Lemma 84.11.3. With notation as above. For an $\mathcal{O}$-module $\mathcal{F}$ on $\mathcal{C}_{total}$ there is a canonical complex

$0 \to a_*\mathcal{F} \to a_{0, *}\mathcal{F}_0 \to a_{1, *}\mathcal{F}_1 \to a_{2, *}\mathcal{F}_2 \to \ldots$

of $\mathcal{O}_\mathcal {D}$-modules which is exact in degrees $-1, 0$. If $\mathcal{F}$ is an injective $\mathcal{O}$-module, then the complex is exact in all degrees and remains exact on applying the functor $\mathop{\mathrm{Hom}}\nolimits _{\mathcal{O}_\mathcal {D}}(\mathcal{G}, -)$ for any $\mathcal{O}_\mathcal {D}$-module $\mathcal{G}$.

Proof. To construct the complex, by the Yoneda lemma, it suffices for any $\mathcal{O}_\mathcal {D}$-modules $\mathcal{G}$ on $\mathcal{D}$ to construct a complex

$0 \to \mathop{\mathrm{Hom}}\nolimits _{\mathcal{O}_\mathcal {D}}(\mathcal{G}, a_*\mathcal{F}) \to \mathop{\mathrm{Hom}}\nolimits _{\mathcal{O}_\mathcal {D}}(\mathcal{G}, a_{0, *}\mathcal{F}_0) \to \mathop{\mathrm{Hom}}\nolimits _{\mathcal{O}_\mathcal {D}}(\mathcal{G}, a_{1, *}\mathcal{F}_1) \to \ldots$

functorially in $\mathcal{G}$. To do this apply $\mathop{\mathrm{Hom}}\nolimits _\mathcal {O}(-, \mathcal{F})$ to the exact complex of Lemma 84.11.2 and use adjointness of pullback and pushforward. The exactness properties in degrees $-1, 0$ follow from the construction as $\mathop{\mathrm{Hom}}\nolimits _\mathcal {O}(-, \mathcal{F})$ is left exact. If $\mathcal{F}$ is an injective $\mathcal{O}$-module, then the complex is exact because $\mathop{\mathrm{Hom}}\nolimits _\mathcal {O}(-, \mathcal{F})$ is exact. $\square$

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