The Stacks project

Lemma 85.11.2. With notation as above. For a $\mathcal{O}_\mathcal {D}$-module $\mathcal{G}$ there is an exact complex

\[ \ldots \to g_{2!}(a_2^*\mathcal{G}) \to g_{1!}(a_1^*\mathcal{G}) \to g_{0!}(a_0^*\mathcal{G}) \to a^*\mathcal{G} \to 0 \]

of sheaves of $\mathcal{O}$-modules on $\mathcal{C}_{total}$. Here $g_{n!}$ is as in Lemma 85.6.1.

Proof. Observe that $a^*\mathcal{G}$ is the $\mathcal{O}$-module on $\mathcal{C}_{total}$ whose restriction to $\mathcal{C}_ m$ is the $\mathcal{O}_ m$-module $a_ m^*\mathcal{G}$. The description of the functors $g_{n!}$ on modules in Lemma 85.6.1 shows that $g_{n!}(a_ n^*\mathcal{G})$ is the $\mathcal{O}$-module on $\mathcal{C}_{total}$ whose restriction to $\mathcal{C}_ m$ is the $\mathcal{O}_ m$-module

\[ \bigoplus \nolimits _{\varphi : [n] \to [m]} f_\varphi ^*a_ n^*\mathcal{G} = \bigoplus \nolimits _{\varphi : [n] \to [m]} a_ m^*\mathcal{G} \]

The rest of the proof is exactly the same as the proof of Lemma 85.9.1, replacing $a_ m^{-1}\mathcal{G}$ by $a_ m^*\mathcal{G}$. $\square$

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