Lemma 50.10.4. With notation as above we have
$\Omega ^ p_{L^\star /S, n} = \Omega ^ p_{L^\star /S, 0} \otimes _{\mathcal{O}_ X} \mathcal{L}^{\otimes n}$ for all $n \in \mathbf{Z}$ as quasi-coherent $\mathcal{O}_ X$-modules,
$\Omega ^\bullet _{X/S} = \Omega ^\bullet _{L/X, 0}$ as complexes, and
for $n > 0$ and $p \geq 0$ we have $\Omega ^ p_{L/X, n} = \Omega ^ p_{L^\star /S, n}$.
Proof. In each case there is a globally defined canonical map which is an isomorphism by local calculations which we omit. $\square$
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