Lemma 23.7.1. Let $R$ be a ring. Let $(A, \text{d}, \gamma )$ be as in Definition 23.6.5. Let $R' \to R$ be a surjection of rings whose kernel has square zero and is generated by one element $f$. If $A$ is a graded divided power polynomial algebra over $R$ with finitely many variables in each degree, then we obtain a derivation $\theta : A/IA \to A/IA$ where $I$ is the annihilator of $f$ in $R$.

Proof. Since $A$ is a divided power polynomial algebra, we can find a divided power polynomial algebra $A'$ over $R'$ such that $A = A' \otimes _ R R'$. Moreover, we can lift $\text{d}$ to an $R$-linear operator $\text{d}$ on $A'$ such that

1. $\text{d}(xy) = \text{d}(x)y + (-1)^{\deg (x)}x \text{d}(y)$ for $x, y \in A'$ homogeneous, and

2. $\text{d}(\gamma _ n(x)) = \text{d}(x) \gamma _{n - 1}(x)$ for $x \in A'_{even, +}$.

We omit the details (hint: proceed one variable at the time). However, it may not be the case that $\text{d}^2$ is zero on $A'$. It is clear that $\text{d}^2$ maps $A'$ into $fA' \cong A/IA$. Hence $\text{d}^2$ annihilates $fA'$ and factors as a map $A \to A/IA$. Since $\text{d}^2$ is $R$-linear we obtain our map $\theta : A/IA \to A/IA$. The verification of the properties of a derivation is immediate. $\square$

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