Lemma 59.6.2. Let $A$ be a ring. Let $(B, J, \delta )$ be a divided power ring and $A \to B$ a ring map.

Consider $B[x]$ with divided power ideal $(JB[x], \delta ')$ where $\delta '$ is the extension of $\delta $ to $B[x]$. Then

\[ \Omega _{B[x]/A, \delta '} = \Omega _{B/A, \delta } \otimes _ B B[x] \oplus B[x]\text{d}x. \]Consider $B\langle x \rangle $ with divided power ideal $(JB\langle x \rangle + B\langle x \rangle _{+}, \delta ')$. Then

\[ \Omega _{B\langle x\rangle /A, \delta '} = \Omega _{B/A, \delta } \otimes _ B B\langle x \rangle \oplus B\langle x\rangle \text{d}x. \]Let $K \subset J$ be an ideal preserved by $\delta _ n$ for all $n > 0$. Set $B' = B/K$ and denote $\delta '$ the induced divided power on $J/K$. Then $\Omega _{B'/A, \delta '}$ is the quotient of $\Omega _{B/A, \delta } \otimes _ B B'$ by the $B'$-submodule generated by $\text{d}k$ for $k \in K$.

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