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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^*} }

    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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