The Stacks project

Lemma 56.5.7. Let $R$ be a ring. Let $X$ and $Y$ be schemes over $R$. Assume $X$ is quasi-compact with affine diagonal. There is an equivalence of categories between

  1. the category of $R$-linear exact functors $F : \mathit{QCoh}(\mathcal{O}_ X) \to \mathit{QCoh}(\mathcal{O}_ Y)$ which commute with arbitrary direct sums, and

  2. the full subcategory of $\mathit{QCoh}(\mathcal{O}_{X \times _ R Y})$ consisting of $\mathcal{K}$ such that

    1. $\mathcal{K}$ is flat over $X$,

    2. for $\mathcal{F} \in \mathit{QCoh}(\mathcal{O}_ X)$ we have $R^ q\text{pr}_{2, *}(\text{pr}_1^*\mathcal{F} \otimes _{\mathcal{O}_{X \times _ R Y}} \mathcal{K}) = 0$ for $q > 0$.

given by sending $\mathcal{K}$ to the functor $F$ in (

Proof. Let $\mathcal{K}$ be as in (2). The functor $F$ in ( commutes with direct sums. Since by (1) (a) the modules $\mathcal{K}$ is $X$-flat, we see that given a short exact sequence $0 \to \mathcal{F}_1 \to \mathcal{F}_2 \to \mathcal{F}_3 \to 0$ we obtain a short exact sequence

\[ 0 \to \text{pr}_1^*\mathcal{F}_1 \otimes _{\mathcal{O}_{X \times _ R Y}} \mathcal{K} \to \text{pr}_1^*\mathcal{F}_2 \otimes _{\mathcal{O}_{X \times _ R Y}} \mathcal{K} \to \text{pr}_1^*\mathcal{F}_3 \otimes _{\mathcal{O}_{X \times _ R Y}} \mathcal{K} \to 0 \]

Since by (2)(b) the higher direct image $R^1\text{pr}_{2, *}$ on the first term is zero, we conclude that $0 \to F(\mathcal{F}_1) \to F(\mathcal{F}_2) \to F(\mathcal{F}_3) \to 0$ is exact and we see that $F$ is as in (1).

Let $F$ be as in (1). Let $\mathcal{K}$ and $t : F \to F_\mathcal {K}$ be as in Lemma 56.5.5. By Lemma 56.5.6 we see that $\mathcal{K}$ is flat over $X$. To finish the proof we have to show that $t$ is an isomorphism and the statement on higher direct images. Both of these follow from the fact that the relative Čech complex

\[ \bigoplus \text{pr}_{i_0, 2, *} (\text{pr}_1^*\mathcal{F} \otimes \mathcal{K})_{i_0} \to \bigoplus \text{pr}_{i_0i_1, 2, *} (\text{pr}_1^*\mathcal{F} \otimes \mathcal{K})_{i_0i_1} \to \bigoplus \text{pr}_{i_0i_1i_2, 2, *} (\text{pr}_1^*\mathcal{F} \otimes \mathcal{K})_{i_0i_1i_2} \to \ldots \]

computes $R\text{pr}_{2, *}(\text{pr}_1^*\mathcal{F} \otimes \mathcal{K})$. Please see proof of Lemma 56.5.5 for notation and for the reason why this is so. In the proof of Lemma 56.5.5 we also found that this complex is equal to $F$ applied to the complex

\[ \bigoplus j_{i_0*}\mathcal{F}_{i_0} \to \bigoplus j_{i_0i_1*}\mathcal{F}_{i_0i_1} \to \bigoplus j_{i_0i_1i_2*}\mathcal{F}_{i_0i_1i_2} \to \ldots \]

This complex is exact except in degree zero with cohomology sheaf equal to $\mathcal{F}$. Hence since $F$ is an exact functor we conclude $F = F_\mathcal {K}$ and that (2)(b) holds.

We omit the proof that the construction that sends $F$ to $\mathcal{K}$ is functorial and a quasi-inverse to the functor sending $\mathcal{K}$ to the functor $F_\mathcal {K}$ determined by ( $\square$

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