The Stacks project

Remark 56.5.8. Let $R$ be a ring. Let $X$ and $Y$ be schemes over $R$. Assume $X$ is quasi-compact with affine diagonal. Lemma 56.5.7 may be generalized as follows: the functors ( associated to quasi-coherent modules on $X \times _ R Y$ are exactly those $F : \mathit{QCoh}(\mathcal{O}_ X) \to \mathit{QCoh}(\mathcal{O}_ Y)$ which have the following properties

  1. $F$ is $R$-linear and commutes with arbitrary direct sums,

  2. $F \circ j_*$ is right exact when $j : U \to X$ is the inclusion of an affine open, and

  3. $0 \to F(\mathcal{F}) \to F(\mathcal{G}) \to F(\mathcal{H})$ is exact whenever $0 \to \mathcal{F} \to \mathcal{G} \to \mathcal{H} \to 0$ is an exact sequence such that for all $x \in X$ the sequence on stalks $0 \to \mathcal{F}_ x \to \mathcal{G}_ x \to \mathcal{H}_ x \to 0$ is a split short exact sequence.

Namely, these assumptions are enough to get construct a transformation $t : F \to F_\mathcal {K}$ as in Lemma 56.5.5 and to show that it is an isomorphism. Moreover, properties (1), (2), and (3) do hold for functors ( If we ever need this we will carefully state and prove this here.

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