The Stacks project

Lemma 70.17.3. Let $k$ be a field. Let $f : Y \to X$ be a morphism of algebraic spaces proper 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 Spaces, Lemmas 67.20.2 and 67.8.1. By Cohomology on Sites, Lemma 21.14.5 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 $p + 1 = n$ (this is the meaning of convergence of the spectral sequence), we conclude. $\square$


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