Lemma 50.10.4 (0FUH)—The Stacks project

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Part 3: Topics in Scheme Theory
Chapter 50: de Rham Cohomology
Section 50.10: de Rham cohomology of a line bundle
Lemma 50.10.4 (cite)

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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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