Lemma 60.12.5 (07J1)—The Stacks project

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Table of contents
Part 3: Topics in Scheme Theory
Chapter 60: Crystalline Cohomology
Section 60.12: Sheaf of differentials
Lemma 60.12.5 (cite)

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Lemma 60.12.5. In Situation 60.7.5. Let $(U, T, \delta )$ be an object of $\text{Cris}(X/S)$ . Let

$(U(1), T(1), \delta (1)) = (U, T, \delta ) \times (U, T, \delta )$

in $\text{Cris}(X/S)$ . Let $\mathcal{K} \subset \mathcal{O}_{T(1)}$ be the quasi-coherent sheaf of ideals corresponding to the closed immersion $\Delta : T \to T(1)$ . Then $\mathcal{K} \subset \mathcal{J}_{T(1)}$ is preserved by the divided structure on $\mathcal{J}_{T(1)}$ and we have

$(\Omega _{X/S})_ T = \mathcal{K}/\mathcal{K}^{[2]}$

Proof. Note that $U = U(1)$ as $U \to X$ is an open immersion and as (60.9.1.1) commutes with products. Hence we see that $\mathcal{K} \subset \mathcal{J}_{T(1)}$ . Given this fact the lemma follows by working affine locally on $T$ and using Lemmas 60.12.4 and 60.6.5. $\square$

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