Let $X$ be an algebraic variety over $\mathbb C$ and let $\mathcal F$ be a constructible sheaf on $X$. It is well-known that the Euler characteristic of the cohomology of $\mathcal F$ is equal to the Euler characteristic of cohomology with compact support. What is the right reference for this result? In $l$-adic cohomology this was proved first by Laumon (I think) but over $\mathbb C$ this should have been known before.
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asked Apr 26, 2016 at 9:55
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$\begingroup$ Surely you want some additional hypotheses, after all $\mathcal F=0$ is constructible. $\endgroup$– Donu ArapuraCommented Apr 26, 2016 at 10:35
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$\begingroup$ @Donu $\mathcal F=0$ has vanishing Euler characteristic with or without compact support. $\endgroup$– Dan PetersenCommented Apr 26, 2016 at 11:41
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$\begingroup$ Never mind, it's too early in the morning here. $\endgroup$– Donu ArapuraCommented Apr 26, 2016 at 11:48
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2$\begingroup$ Sullivan, in "Combinatorial invariants of analytic spaces" (1971), comes close to stating this result. To get the result from what Sullivan does, one needs to use that algebraic varieties have a Whitney stratification, and that a tubular neighborhood of a stratum is a fiber bundle whose fiber is a cone over the link. Sullivan's argument shows that the tubular neighborhood has vanishing Euler characteristic, and then the result follows from Mayer-Vietoris. So it does indeed seem plausible that the result over $\mathbb C$ was known to experts before Laumon's result in the $\ell$-adic case. $\endgroup$– Dan PetersenCommented Apr 26, 2016 at 13:24
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The paper you refer to by Gérard Laumon appears to be:
Comparaison de caractéristiques d'Euler-Poincaré en cohomologie l-adique, C. R. Acad. Sci. Paris Sér. I Math. 292 (1981), no. 3, 209–212.
According to the MathSciNet Review of this paper (reference MR0610321) by William E. Lang:
"As a corollary, the author deduces that if $k$ is algebraically closed, and if $X$ is a separated scheme of finite type over $k$, then for all $E_\lambda$-sheaves $\mathcal{F}$ on $X$, $\chi(X,\mathcal{F})=\chi_c(X,\mathcal{F})$. This corollary had been previously proved by A. Grothendieck for curves in all characteristics [ibid., (SGA 5 X), pp. 372–406], and in all dimensions when the characteristic of $k$ is zero."
According to the reference, that was in 1965–1966.
answered Apr 26, 2016 at 12:26