The Stacks project

Example 56.5.1. Let $R$ be a ring. Let $X$ and $Y$ be schemes over $R$ with $X$ quasi-compact and quasi-separated. Let $\mathcal{K}$ be a quasi-coherent $\mathcal{O}_{X \times _ R Y}$-module. Then we can consider the functor
\begin{equation} \label{functors-equation-FM-QCoh} F : \mathit{QCoh}(\mathcal{O}_ X) \longrightarrow \mathit{QCoh}(\mathcal{O}_ Y),\quad \mathcal{F} \longmapsto \text{pr}_{2, *}(\text{pr}_1^*\mathcal{F} \otimes _{\mathcal{O}_{X \times _ R Y}} \mathcal{K}) \end{equation}

The morphism $\text{pr}_2$ is quasi-compact and quasi-separated (Schemes, Lemmas 26.19.3 and 26.21.12). Hence pushforward along this morphism preserves quasi-coherent modules, see Schemes, Lemma 26.24.1. Moreover, our functor is $R$-linear and commutes with arbitrary direct sums, see Cohomology of Schemes, Lemma 30.6.1.

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