Lemma 33.33.5. Let $k$ be a field. Let $f : Y \to X$ be a morphism of proper schemes over $k$. Let $\mathcal{G}$ be a coherent $\mathcal{O}_ Y$-module. Then

$\chi (Y, \mathcal{G}) = \sum (-1)^ i \chi (X, R^ if_*\mathcal{G})$

Proof. The formula makes sense: the sheaves $R^ if_*\mathcal{G}$ are coherent and only a finite number of them are nonzero, see Cohomology of Schemes, Proposition 30.19.1 and Lemma 30.4.5. By Cohomology, Lemma 20.13.4 there is a spectral sequence with

$E_2^{p, q} = H^ p(X, R^ qf_*\mathcal{G})$

converging to $H^{p + q}(Y, \mathcal{G})$. By finiteness of cohomology on $X$ we see that only a finite number of $E_2^{p, q}$ are nonzero and each $E_2^{p, q}$ is a finite dimensional vector space. It follows that the same is true for $E_ r^{p, q}$ for $r \geq 2$ and that

$\sum (-1)^{p + q} \dim _ k E_ r^{p, q}$

is independent of $r$. Since for $r$ large enough we have $E_ r^{p, q} = E_\infty ^{p, q}$ and since convergence means there is a filtration on $H^ n(Y, \mathcal{G})$ whose graded pieces are $E_\infty ^{p, q}$ with $p + q = n$ (this is the meaning of convergence of the spectral sequence), we conclude. Compare also with the more general Homology, Lemma 12.24.12. $\square$

Comment #3214 by Junho Won on

(Typo) In the last sentence, should it be $p+q=n$, so if $F^p$ is the filtration on $H^n(Y,\mathcal{G})$ then $Gr^p_F(H^{p+q}(Y,\mathcal{G})) = E^{p,q}_{\infty}$)?

Comment #3316 by on

Fixed this and also added a reference to a general result of this kind in the chapter on homology. See this change.

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