Lemma 75.12.5. Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of Noetherian algebraic spaces over $S$. 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 the morphism is quasi-compact and quasi-separated (see Morphisms of Spaces, Lemma 67.8.10). Thus $f_*$ preserve quasi-coherence, see Morphisms of Spaces, Lemma 67.11.2. 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 of Spaces, Proposition 66.32.2), 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 it suffices to prove the map induces an isomorphism on cohomology sheaves. Pick any $m \in \mathbf{Z}$. Let $N = N(X, Y, f)$ be as in Lemma 75.6.1. Consider the short exact sequence

\[ 0 \to \sigma _{\geq m - N - 1}\mathcal{I}^\bullet \to \mathcal{I}^\bullet \to \sigma _{\leq m - N - 2}\mathcal{I}^\bullet \to 0 \]

of complexes of quasi-coherent sheaves on $X$. By Lemma 75.6.1 we see that the cohomology sheaves of $Rf_*\sigma _{\leq m - N - 2}\mathcal{I}^\bullet $ are zero in degrees $\geq m - 1$. Thus we see that $R^ mf_*\mathcal{I}^\bullet $ is isomorphic to $R^ mf_*\sigma _{\geq m - N - 1}\mathcal{I}^\bullet $. In other words, we may assume that $\mathcal{I}^\bullet $ is a bounded below complex of injective objects of $\mathit{QCoh}(\mathcal{O}_ X)$. This case follows from Leray's acyclicity lemma (Derived Categories, Lemma 13.16.7) with required vanishing because of Lemma 75.12.4.
$\square$

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