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

\[ \xymatrix{ D(B) \ar[r] & D_\mathit{QCoh}(\mathcal{O}_ Y) \\ D(A) \ar[r] \ar[u]^{- \otimes _ A^\mathbf {L} B} & D_\mathit{QCoh}(\mathcal{O}_ X) \ar[u]_{Lf^*} } \]commutes.

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