The Stacks project

Lemma 56.5.5. Let $R$ be a ring. Let $X$ and $Y$ be schemes over $R$. Assume $X$ is quasi-compact with affine diagonal. Let $F : \mathit{QCoh}(\mathcal{O}_ X) \to \mathit{QCoh}(\mathcal{O}_ Y)$ be an $R$-linear, right exact functor which commutes with arbitrary direct sums. Then we can construct

  1. a quasi-coherent module $\mathcal{K}$ on $X \times _ R Y$, and

  2. a natural transformation $t : F \to F_\mathcal {K}$ where $F_\mathcal {K}$ denotes the functor (

such that $t : F \circ f_* \to F_\mathcal {K} \circ f_*$ is an isomorphism for every morphism $f : X' \to X$ whose source is an affine scheme.

Proof. Consider a morphism $f' : X' \to X$ with $X'$ affine. Since the diagonal of $X$ is affine, we see that $f'$ is an affine morphism (Morphisms, Lemma 29.11.11). Thus $f'_* : \mathit{QCoh}(\mathcal{O}_{X'}) \to \mathit{QCoh}(\mathcal{O}_ X)$ is an $R$-linear exact functor (Cohomology of Schemes, Lemma 30.2.3) which commutes with direct sums (Cohomology of Schemes, Lemma 30.6.1). Thus $F \circ f'_*$ is an $R$-linear, right exact functor which commutes with arbitrary direct sums. Whence $F \circ f'_* = F_{\mathcal{K}'}$ for some $\mathcal{K}'$ on $X' \times _ R Y$ by Lemma 56.5.2. Moreover, given a morphism $f'' : X'' \to X'$ with $X''$ affine we obtain a canonical identification $(f'' \times \text{id}_ Y)^*\mathcal{K}' = \mathcal{K}''$ by the references already given combined with Lemma 56.5.4. These identifications satisfy a cocycle condition given another morphism $f''' : X''' \to X''$ which we leave it to the reader to spell out.

Choose an affine open covering $X = \bigcup _{i = 1, \ldots , n} U_ i$. Since the diagonal of $X$ is affine, we see that the intersections $U_{i_0 \ldots i_ p} = U_{i_0} \cap \ldots \cap U_{i_ p}$ are affine. As above the inclusion morphisms $j_{i_0 \ldots i_ p} : U_{i_0 \ldots i_ p} \to X$ are affine. Denote $\mathcal{K}_{i_0 \ldots i_ p}$ the quasi-coherent module on $U_{i_0 \ldots i_ p} \times _ R Y$ corresponding to $F \circ j_{i_0 \ldots i_ p *}$ as above. By the above we obtain identifications

\[ \mathcal{K}_{i_0 \ldots i_ p} = \mathcal{K}_{i_0 \ldots \hat i_ j \ldots i_ p}|_{U_{i_0 \ldots i_ p} \times _ R Y} \]

which satisfy the usual compatibilites for glueing. In other words, we obtain a unique quasi-coherent module $\mathcal{K}$ on $X \times _ R Y$ whose restriction to $U_{i_0 \ldots i_ p} \times _ R Y$ is $\mathcal{K}_{i_0 \ldots i_ p}$ compatible with the displayed identifications.

Next, we construct the transformation $t$. Given a quasi-coherent $\mathcal{O}_ X$-module $\mathcal{F}$ denote $\mathcal{F}_{i_0 \ldots i_ p}$ the restriction of $\mathcal{F}$ to $U_{i_0 \ldots i_ p}$ and denote $(\text{pr}_1^*\mathcal{F} \otimes \mathcal{K})_{i_0 \ldots i_ p}$ the restriction of $\text{pr}_1^*\mathcal{F} \otimes \mathcal{K}$ to $U_{i_0 \ldots i_ p} \times _ R Y$. Observe that

\begin{align*} F(j_{i_0 \ldots i_ p *}\mathcal{F}_{i_0 \ldots i_ p}) & = \text{pr}_{i_0 \ldots i_ p, 2, *}( \text{pr}_{i_0 \ldots i_ p, 1}^*\mathcal{F}_{i_0 \ldots i_ p} \otimes \mathcal{K}_{i_0 \ldots i_ p}) \\ & = \text{pr}_{i_0 \ldots i_ p, 2, *} (\text{pr}_1^*\mathcal{F} \otimes \mathcal{K})_{i_0 \ldots i_ p} \end{align*}

where $\text{pr}_{i_0 \ldots i_ p, 2} : U_{i_0 \ldots i_ p} \times _ R Y \to Y$ is the projection and similarly for the other projection. Moreover, these identifications are compatible with the displayed identifications in the previous paragraph. Recall, from Cohomology of Schemes, Lemma 30.7.1 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 $R\text{pr}_{2, *}(\text{pr}_1^*\mathcal{F} \otimes \mathcal{K})$. Hence the cohomology sheaf in degree $0$ is $F_\mathcal {K}(\mathcal{F})$. Thus we obtain the desired map $t : F(\mathcal{F}) \to F_\mathcal {K}(\mathcal{F})$ by contemplating the following commutative diagram

\[ \xymatrix{ & F(\mathcal{F}) \ar[r] \ar@{..>}[d] & \bigoplus F(j_{i_0*}\mathcal{F}_{i_0}) \ar[r] \ar[d] & \bigoplus F(j_{i_0i_1*}\mathcal{F}_{i_0i_1}) \ar[d] \\ 0 \ar[r] & F_\mathcal {K}(\mathcal{F}) \ar[r] & \bigoplus \text{pr}_{i_0, 2, *} (\text{pr}_1^*\mathcal{F} \otimes \mathcal{K})_{i_0} \ar[r] & \bigoplus \text{pr}_{i_0i_1, 2, *} (\text{pr}_1^*\mathcal{F} \otimes \mathcal{K})_{i_0i_1} } \]

We obtain the top row by applying $F$ to the (exact) complex $0 \to \mathcal{F} \to \bigoplus j_{i_0*}\mathcal{F}_{i_0} \to \bigoplus j_{i_0i_1*}\mathcal{F}_{i_0i_1}$ (but since $F$ is not exact, the top row is just a complex and not necessarily exact). The solid vertical arrows are the identifications above. This does indeed define the dotted arrow as desired. The arrow is functorial in $\mathcal{F}$; we omit the details.

We still have to prove the final assertion. Let $f : X' \to X$ be as in the statement of the lemma and let $\mathcal{K}'$ be the quasi-coherent module on $X' \times _ R Y$ constructed in the first paragraph of the proof. If the morphism $f : X' \to X$ maps into one of the opens $U_ i$, then the result follows from Lemma 56.5.4 because in this case we know that $\mathcal{K}_ i = \mathcal{K}|_{U_ i \times _ R Y}$ pulls back to $\mathcal{K}$. In general, we obtain an affine open covering $X' = \bigcup U'_ i$ with $U'_ i = f^{-1}(U_ i)$ and we obtain isomorphisms $\mathcal{K}'|_{U'_ i} = f_ i^*\mathcal{K}_ i$ where $f_ i : U'_ i \to U_ i$ is the induced morphism. These morphisms satisfy the compatibility conditions needed to glue to an isomorphism $\mathcal{K}' = f^*\mathcal{K}$ and we conclude. Some details omitted. $\square$

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