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The Stacks project

Lemma 36.7.2. Let f : X \to Y be a morphism of schemes. Assume f is quasi-compact, quasi-separated, and flat. Then, denoting

\Phi : D(\mathit{QCoh}(\mathcal{O}_ X)) \to D(\mathit{QCoh}(\mathcal{O}_ Y))

the right derived functor of f_* : \mathit{QCoh}(\mathcal{O}_ X) \to \mathit{QCoh}(\mathcal{O}_ Y) we have RQ_ Y \circ Rf_* = \Phi \circ RQ_ X.

Proof. We will prove this by showing that RQ_ Y \circ Rf_* and \Phi \circ RQ_ X are right adjoint to the same functor D(\mathit{QCoh}(\mathcal{O}_ Y)) \to D(\mathcal{O}_ X).

Since f is quasi-compact and quasi-separated, we see that f_* preserves quasi-coherence, see Schemes, Lemma 26.24.1. Recall that \mathit{QCoh}(\mathcal{O}_ X) is a Grothendieck abelian category (Properties, Proposition 28.23.4). Hence any K in D(\mathit{QCoh}(\mathcal{O}_ X)) can be represented by a K-injective complex \mathcal{I}^\bullet of \mathit{QCoh}(\mathcal{O}_ X), see Injectives, Theorem 19.12.6. Then we can define \Phi (K) = f_*\mathcal{I}^\bullet .

Since f is flat, the functor f^* is exact. Hence f^* defines f^* : D(\mathcal{O}_ Y) \to D(\mathcal{O}_ X) and also f^* : D(\mathit{QCoh}(\mathcal{O}_ Y)) \to D(\mathit{QCoh}(\mathcal{O}_ X)). The functor f^* = Lf^* : D(\mathcal{O}_ Y) \to D(\mathcal{O}_ X) is left adjoint to Rf_* : D(\mathcal{O}_ X) \to D(\mathcal{O}_ Y), see Cohomology, Lemma 20.28.1. Similarly, the functor f^* : D(\mathit{QCoh}(\mathcal{O}_ Y)) \to D(\mathit{QCoh}(\mathcal{O}_ X)) is left adjoint to \Phi : D(\mathit{QCoh}(\mathcal{O}_ X)) \to D(\mathit{QCoh}(\mathcal{O}_ Y)) by Derived Categories, Lemma 13.30.3.

Let A be an object of D(\mathit{QCoh}(\mathcal{O}_ Y)) and E an object of D(\mathcal{O}_ X). Then

\begin{align*} \mathop{\mathrm{Hom}}\nolimits _{D(\mathit{QCoh}(\mathcal{O}_ Y))}(A, RQ_ Y(Rf_*E)) & = \mathop{\mathrm{Hom}}\nolimits _{D(\mathcal{O}_ Y)}(A, Rf_*E) \\ & = \mathop{\mathrm{Hom}}\nolimits _{D(\mathcal{O}_ X)}(f^*A, E) \\ & = \mathop{\mathrm{Hom}}\nolimits _{D(\mathit{QCoh}(\mathcal{O}_ X))}(f^*A, RQ_ X(E)) \\ & = \mathop{\mathrm{Hom}}\nolimits _{D(\mathit{QCoh}(\mathcal{O}_ Y))}(A, \Phi (RQ_ X(E))) \end{align*}

This implies what we want. \square


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