Remark 60.6.4. Let $A \to B$ be a ring map and let $(J, \delta )$ be a divided power structure on $B$. The universal module $\Omega _{B/A, \delta }$ comes with a little bit of extra structure, namely the $B$-submodule $N$ of $\Omega _{B/A, \delta }$ generated by $\text{d}_{B/A, \delta }(J)$. In terms of the isomorphism given in Lemma 60.6.3 this corresponds to the image of $K \cap J(1)$ in $\Omega _{B/A, \delta }$. Consider the $A$-algebra $D = B \oplus \Omega ^1_{B/A, \delta }$ with ideal $\bar J = J \oplus N$ and divided powers $\bar\delta$ as in the proof of the lemma. Then $(D, \bar J, \bar\delta )$ is a divided power ring and the two maps $B \to D$ given by $b \mapsto b$ and $b \mapsto b + \text{d}_{B/A, \delta }(b)$ are homomorphisms of divided power rings over $A$. Moreover, $N$ is the smallest submodule of $\Omega _{B/A, \delta }$ such that this is true.

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