The Stacks project

Lemma 56.12.3. Let $k$ be a field. Let $X$, $Y$ be finite type schemes over $k$ with $X$ separated. There is an equivalence of categories between

  1. the category of $k$-linear exact functors $F : \textit{Coh}(\mathcal{O}_ X) \to \textit{Coh}(\mathcal{O}_ Y)$, and

  2. the category of coherent $\mathcal{O}_{X \times Y}$-modules $\mathcal{K}$ which are flat over $X$ and have support finite over $Y$

given by sending $\mathcal{K}$ to the restriction of the functor (56.11.1.1) to $\textit{Coh}(\mathcal{O}_ X)$.

Proof. Let $\mathcal{K}$ be as in (2). By Lemma 56.11.6 the functor $F$ given by (56.11.1.1) is exact and $k$-linear. Moreover, $F$ sends $\textit{Coh}(\mathcal{O}_ X)$ into $\textit{Coh}(\mathcal{O}_ Y)$ for example by Cohomology of Schemes, Lemma 30.26.10.

Let us construct the quasi-inverse to the construction. Let $F$ be as in (1). By Lemma 56.12.1 we can extend $F$ to a $k$-linear exact functor on the categories of quasi-coherent modules which commutes with arbitrary direct sums. By Lemma 56.11.6 the extension corresponds to a unique quasi-coherent module $\mathcal{K}$, flat over $X$, such that $R^ q\text{pr}_{2, *}(\text{pr}_1^*\mathcal{F} \otimes _{\mathcal{O}_{X \times Y}} \mathcal{K}) = 0$ for $q > 0$ for all quasi-coherent $\mathcal{O}_ X$-modules $\mathcal{F}$. Since $F(\mathcal{O}_ X)$ is a coherent $\mathcal{O}_ Y$-module, we conclude from Lemma 56.11.8 that $\mathcal{K}$ is coherent.

For a closed point $x \in X$ denote $\mathcal{O}_ x$ the skyscraper sheaf at $x$ with value the residue field of $x$. We have

\[ F(\mathcal{O}_ x) = \text{pr}_{2, *}(\text{pr}_1^*\mathcal{O}_ x \otimes \mathcal{K}) = (x \times Y \to Y)_*(\mathcal{K}|_{x \times Y}) \]

Since $x \times Y \to Y$ is finite, we see that the pushforward along this morphism is faithful. Hence if $y \in Y$ is in the image of the support of $\mathcal{K}|_{x \times Y}$, then $y$ is in the support of $F(\mathcal{O}_ x)$.

Let $Z \subset X \times Y$ be the scheme theoretic support $Z$ of $\mathcal{K}$, see Morphisms, Definition 29.5.5. We first prove that $Z \to Y$ is quasi-finite, by proving that its fibres over closed points are finite. Namely, if the fibre of $Z \to Y$ over a closed point $y \in Y$ has dimension $> 0$, then we can find infinitely many pairwise distinct closed points $x_1, x_2, \ldots $ in the image of $Z_ y \to X$. Since we have a surjection $\mathcal{O}_ X \to \bigoplus _{i = 1, \ldots , n} \mathcal{O}_{x_ i}$ we obtain a surjection

\[ F(\mathcal{O}_ X) \to \bigoplus \nolimits _{i = 1, \ldots , n} F(\mathcal{O}_{x_ i}) \]

By what we said above, the point $y$ is in the support of each of the coherent modules $F(\mathcal{O}_{x_ i})$. Since $F(\mathcal{O}_ X)$ is a coherent module, this will lead to a contradiction because the stalk of $F(\mathcal{O}_ X)$ at $y$ will be generated by $< n$ elements if $n$ is large enough. Hence $Z \to Y$ is quasi-finite. Since $\text{pr}_{2, *}\mathcal{K}$ is coherent and $R^ q\text{pr}_{2, *}\mathcal{K} = 0$ for $q > 0$ we conclude that $Z \to Y$ is finite by Lemma 56.12.2. $\square$


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