The Stacks project

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 √Čtale 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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