Lemma 60.6.5 (07HV)—The Stacks project

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Table of contents
Part 3: Topics in Scheme Theory
Chapter 60: Crystalline Cohomology
Section 60.6: Module of differentials
Lemma 60.6.5 (cite)

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Lemma 60.6.5. In Situation 60.5.1. Let $(B, J, \delta )$ be an object of $\text{CRIS}(C/A)$ . Let $(B(1), J(1), \delta (1))$ be the coproduct of $(B, J, \delta )$ with itself in $\text{CRIS}(C/A)$ . Denote $K = \mathop{\mathrm{Ker}}(B(1) \to B)$ . Then $K \cap J(1) \subset J(1)$ is preserved by the divided power structure and

$\Omega _{B/A, \delta } = K/ \left(K^2 + (K \cap J(1))^{[2]}\right)$

canonically.

Proof. Word for word the same as the proof of Lemma 60.6.3. The only point that has to be checked is that the divided power ring $D = B \oplus M$ is an object of $\text{CRIS}(C/A)$ and that the two maps $B \to D$ are morphisms of $\text{CRIS}(C/A)$ . Since $D/(J \oplus M) = B/J$ we can use $C \to B/J$ to view $D$ as an object of $\text{CRIS}(C/A)$ and the statement on morphisms is clear from the construction. $\square$

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