Lemma 36.3.8. Let $f : Y \to X$ be a morphism of schemes.
The functor $Lf^*$ sends $D_\mathit{QCoh}(\mathcal{O}_ X)$ into $D_\mathit{QCoh}(\mathcal{O}_ Y)$.
If $X$ and $Y$ are affine and $f$ is given by the ring map $A \to B$, then the diagram
commutes.
Proof. We first prove the diagram
commutes. This is clear from Lemma 36.3.6 and the constructions of the functors in question. To see (1) let $E$ be an object of $D_\mathit{QCoh}(\mathcal{O}_ X)$. To see that $Lf^*E$ has quasi-coherent cohomology sheaves we may work locally on $X$. Note that $Lf^*$ is compatible with restricting to open subschemes. Hence we can assume that $f$ is a morphism of affine schemes as in (2). Then we can apply Lemma 36.3.5 to see that $E$ comes from a complex of $A$-modules. By the commutativity of the first diagram of the proof the same holds for $Lf^*E$ and we conclude (1) is true. $\square$
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