The Stacks project

Proof. Let $U \subset X$ be an open subset and let $s \in \mathcal{J}(U)$ be a section. Let $\mathcal{I} \subset \mathcal{O}_ X$ be the quasi-coherent sheaf of ideals defining the reduced induced scheme structure on $X \setminus U$ (see Schemes, Definition 26.12.5). By Cohomology of Schemes, Lemma 30.10.5 the section $s$ corresponds to a map $\sigma : \mathcal{I}^ n \to \mathcal{J}$ for some $n$. As $\mathcal{J}$ is an injective object of $\mathit{QCoh}(\mathcal{O}_ X)$ we can extend $\sigma $ to a map $\tilde s : \mathcal{O}_ X \to \mathcal{J}$. Then $\tilde s$ corresponds to a global section of $\mathcal{J}$ restricting to $s$. $\square$

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

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

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. $\square$

Proof. This follows from Lemma 36.7.4 and Lemma 36.8.2. $\square$