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

Note that

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

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