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Example 60.25.2. A standard example of the situation above occurs when $B' = B\langle z \rangle $ is the divided power polynomial ring over a divided power ring $(B, J, \delta )$ with divided powers $\delta '$ on $J' = B'_{+} + JB' \subset B'$. Namely, we take $\Omega = \Omega _{B, \delta }$ and $\Omega ' = \Omega _{B', \delta '}$. In this case we can take $a = 1$ and

\[ \theta ( \sum b_ m z^{[m]} ) = \sum b_ m z^{[m + 1]} \]

Note that

\[ f - \theta (\partial _ z(f)) = f(0) \]

equals the constant term. It follows that in this case Lemma 60.25.1 recovers the crystalline Poincaré lemma (Lemma 60.20.2).

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