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Euler char and proper morphism of spaces

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aisejohan committed Dec 6, 2017
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@@ -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}

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