Lemma 60.5.6 (07HP)—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 60.5.6 (cite)

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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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Comment #208 by Rex on May 16, 2013 at 23:49

Typo: "and a morphsm"

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