Lemma 50.10.5. In the situation above, assume there is a morphism S \to \mathop{\mathrm{Spec}}(\mathbf{Q}). Then \Omega ^\bullet _{X/S} \to \pi _*\Omega ^\bullet _{L/S} is a quasi-isomorphism and H_{dR}^*(X/S) = H_{dR}^*(L/S).
Proof. Let R be a \mathbf{Q}-algebra. Let A be an R-algebra. The affine local statement is that the map
\Omega ^\bullet _{A/R} \longrightarrow \Omega ^\bullet _{A[t]/R}
is a quasi-isomorphism of complexes of R-modules. In fact it is a homotopy equivalence with homotopy inverse given by the map sending g \omega + g' \text{d}t \wedge \omega ' to g(0)\omega for g, g' \in A[t] and \omega , \omega ' \in \Omega ^\bullet _{A/R}. The homotopy sends g \omega + g' \text{d}t \wedge \omega ' to (\int g') \omega ' were \int g' \in A[t] is the polynomial with vanishing constant term whose derivative with respect to t is g'. Of course, here we use that R contains \mathbf{Q} as \int t^ n = (1/n)t^{n + 1}. \square
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