# stacks/stacks-project

Projective space bundle formula

 @@ -1595,6 +1595,48 @@ \section{The spectral sequence for a smooth morphism} \section{Projective space bundle formula} \label{section-projective-space-bundle-formula} \noindent The title says it all. \begin{proposition} \label{proposition-projective-space-bundle-formula} 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 $$\bigoplus\nolimits_{i = 0, \ldots, r - 1} H^*_{dR}(X/S) \longrightarrow H^*_{dR}(P/S), \quad (a_0, \ldots, a_{r - 1}) \longmapsto \sum p^*(a_i) \cup c_1^{dR}(\mathcal{O}_P(1))^i$$ is an isomorphism. \end{proposition} \begin{proof} 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. \end{proof} \section{Comparing sheaves of differential forms} \label{section-quasi-finite-syntomic}