The Stacks project

Lemma 75.12.3. Let $S$ be a scheme. Let $f : X \to Y$ be a flat, quasi-compact, and quasi-separated morphism of algebraic spaces over $S$. If $\mathcal{J}$ is an injective object of $\mathit{QCoh}(\mathcal{O}_ X)$, then $f_*\mathcal{J}$ is an injective object of $\mathit{QCoh}(\mathcal{O}_ Y)$.

Proof. Since $f$ is quasi-compact and quasi-separated, the functor $f_*$ transforms quasi-coherent sheaves into quasi-coherent sheaves (Morphisms of Spaces, Lemma 67.11.2). The functor $f^*$ is a left adjoint to $f_*$ which transforms injections into injections. Hence the result follows from Homology, Lemma 12.29.1 $\square$

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