Lemma 36.8.2. Let $f : X \to Y$ be a morphism of Noetherian schemes. Then $f_*$ on quasi-coherent sheaves has a right derived extension $\Phi : D(\mathit{QCoh}(\mathcal{O}_ X)) \to D(\mathit{QCoh}(\mathcal{O}_ Y))$ such that the diagram

$\xymatrix{ D(\mathit{QCoh}(\mathcal{O}_ X)) \ar[d]_{\Phi } \ar[r] & D_\mathit{QCoh}(\mathcal{O}_ X) \ar[d]^{Rf_*} \\ D(\mathit{QCoh}(\mathcal{O}_ Y)) \ar[r] & D_\mathit{QCoh}(\mathcal{O}_ Y) }$

commutes.

Proof. Since $X$ and $Y$ are Noetherian schemes the morphism is quasi-compact and quasi-separated (see Properties, Lemma 28.5.4 and Schemes, Remark 26.21.18). Thus $f_*$ preserve quasi-coherence, see Schemes, Lemma 26.24.1. Next, let $K$ be an object of $D(\mathit{QCoh}(\mathcal{O}_ X))$. Since $\mathit{QCoh}(\mathcal{O}_ X)$ is a Grothendieck abelian category (Properties, Proposition 28.23.4), we can represent $K$ by a K-injective complex $\mathcal{I}^\bullet$ such that each $\mathcal{I}^ n$ is an injective object of $\mathit{QCoh}(\mathcal{O}_ X)$, see Injectives, Theorem 19.12.6. Thus we see that the functor $\Phi$ is defined by setting

$\Phi (K) = f_*\mathcal{I}^\bullet$

where the right hand side is viewed as an object of $D(\mathit{QCoh}(\mathcal{O}_ Y))$. To finish the proof of the lemma it suffices to show that the canonical map

$f_*\mathcal{I}^\bullet \longrightarrow Rf_*\mathcal{I}^\bullet$

is an isomorphism in $D(\mathcal{O}_ Y)$. To see this by Lemma 36.4.2 it suffices to show that $\mathcal{I}^ n$ is right $f_*$-acyclic for all $n \in \mathbf{Z}$. This is true because $\mathcal{I}^ n$ is flasque by Lemma 36.8.1 and flasque modules are right $f_*$-acyclic by Cohomology, Lemma 20.12.5. $\square$

Comment #8132 by Dennis Eriksson on

It says "This is true because f_∗ I^n is flasque", but I think there should be no direct image there.

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