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 verification that these isomorphisms are compatible with restriction mappings.
\square
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