The Stacks project

Lemma 50.3.1. Let $X \to S$ be a morphism of affine schemes given by the ring map $R \to A$. Then $R\Gamma (X, \Omega ^\bullet _{X/S}) = \Omega ^\bullet _{A/R}$ in $D(R)$ and $H^ i_{dR}(X/S) = H^ i(\Omega ^\bullet _{A/R})$.

Proof. This follows from Cohomology of Schemes, Lemma 30.2.2 and Leray's acyclicity lemma (Derived Categories, Lemma 13.16.7). $\square$

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