The Stacks project

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. $\square$

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