The Stacks project

Proof. Recall that $\epsilon $ is flat (Lemma 75.4.1) and hence $\epsilon ^* = L\epsilon ^*$. There is a canonical map from left to right by Cohomology on Sites, Remark 21.35.11. To see this is an isomorphism we can work locally, i.e., we may assume $X$ is an affine scheme.

In case (1) we can represent $E$ by a bounded above complex $\mathcal{E}^\bullet $ of finite free $\mathcal{O}_ X$-modules, see Derived Categories of Schemes, Lemma 36.13.3. We may also represent $F$ by a bounded below complex $\mathcal{F}^\bullet $ of $\mathcal{O}_ X$-modules. Applying Cohomology, Lemma 20.46.11 we see that $R\mathop{\mathcal{H}\! \mathit{om}}\nolimits (E, F)$ is represented by the complex with terms

Applying Cohomology on Sites, Lemma 21.44.10 we see that $R\mathop{\mathcal{H}\! \mathit{om}}\nolimits (\epsilon ^*E, \epsilon ^*F)$ is represented by the complex with terms

Thus the statement of the lemma boils down to the true fact that the canonical map

is an isomorphism for any $\mathcal{O}_ X$-module $\mathcal{F}$ and finite free $\mathcal{O}_ X$-module $\mathcal{E}$.

In case (2) we can represent $E$ by a strictly perfect complex $\mathcal{E}^\bullet $ of $\mathcal{O}_ X$-modules, use Derived Categories of Schemes, Lemmas 36.3.5 and 36.10.7 and the fact that a perfect complex of modules is represented by a finite complex of finite projective modules. Thus we can do the exact same proof as above, replacing the reference to Cohomology, Lemma 20.46.11 by a reference to Cohomology, Lemma 20.46.9. $\square$