# stacks/stacks-project

Euler char and proper morphism of spaces

 @@ -2020,6 +2020,41 @@ \section{Euler characteristics} This immediately implies the lemma. \end{proof} \begin{lemma} \label{lemma-euler-characteristic-morphism} 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 $$\chi(Y, \mathcal{G}) = \sum (-1)^i \chi(X, R^if_*\mathcal{G})$$ \end{lemma} \begin{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 \ref{spaces-cohomology-lemma-proper-pushforward-coherent} and \ref{spaces-cohomology-lemma-vanishing-higher-direct-images}. By Cohomology on Sites, Lemma \ref{sites-cohomology-lemma-Leray} 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. \end{proof}