Lemma 23.6.7. Let $(A, \text{d}, \gamma )$ be as in Definition 23.6.5. Let $R \to R'$ be a ring map. Then $\text{d}$ and $\gamma$ induce similar structures on $A' = A \otimes _ R R'$ such that $(A', \text{d}, \gamma )$ is as in Definition 23.6.5.

Proof. Observe that $A'_{even} = A_{even} \otimes _ R R'$ and $A'_{even, +} = A_{even, +} \otimes _ R R'$. Hence we are trying to show that the divided powers $\gamma$ extend to $A'_{even}$ (terminology as in Definition 23.4.1). Once we have shown $\gamma$ extends it follows easily that this extension has all the desired properties.

Choose a polynomial $R$-algebra $P$ (on any set of generators) and a surjection of $R$-algebras $P \to R'$. The ring map $A_{even} \to A_{even} \otimes _ R P$ is flat, hence the divided powers $\gamma$ extend to $A_{even} \otimes _ R P$ uniquely by Lemma 23.4.2. Let $J = \mathop{\mathrm{Ker}}(P \to R')$. To show that $\gamma$ extends to $A \otimes _ R R'$ it suffices to show that $I' = \mathop{\mathrm{Ker}}(A_{even, +} \otimes _ R P \to A_{even, +} \otimes _ R R')$ is generated by elements $z$ such that $\gamma _ n(z) \in I'$ for all $n > 0$. This is clear as $I'$ is generated by elements of the form $x \otimes f$ with $x \in A_{even, +}$ and $f \in \mathop{\mathrm{Ker}}(P \to R')$. $\square$

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