The Stacks project

Lemma 21.40.3. Assumptions and notation as in Situation 21.38.3. For an abelian sheaf $\mathcal{F}'$ on $\mathcal{C}'$ the sheaf $L_ ng_!(\mathcal{F}')$ is the sheaf associated to the presheaf

\[ U \longmapsto H_ n(\mathcal{I}_ U, \mathcal{F}'_ U) \]

For notation and restriction maps see proof.

Proof. Say $p(U) = V$. The category $\mathcal{I}_ U$ is the category of pairs $(U', \varphi )$ where $\varphi : U \to u(U')$ is a morphism of $\mathcal{C}$ with $p(\varphi ) = \text{id}_ V$, i.e., $\varphi $ is a morphism of the fibre category $\mathcal{C}_ V$. Morphisms $(U'_1, \varphi _1) \to (U'_2, \varphi _2)$ are given by morphisms $a : U'_1 \to U'_2$ of the fibre category $\mathcal{C}'_ V$ such that $\varphi _2 = u(a) \circ \varphi _1$. The presheaf $\mathcal{F}'_ U$ sends $(U', \varphi )$ to $\mathcal{F}'(U')$. We will construct the restriction mappings below.

Choose a factorization

\[ \xymatrix{ \mathcal{C}' \ar@<1ex>[r]^{u'} & \mathcal{C}'' \ar[r]^{u''} \ar@<1ex>[l]^ w & \mathcal{C} } \]

of $u$ as in Categories, Lemma 4.33.14. Then $g_! = g''_! \circ g'_!$ and similarly for derived functors. On the other hand, the functor $g'_!$ is exact, see Modules on Sites, Lemma 18.16.6. Thus we get $Lg_!(\mathcal{F}') = Lg''_!(\mathcal{F}'')$ where $\mathcal{F}'' = g'_!\mathcal{F}'$. Note that $\mathcal{F}'' = h^{-1}\mathcal{F}'$ where $h : \mathop{\mathit{Sh}}\nolimits (\mathcal{C}'') \to \mathop{\mathit{Sh}}\nolimits (\mathcal{C}')$ is the morphism of topoi associated to $w$, see Sites, Lemma 7.23.1. The functor $u''$ turns $\mathcal{C}''$ into a fibred category over $\mathcal{C}$, hence Lemma 21.40.2 applies to the computation of $L_ ng''_!$. The result follows as the construction of $\mathcal{C}''$ in the proof of Categories, Lemma 4.33.14 shows that the fibre category $\mathcal{C}''_ U$ is equal to $\mathcal{I}_ U$. Moreover, $h^{-1}\mathcal{F}'|_{\mathcal{C}''_ U}$ is given by the rule described above (as $w$ is continuous and cocontinuous by Stacks, Lemma 8.10.3 so we may apply Sites, Lemma 7.21.5). $\square$

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