The Stacks project

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 ( 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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