wedge : I^\wedge \to I^\wedge$ defining a divided power structure on $I^\wedge$.

If moreover $A$ is a $\mathbf{Z}_{(p)}$-algebra, then

1. (4)    for $e$ large enough the ideal $p^eA \subset I$ is preserved by the divided power structure $\gamma$ and $$(A^\wedge, I^\wedge, \gamma^\wedge) = \mathop{\rm lim}\nolimits_e (A/p^eA, I/p^eA, \bar\gamma)$$ in the category of divided power rings.

Proof. Let $t \geq 1$ be an integer such that $p^tA/I = 0$, i.e., $p^tA \subset I$. The map $A^\wedge \to A/I$ is the composition $A^\wedge \to A/p^tA \to A/I$ which is surjective (for example by Algebra, Lemma 10.95.1). As $p^eI \subset p^eA \cap I \subset p^{e - t}I$ for $e \geq t$ we see that the kernel of the composition $A^\wedge \to A/I$ is the $p$-adic completion of $I$. The map $\gamma_n$ is continuous because $$\gamma_n(x + p^ey) = \sum\nolimits_{i + j = n} p^{je}\gamma_i(x)\gamma_j(y) = \gamma_n(x) \bmod p^eI$$ by the axioms of a divided power structure. It is clear that the axioms for divided power structures are inherited by the maps $\gamma_n^\wedge$ from the maps $\gamma_n$. Finally, to see the last statement say $e > t$. Then $p^eA \subset I$ and $\gamma_1(p^eA) \subset p^eA$ and for $n > 1$ we have $$\gamma_n(p^ea) = p^n \gamma_n(p^{e - 1}a) = \frac{p^n}{n!} p^{n(e - 1)}a^n \in p^e A$$ as $p^n/n! \in \mathbf{Z}_{(p)}$ and as $n \geq 2$ and $e \geq 2$ so $n(e - 1) \geq e$. This proves that $\gamma$ extends to $A/p^eA$, see Lemma 23.4.3. The statement on limits is clear from the construction of limits in the proof of Lemma 23.3.2. $\square$

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\begin{lemma}
\label{lemma-extend-to-completion}
Let $(A, I, \gamma)$ be a divided power ring. Let $p$ be a prime.
If $p$ is nilpotent in $A/I$, then
\begin{enumerate}
\item the $p$-adic completion $A^\wedge = \lim_e A/p^eA$ surjects onto $A/I$,
\item the kernel of this map is the $p$-adic completion $I^\wedge$ of $I$, and
\item each $\gamma_n$ is continuous for the $p$-adic topology and extends
to $\gamma_n^\wedge : I^\wedge \to I^\wedge$ defining a divided power
structure on $I^\wedge$.
\end{enumerate}
If moreover $A$ is a $\mathbf{Z}_{(p)}$-algebra, then
\begin{enumerate}
\item[(4)] for $e$ large enough the ideal $p^eA \subset I$ is preserved by the
divided power structure $\gamma$ and
$$(A^\wedge, I^\wedge, \gamma^\wedge) = \lim_e (A/p^eA, I/p^eA, \bar\gamma)$$
in the category of divided power rings.
\end{enumerate}
\end{lemma}

\begin{proof}
Let $t \geq 1$ be an integer such that $p^tA/I = 0$, i.e., $p^tA \subset I$.
The map $A^\wedge \to A/I$ is the composition $A^\wedge \to A/p^tA \to A/I$
which is surjective (for example by
Algebra, Lemma \ref{algebra-lemma-completion-generalities}).
As $p^eI \subset p^eA \cap I \subset p^{e - t}I$ for $e \geq t$ we see
that the kernel of the composition $A^\wedge \to A/I$ is the $p$-adic
completion of $I$. The map $\gamma_n$ is continuous because
$$\gamma_n(x + p^ey) = \sum\nolimits_{i + j = n} p^{je}\gamma_i(x)\gamma_j(y) = \gamma_n(x) \bmod p^eI$$
by the axioms of a divided power structure. It is clear that the axioms
for divided power structures are inherited by the maps $\gamma_n^\wedge$
from the maps $\gamma_n$. Finally, to see the last statement say $e > t$.
Then $p^eA \subset I$ and $\gamma_1(p^eA) \subset p^eA$ and for $n > 1$
we have
$$\gamma_n(p^ea) = p^n \gamma_n(p^{e - 1}a) = \frac{p^n}{n!} p^{n(e - 1)}a^n \in p^e A$$
as $p^n/n! \in \mathbf{Z}_{(p)}$ and as $n \geq 2$ and $e \geq 2$ so
$n(e - 1) \geq e$.
This proves that $\gamma$ extends to $A/p^eA$, see Lemma \ref{lemma-kernel}.
The statement on limits is clear from the construction of limits in
the proof of Lemma \ref{lemma-limits}.
\end{proof}

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