Lemma 85.11.3. 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 85.11.2 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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