Lemma 50.13.1 (0FMP)—The Stacks project

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Table of contents
Part 3: Topics in Scheme Theory
Chapter 50: de Rham Cohomology
Section 50.13: Leray-Hirsch type theorems
Lemma 50.13.1 (cite)

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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

$\bigoplus \nolimits _{i = 1}^ N \mathcal{O}_ Y[-n_ i] \longrightarrow Rf_*\Omega ^\bullet _{X/Y}$

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.) $\square$

The Stacks project

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