Lemma 60.2.5. Let $(A, I, \gamma )$ be a divided power ring. Let $B$ be an $A$-algebra and $IB \subset J \subset B$ an ideal. Let $x_ i$ be a set of variables. Then

$D_{B[x_ i], \gamma }(JB[x_ i] + (x_ i)) = D_{B, \gamma }(J) \langle x_ i \rangle$

Proof. One possible proof is to deduce this from Lemma 60.2.4 as any relation between $x_ i$ in $B[x_ i]$ is trivial. On the other hand, the lemma follows from the universal property of the divided power polynomial algebra and the universal property of divided power envelopes. $\square$

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