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
the p-adic completion A^\wedge = \mathop{\mathrm{lim}}\nolimits _ e A/p^ eA surjects onto A/I,
the kernel of this map is the p-adic completion I^\wedge of I, and
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
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{\mathrm{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 10.96.1). 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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