Lemma 50.13.1. Let $f : X \to Y$ be a smooth proper morphism of schemes. Let $N$ and $n_1, \ldots , n_ N \geq 0$ be integers and let $\xi _ i \in H^{n_ i}_{dR}(X/Y)$, $1 \leq i \leq N$. Assume for all points $y \in Y$ the images of $\xi _1, \ldots , \xi _ N$ in $H^*_{dR}(X_ y/y)$ form a basis over $\kappa (y)$. Then the map
associated to $\xi _1, \ldots , \xi _ N$ is an isomorphism.
Proof. By Lemma 50.3.5 $Rf_*\Omega ^\bullet _{X/Y}$ is a perfect object of $D(\mathcal{O}_ Y)$ whose formation commutes with arbitrary base change. Thus the map of the lemma is a map $a : K \to L$ between perfect objects of $D(\mathcal{O}_ Y)$ whose derived restriction to any point is an isomorphism by our assumption on fibres. Then the cone $C$ on $a$ is a perfect object of $D(\mathcal{O}_ Y)$ (Cohomology, Lemma 20.49.7) whose derived restriction to any point is zero. It follows that $C$ is zero by More on Algebra, Lemma 15.75.7 and $a$ is an isomorphism. (This also uses Derived Categories of Schemes, Lemmas 36.3.5 and 36.10.7 to translate into algebra.) $\square$
Your email address will not be published. Required fields are marked.
In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).
All contributions are licensed under the GNU Free Documentation License.
Comments (0)