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Projective space bundle formula

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aisejohan committed Oct 1, 2019
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@@ -1595,6 +1595,48 @@ \section{The spectral sequence for a smooth morphism}

\section{Projective space bundle formula}

The title says it all.

Let $X \to S$ be a morphism of schemes. Let $\mathcal{E}$ be a locally
free $\mathcal{O}_X$-module of constant rank $r$. Consider the morphism
$p : P = \mathbf{P}(\mathcal{E}) \to X$.
Then the map
\bigoplus\nolimits_{i = 0, \ldots, r - 1} H^*_{dR}(X/S)
H^*_{dR}(P/S), \quad
(a_0, \ldots, a_{r - 1}) \longmapsto
\sum p^*(a_i) \cup c_1^{dR}(\mathcal{O}_P(1))^i
is an isomorphism.

Choose an affine open $\Spec(A) \subset X$ such that $\mathcal{E}$ restricts
to the trivial locally free module $\mathcal{O}_{\Spec(A)}^{\oplus r}$.
Then $P \times_X \Spec(A) = \mathbf{P}^{r - 1}_A$. Thus we see that
$p$ is proper and smooth, see Section \ref{section-projective-space}.
Moreover, the classes $c_1^{dR}(\mathcal{O}_P(1))^i$, $i = 0, 1, \ldots, r - 1$
restricted to a fibre $X_y = \mathbf{P}^{r - 1}_y$ freely generate the
de Rham cohomology $H^*_{dR}(X_y/y)$ over $\kappa(y)$, see
Lemma \ref{lemma-de-rham-cohomology-projective-space}. Thus we've verified the
conditions of Proposition \ref{proposition-global-generation-on-fibres}
and we win.

\section{Comparing sheaves of differential forms}

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