Lemma 60.5.6. In Situation 60.5.1. Let $P$ be a polynomial algebra over $A$ and let $P \to C$ be a surjection of $A$-algebras with kernel $J$. With $(D_ e, \bar J_ e, \bar\gamma )$ as in Lemma 60.5.5: for every object $(B, J_ B, \delta )$ of $\text{CRIS}(C/A)$ there exists an $e$ and a morphism $D_ e \to B$ of $\text{CRIS}(C/A)$.
Proof. We can find an $A$-algebra homomorphism $P \to B$ lifting the map $C \to B/J_ B$. By our definition of $\text{CRIS}(C/A)$ we see that $p^ eB = 0$ for some $e$ hence $P \to B$ factors as $P \to P_ e \to B$. By the universal property of the divided power envelope we conclude that $P_ e \to B$ factors through $D_ e$. $\square$
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