The Stacks project

56.5 Functors between categories of quasi-coherent modules

In this section we briefly study functors between categories of quasi-coherent modules.

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

56.5.1.1
\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.

The following lemma is a natural generalization of Lemma 56.3.3.

Lemma 56.5.2. Let $R$ be a ring. Let $X$ and $Y$ be schemes over $R$ with $X$ affine. There is an equivalence of categories between

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

  2. the category $\mathit{QCoh}(\mathcal{O}_{X \times _ R Y})$

given by sending $\mathcal{K}$ to the functor $F$ in (56.5.1.1).

Proof. Let $\mathcal{K}$ be an object of $\mathit{QCoh}(\mathcal{O}_{X \times _ R Y})$ and $F_\mathcal {K}$ the functor (56.5.1.1). By the discussion in Example 56.5.1 we already know that $F$ is $R$-linear and commutes with arbitrary direct sums. Since $\text{pr}_2 : X \times _ R Y \to Y$ is affine (Morphisms, Lemma 29.11.8) the functor $\text{pr}_{2, *}$ is exact, see Cohomology of Schemes, Lemma 30.2.3. Hence $F$ is right exact as well, in other words $F$ is as in (1).

Let $F$ be as in (1). Say $X = \mathop{\mathrm{Spec}}(A)$. Consider the quasi-coherent $\mathcal{O}_ Y$-module $\mathcal{G} = F(\mathcal{O}_ X)$. The functor $F$ induces an $R$-linear map $A \to \text{End}_{\mathcal{O}_ Y}(\mathcal{G})$, $a \mapsto F(a \cdot \text{id})$. Thus $\mathcal{G}$ is a sheaf of modules over

\[ A \otimes _ R \mathcal{O}_ Y = \text{pr}_{2, *}\mathcal{O}_{X \times _ R Y} \]

By Morphisms, Lemma 29.11.6 we find that there is a unique quasi-coherent module $\mathcal{K}$ on $X \times _ R Y$ such that $F(\mathcal{O}_ X) = \mathcal{G} = \text{pr}_{2, *}\mathcal{K}$ compatible with action of $A$ and $\mathcal{O}_ Y$. Denote $F_\mathcal {K}$ the functor given by (56.5.1.1). There is an equivalence $\text{Mod}_ A \to \mathit{QCoh}(\mathcal{O}_ X)$ sending $A$ to $\mathcal{O}_ X$, see Schemes, Lemma 26.7.5. Hence we find an isomorphism $F \cong F_\mathcal {K}$ by Lemma 56.2.6 because we have an isomorphism $F(\mathcal{O}_ X) \cong F_\mathcal {K}(\mathcal{O}_ X)$ compatible with $A$-action by construction.

This shows that the functor sending $\mathcal{K}$ to $F_\mathcal {K}$ is essentially surjective. We omit the verification of fully faithfulness. $\square$

Remark 56.5.3. Below we will use that for an affine morphism $h : T \to S$ we have $h_*\mathcal{G} \otimes _{\mathcal{O}_ S} \mathcal{H} = h_*(\mathcal{G} \otimes _{\mathcal{O}_ T} h^*\mathcal{H})$ for $\mathcal{G} \in \mathit{QCoh}(\mathcal{O}_ T)$ and $\mathcal{H} \in \mathit{QCoh}(\mathcal{O}_ S)$. This follows immediately on translating into algebra.

Lemma 56.5.4. In Lemma 56.5.2 let $F$ correspond to $\mathcal{K}$ in $\mathit{QCoh}(\mathcal{O}_{X \times _ R Y})$. We have

  1. If $f : X' \to X$ is an affine morphism, then $F \circ f_*$ corresponds to $(f \times \text{id}_ Y)^*\mathcal{K}$.

  2. If $g : Y' \to Y$ is a flat morphism, then $g^* \circ F$ corresponds to $(\text{id}_ X \times g)^*\mathcal{K}$.

  3. If $j : V \to Y$ is an open immersion, then $j^* \circ F$ corresponds to $\mathcal{K}|_{X \times _ R V}$.

Proof. Proof of (1). Consider the commutative diagram

\[ \xymatrix{ X' \times _ R Y \ar[rrd]^{\text{pr}'_2} \ar[rd]_{f \times \text{id}_ Y} \ar[dd]_{\text{pr}'_1} \\ & X \times _ R Y \ar[r]_{\text{pr}_2} \ar[d]_{\text{pr}_1} & Y \\ X' \ar[r]^ f & X } \]

Let $\mathcal{F}'$ be a quasi-coherent module on $X'$. We have

\begin{align*} \text{pr}_{2, *}(\text{pr}_1^*f_*\mathcal{F}' \otimes _{\mathcal{O}_{X \times _ R Y}} \mathcal{K}) & = \text{pr}_{2, *}((f \times \text{id}_ Y)_* (\text{pr}'_1)^*\mathcal{F}' \otimes _{\mathcal{O}_{X \times _ R Y}} \mathcal{K}) \\ & = \text{pr}_{2, *}(f \times \text{id}_ Y)_* \left((\text{pr}'_1)^*\mathcal{F}' \otimes _{\mathcal{O}_{X' \times _ R Y}} (f \times \text{id}_ Y)^*\mathcal{K})\right) \\ & = \text{pr}'_{2, *}((\text{pr}'_1)^*\mathcal{F}' \otimes _{\mathcal{O}_{X' \times _ R Y}} (f \times \text{id}_ Y)^*\mathcal{K}) \end{align*}

Here the first equality is affine base change for the left hand square in the diagram, see Cohomology of Schemes, Lemma 30.5.1. The second equality hold by Remark 56.5.3. The third equality is functoriality of pushforwards for modules. This proves (1).

Proof of (2). Consider the commutative diagram

\[ \xymatrix{ X \times _ R Y' \ar[rr]_-{\text{pr}'_2} \ar[rd]^{\text{id}_ X \times g} \ar[rdd]_{\text{pr}'_1} & & Y' \ar[d]^ g \\ & X \times _ R Y \ar[r]_-{\text{pr}_2} \ar[d]^{\text{pr}_1} & Y \\ & X } \]

We have

\begin{align*} g^*\text{pr}_{2, *}(\text{pr}_1^*\mathcal{F} \otimes _{\mathcal{O}_{X \times _ R Y}} \mathcal{K}) & = \text{pr}'_{2, *}( (\text{id}_ X \times g)^*( \text{pr}_1^*\mathcal{F} \otimes _{\mathcal{O}_{X \times _ R Y}} \mathcal{K})) \\ & = \text{pr}'_{2, *}((\text{pr}'_1)^*\mathcal{F} \otimes _{\mathcal{O}_{X \times _ R Y'}} (\text{id}_ X \times g)^*\mathcal{K}) \end{align*}

The first equality by flat base change for the square in the diagram, see Cohomology of Schemes, Lemma 30.5.2. The second equality by functoriality of pullback and the fact that a pullback of tensor products it the tensor product of the pullbacks.

Part (3) is a special case of (2). $\square$

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 (56.5.1.1)

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. $\square$

Lemma 56.5.6. In Lemma 56.5.2 or in Lemma 56.5.5 if $F$ is an exact functor, then the corresponding object $\mathcal{K}$ of $\mathit{QCoh}(\mathcal{O}_{X \times _ R Y})$ is flat over $X$.

Proof. We may assume $X$ is affine, so we are in the case of Lemma 56.5.2. By Lemma 56.5.4 we may assume $Y$ is affine. In the affine case the statement translates into Remark 56.3.5. $\square$

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 (56.5.1.1).

Proof. Let $\mathcal{K}$ be as in (2). The functor $F$ in (56.5.1.1) 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 (56.5.1.1). $\square$

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 (56.5.1.1) 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 (56.5.1.1). If we ever need this we will carefully state and prove this here.

Lemma 56.5.9. Let $R$ be a ring. Let $X$, $Y$, $Z$ be schemes over $R$. Assume $X$ and $Y$ are quasi-compact and have affine diagonal. Let

\[ F : \mathit{QCoh}(\mathcal{O}_ X) \to \mathit{QCoh}(\mathcal{O}_ Y) \quad \text{and}\quad G : \mathit{QCoh}(\mathcal{O}_ Y) \to \mathit{QCoh}(\mathcal{O}_ Z) \]

be $R$-linear exact functors which commute with arbitrary direct sums. Let $\mathcal{K}$ in $\mathit{QCoh}(\mathcal{O}_{X \times _ R Y})$ and $\mathcal{L}$ in $\mathit{QCoh}(\mathcal{O}_{Y \times _ R Z})$ be the corresponding “kernels”, see Lemma 56.5.7. Then $G \circ F$ corresponds to $\text{pr}_{13, *}(\text{pr}_{12}^*\mathcal{K} \otimes _{\mathcal{O}_{X \times _ R Y \times _ R Z}} \text{pr}_{23}^*\mathcal{L})$ in $\mathit{QCoh}(\mathcal{O}_{X \times _ R Z})$.

Proof. Since $G \circ F : \mathit{QCoh}(\mathcal{O}_ X) \to \mathit{QCoh}(\mathcal{O}_ Z)$ is $R$-linear, exact, and commutes with arbitrary direct sums, we find by Lemma 56.5.7 that there exists an $\mathcal{M}$ in $\mathit{QCoh}(\mathcal{O}_{X \times _ R Z})$ corresponding to $G \circ F$. On the other hand, denote $\mathcal{E} = \text{pr}_{13, *}(\text{pr}_{12}^*\mathcal{K} \otimes \text{pr}_{23}^*\mathcal{L})$. Here and in the rest of the proof we omit the subscript from the tensor products. Let $U \subset X$ and $W \subset Z$ be affine open subschemes. To prove the lemma, we will construct an isomorphism

\[ \Gamma (U \times _ R W, \mathcal{E}) \cong \Gamma (U \times _ R W, \mathcal{M}) \]

compatible with restriction mappings for varying $U$ and $W$.

First, we observe that

\[ \Gamma (U \times _ R W, \mathcal{E}) = \Gamma (U \times _ R Y \times _ R W, \text{pr}_{12}^*\mathcal{K} \otimes \text{pr}_{23}^*\mathcal{L}) \]

by construction. Thus we have to show that the same thing is true for $\mathcal{M}$.

Write $U = \mathop{\mathrm{Spec}}(A)$ and denote $j : U \to X$ the inclusion morphism. Recall from the construction of $\mathcal{M}$ in the proof of Lemma 56.5.2 that

\[ \Gamma (U \times _ R W, \mathcal{M}) = \Gamma (W, G(F(j_*\mathcal{O}_ U))) \]

where the $A$-module action on the right hand side is given by the action of $A$ on $\mathcal{O}_ U$. The correspondence between $F$ and $\mathcal{K}$ tells us that $F(j_*\mathcal{O}_ U) = b_*(a^*j_*\mathcal{O}_ U \otimes \mathcal{K})$ where $a : X \times _ R Y \to X$ and $b : X \times _ R Y \to Y$ are the projection morphisms. Since $j$ is an affine morphism, we have $a^*j_*\mathcal{O}_ U = (j \times \text{id}_ Y)_*\mathcal{O}_{U \times _ R Y}$ by Cohomology of Schemes, Lemma 30.5.1. Next, we have $(j \times \text{id}_ Y)_*\mathcal{O}_{U \times _ R Y} \otimes \mathcal{K} = (j \times \text{id}_ Y)_*\mathcal{K}|_{U \times _ R Y}$ by Remark 56.5.3 for example. Putting what we have found together we find

\[ F(j_*\mathcal{O}_ U) = (U \times _ R Y \to Y)_*\mathcal{K}|_{U \times _ R Y} \]

with obvious $A$-action. (This formula is implicit in the proof of Lemma 56.5.2.) Applying the functor $G$ we obtain

\[ G(F(j_*\mathcal{O}_ U)) = t_*(s^*((U \times _ R Y \to Y)_*\mathcal{K}|_{U \times _ R Y}) \otimes \mathcal{L}) \]

where $s : Y \times _ R Z \to Y$ and $t : Y \times _ R Z \to Z$ are the projection morphisms. Again using affine base change (Cohomology of Schemes, Lemma 30.5.1) but this time for the square

\[ \xymatrix{ U \times _ R Y \times _ R Z \ar[r] \ar[d] & U \times _ R Y \ar[d] \\ Y \times _ R Z \ar[r] & Y } \]

we obtain

\[ s^*((U \times _ R Y \to Y)_*\mathcal{K}|_{U \times _ R Y}) = (U \times _ R Y \times _ R Z \to Y \times _ R Z)_* \text{pr}_{12}^*\mathcal{K}|_{U \times _ R Y \times _ R Z} \]

Using Remark 56.5.3 again we find

\begin{align*} (U \times _ R Y \times _ R Z \to Y \times _ R Z)_* \text{pr}_{12}^*\mathcal{K}|_{U \times _ R Y \times _ R Z} \otimes \mathcal{L} \\ = (U \times _ R Y \times _ R Z \to Y \times _ R Z)_* \left(\text{pr}_{12}^*\mathcal{K} \otimes \text{pr}_{23}^*\mathcal{L}\right)|_{U \times _ R Y \times _ R Z} \end{align*}

Applying the functor $\Gamma (W, t_*(-)) = \Gamma (Y \times _ R W, -)$ to this we obtain

\begin{align*} \Gamma (U \times _ R W, \mathcal{M}) & = \Gamma (W, G(F(j_*\mathcal{O}_ U))) \\ & = \Gamma (Y \times _ R W, (U \times _ R Y \times _ R Z \to Y \times _ R Z)_* (\text{pr}_{12}^*\mathcal{K} \otimes \text{pr}_{23}^*\mathcal{L})|_{U \times _ R Y \times _ R Z}) \\ & = \Gamma (U \times _ R Y \times _ R W, \text{pr}_{12}^*\mathcal{K} \otimes \text{pr}_{23}^*\mathcal{L}) \end{align*}

as desired. We omit the verication that these isomorphisms are compatible with restriction mappings. $\square$

Lemma 56.5.10. Let $R$, $X$, $Y$, and $\mathcal{K}$ be as in Lemma 56.5.7 part (2). Then for any scheme $T$ over $R$ we have

\[ R^ q\text{pr}_{13, *}(\text{pr}_{12}^*\mathcal{F} \otimes _{\mathcal{O}_{T \times _ R X \times _ R Y}} \text{pr}_{23}^*\mathcal{K}) = 0 \]

for $\mathcal{F}$ quasi-coherent on $T \times _ R X$ and $q > 0$.

Proof. The question is local on $T$ hence we may assume $T$ is affine. In this case we can consider the diagram

\[ \xymatrix{ T \times _ R X \ar[d] & T \times _ R X \times _ R Y \ar[d] \ar[l] \ar[r] & T \times _ R Y \ar[d] \\ X & X \times _ R Y \ar[l] \ar[r] & Y } \]

whose vertical arrows are affine. In particular the pushforward along $T \times _ R Y \to Y$ is faithful and exact (Cohomology of Schemes, Lemma 30.2.3 and Morphisms, Lemma 29.11.6). Chasing around in the diagram using that higher direct images along affine morphisms vanish (see reference above) we see that it suffices to prove

\[ R^ q\text{pr}_{2, *}( \text{pr}_{23, *}(\text{pr}_{12}^*\mathcal{F} \otimes _{\mathcal{O}_{T \times _ R X \times _ R Y}} \text{pr}_{23}^*\mathcal{K})) = R^ q\text{pr}_{2, *}( \text{pr}_{23, *}(\text{pr}_{12}^*\mathcal{F}) \otimes _{\mathcal{O}_{X \times _ R Y}} \mathcal{K})) \]

is zero which is true by assumption on $\mathcal{K}$. The equality holds by Remark 56.5.3. $\square$

Lemma 56.5.11. In Lemma 56.5.7 let $F$ and $\mathcal{K}$ correspond. If $X$ is separated and flat over $R$, then there is a surjection $\mathcal{O}_ X \boxtimes F(\mathcal{O}_ X) \to \mathcal{K}$.

Proof. Let $\Delta : X \to X \times _ R X$ be the diagonal morphism and set $\mathcal{O}_\Delta = \Delta _*\mathcal{O}_ X$. Since $\Delta $ is a closed immersion have a short exact sequence

\[ 0 \to \mathcal{I} \to \mathcal{O}_{X \times _ R X} \to \mathcal{O}_\Delta \to 0 \]

Since $\mathcal{K}$ is flat over $X$, the pullback $\text{pr}_{23}^*\mathcal{K}$ to $X \times _ R X \times _ R Y$ is flat over $X \times _ R X$. We obtain a short exact sequence

\[ 0 \to \text{pr}_{12}^*\mathcal{I} \otimes \text{pr}_{23}^*\mathcal{K} \to \text{pr}_{23}^*\mathcal{K} \to \text{pr}_{12}^*\mathcal{O}_\Delta \otimes \text{pr}_{23}^*\mathcal{K} \to 0 \]

on $X \times _ R X \times _ R Y$, see Modules, Lemma 17.20.4. Thus, by Lemma 56.5.10 we obtain a surjection

\[ \text{pr}_{13, *}(\text{pr}_{23}^*\mathcal{K}) \to \text{pr}_{13, *}( \text{pr}_{12}^*\mathcal{O}_\Delta \otimes \text{pr}_{23}^*\mathcal{K}) \]

By flat base change (Cohomology of Schemes, Lemma 30.5.2) the source of this arrow is equal to $\text{pr}_2^*\text{pr}_{2, *}\mathcal{K} = \mathcal{O}_ X \boxtimes F(\mathcal{O}_ X)$. On the other hand the target is equal to

\[ \text{pr}_{13, *}( \text{pr}_{12}^*\mathcal{O}_\Delta \otimes \text{pr}_{23}^*\mathcal{K}) = \text{pr}_{13, *} (\Delta \times \text{id}_ Y)_* \mathcal{K} = \mathcal{K} \]

which finishes the proof. The first equality holds for example by Cohomology, Lemma 20.52.4 and the fact that $\text{pr}_{12}^*\mathcal{O}_\Delta = (\Delta \times \text{id}_ Y)_*\mathcal{O}_{X \times _ R Y}$. $\square$


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