Lemma 60.5.7. 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 $J$. Let $(D, \bar J, \bar\gamma )$ be the $p$-adic completion of $D_{P, \gamma }(J)$. For every object $(B \to C, \delta )$ of $\text{Cris}^\wedge (C/A)$ there exists a morphism $D \to B$ of $\text{Cris}^\wedge (C/A)$.

**Proof.**
We can find an $A$-algebra homomorphism $P \to B$ compatible with maps to $C$. By our definition of $\text{Cris}(C/A)$ we see that $P \to B$ factors as $P \to D_{P, \gamma }(J) \to B$. As $B$ is $p$-adically complete we can factor this map through $D$.
$\square$

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## Comments (1)

Comment #209 by Rex on

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