The Stacks project

Lemma 73.11.2. Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. 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 Morphisms of Spaces, Lemma 65.11.2. Recall that $\mathit{QCoh}(\mathcal{O}_ X)$ is a Grothendieck abelian category (Properties of Spaces, Proposition 64.32.2). 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 on Sites, Lemma 21.19.1. 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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