Lemma 36.3.8 (08DW)—The Stacks project

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Table of contents
Part 2: Schemes
Chapter 36: Derived Categories of Schemes
Section 36.3: Derived category of quasi-coherent modules
Lemma 36.3.8 (cite)

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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.

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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