The Stacks project

Lemma 36.3.8. Let $f : Y \to X$ be a morphism of schemes.

  1. The functor $Lf^*$ sends $D_\mathit{QCoh}(\mathcal{O}_ X)$ into $D_\mathit{QCoh}(\mathcal{O}_ Y)$.

  2. If $X$ and $Y$ are affine and $f$ is given by the ring map $A \to B$, then the diagram

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


Proof. We first prove the diagram

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

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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