Lemma 23.6.8. Let $(A, \text{d}, \gamma )$ be as in Definition 23.6.5. Let $d \geq 1$ be an integer. Let $A\langle T \rangle $ be the graded divided power polynomial algebra on $T$ with $\deg (T) = d$ constructed in Example 23.6.2 or 23.6.3. Let $f \in A_{d - 1}$ be an element with $\text{d}(f) = 0$. There exists a unique differential $\text{d}$ on $A\langle T\rangle $ such that $\text{d}(T) = f$ and such that $\text{d}$ is compatible with the divided power structure on $A\langle T \rangle $.

**Proof.**
This is proved by a direct computation which is omitted.
$\square$

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