The Stacks project

Lemma 36.7.6. Let $f : X \to Y$ be a morphism of schemes. Assume $X$ and $Y$ are quasi-compact and have affine diagonal. Then, denoting

\[ \Phi : D(\mathit{QCoh}(\mathcal{O}_ X)) \to D(\mathit{QCoh}(\mathcal{O}_ Y)) \]

the right derived functor of $f_* : \mathit{QCoh}(\mathcal{O}_ X) \to \mathit{QCoh}(\mathcal{O}_ Y)$ the diagram

\[ \xymatrix{ D(\mathit{QCoh}(\mathcal{O}_ X)) \ar[d]_\Phi \ar[r] & D_\mathit{QCoh}(\mathcal{O}_ X) \ar[d]^{Rf_*} \\ D(\mathit{QCoh}(\mathcal{O}_ Y)) \ar[r] & D_\mathit{QCoh}(\mathcal{O}_ Y) } \]

is commutative.

Proof. Observe that the horizontal arrows in the diagram are equivalences of categories by Proposition 36.7.5. Hence we can identify these categories (and similarly for other quasi-compact schemes with affine diagonal). The statement of the lemma is that the canonical map $\Phi (K) \to Rf_*(K)$ is an isomorphism for all $K$ in $D(\mathit{QCoh}(\mathcal{O}_ X))$. Note that if $K_1 \to K_2 \to K_3 \to K_1[1]$ is a distinguished triangle in $D(\mathit{QCoh}(\mathcal{O}_ X))$ and the statement is true for two-out-of-three, then it is true for the third.

Let $U \subset X$ be an affine open. Since the diagonal of $X$ is affine, the inclusion morphism $j : U \to X$ is affine (Morphisms, Lemma 29.11.11). Similarly, the composition $g = f \circ j : U \to Y$ is affine. Let $\mathcal{I}^\bullet $ be a K-injective complex in $\mathit{QCoh}(\mathcal{O}_ U)$. Since $j_* : \mathit{QCoh}(\mathcal{O}_ U) \to \mathit{QCoh}(\mathcal{O}_ X)$ has an exact left adjoint $j^* : \mathit{QCoh}(\mathcal{O}_ X) \to \mathit{QCoh}(\mathcal{O}_ U)$ we see that $j_*\mathcal{I}^\bullet $ is a K-injective complex in $\mathit{QCoh}(\mathcal{O}_ X)$, see Derived Categories, Lemma 13.31.9. It follows that

\[ \Phi (j_*\mathcal{I}^\bullet ) = f_*j_*\mathcal{I}^\bullet = g_*\mathcal{I}^\bullet \]

By Lemma 36.7.1 we see that $j_*\mathcal{I}^\bullet $ represents $Rj_*\mathcal{I}^\bullet $ and $g_*\mathcal{I}^\bullet $ represents $Rg_*\mathcal{I}^\bullet $. On the other hand, we have $Rf_* \circ Rj_* = Rg_*$. Hence $f_*j_*\mathcal{I}^\bullet $ represents $Rf_*(j_*\mathcal{I}^\bullet )$. We conclude that the lemma is true for any complex of the form $j_*\mathcal{G}^\bullet $ with $\mathcal{G}^\bullet $ a complex of quasi-coherent modules on $U$. (Note that if $\mathcal{G}^\bullet \to \mathcal{I}^\bullet $ is a quasi-isomorphism, then $j_*\mathcal{G}^\bullet \to j_*\mathcal{I}^\bullet $ is a quasi-isomorphism as well since $j_*$ is an exact functor on quasi-coherent modules.)

Let $\mathcal{F}^\bullet $ be a complex of quasi-coherent $\mathcal{O}_ X$-modules. Let $T \subset X$ be a closed subset such that the support of $\mathcal{F}^ p$ is contained in $T$ for all $p$. We will use induction on the minimal number $n$ of affine opens $U_1, \ldots , U_ n$ such that $T \subset U_1 \cup \ldots \cup U_ n$. The base case $n = 0$ is trivial. If $n \geq 1$, then set $U = U_1$ and denote $j : U \to X$ the open immersion as above. We consider the map of complexes $c : \mathcal{F}^\bullet \to j_*j^*\mathcal{F}^\bullet $. We obtain two short exact sequences of complexes:

\[ 0 \to \mathop{\mathrm{Ker}}(c) \to \mathcal{F}^\bullet \to \mathop{\mathrm{Im}}(c) \to 0 \]


\[ 0 \to \mathop{\mathrm{Im}}(c) \to j_*j^*\mathcal{F}^\bullet \to \mathop{\mathrm{Coker}}(c) \to 0 \]

The complexes $\mathop{\mathrm{Ker}}(c)$ and $\mathop{\mathrm{Coker}}(c)$ are supported on $T \setminus U \subset U_2 \cup \ldots \cup U_ n$ and the result holds for them by induction. The result holds for $j_*j^*\mathcal{F}^\bullet $ by the discussion in the preceding paragraph. We conclude by looking at the distinguished triangles associated to the short exact sequences and using the initial remark of the proof. $\square$

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