Lemma 63.10.3. Let f : X \to Y be a quasi-finite separated morphism of quasi-compact and quasi-separated schemes. Then the functors Rf_! constructed in Section 63.9 agree with the restriction of the functor f_! : D(X_{\acute{e}tale}, \Lambda ) \to D(Y_{\acute{e}tale}, \Lambda ) constructed in Section 63.7 to their common domains of definition.
Proof. By Zariski's main theorem (More on Morphisms, Lemma 37.43.3) we can find an open immersion j : X \to \overline{X} and a finite morphism \overline{f} : \overline{X} \to Y with f = \overline{f} \circ j. By construction we have Rf_! = R\overline{f}_* \circ j_!. Since \overline{f} is finite, we have R\overline{f}_* = \overline{f}_* by Étale Cohomology, Proposition 59.55.2. The lemma follows because \overline{f}_* \circ j_! = f_! for example by Lemma 63.3.6. \square
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