I saw it written that, for $j : U \rightarrow X$ an open embedding and $\mathcal{F}$ a sheaf of complex vector spaces on $U,$ that $\chi(X, Rj_*\mathcal{F}) = \chi(U, \mathcal{F}).$ I am a little confused. Firstly, I have never seen a source explicitly use the Euler characteristic of a complex--I know that $Rj_*\mathcal{F}$ has hypercohomology sheaves, and so I can imagine we can just take global sections of those sheaves and use the alternating sum of their ranks to comptue $\chi(X, Rj_*\mathcal{F}),$ but perhaps that is the wrong definition. Also, I do not see why this is equal to the usual $\chi(U, \mathcal{F}).$ Can anyone provide a reference or explain this to me? Thanks.
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1I would have defined $\chi(X,Rj_\mathcal{F})$ as the alternative sum of the dimensions of the groups $\mathbb{H}^i(X,Rj_\mathcal{F})$ which makes the statement trivial as $\mathbb{H}^i(X,Rj_*\mathcal{F})=H^i(U,\mathcal{F})$. – Roland Jul 31 '22 at 23:00