Lemma 60.5.7 (07KI)—The Stacks project

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Table of contents
Part 3: Topics in Scheme Theory
Chapter 60: Crystalline Cohomology
Section 60.5: Affine crystalline site
Lemma 60.5.7 (cite)

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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 $\text{Cris}^\wedge (C/A)$ 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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Comment #209 by Rex on May 16, 2013 at 23:50

Typo: "there exists a morphsm"

There are also:

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