Lemma 21.7.2 (03YU)—The Stacks project

Lemma 21.7.2. Let $\mathcal{C}$ and $\mathcal{D}$ be sites. Let $u : \mathcal{C} \to \mathcal{D}$ be a functor. Assume $u$ satisfies the hypotheses of Sites, Lemma 7.21.8. Let $g : \mathop{\mathit{Sh}}\nolimits (\mathcal{C}) \to \mathop{\mathit{Sh}}\nolimits (\mathcal{D})$ be the associated morphism of topoi. For any abelian sheaf $\mathcal{F}$ on $\mathcal{D}$ we have isomorphisms

\[ R\Gamma (\mathcal{C}, g^{-1}\mathcal{F}) = R\Gamma (\mathcal{D}, \mathcal{F}), \]

in particular $H^ p(\mathcal{C}, g^{-1}\mathcal{F}) = H^ p(\mathcal{D}, \mathcal{F})$ and for any $U \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$ we have isomorphisms

\[ R\Gamma (U, g^{-1}\mathcal{F}) = R\Gamma (u(U), \mathcal{F}), \]

in particular $H^ p(U, g^{-1}\mathcal{F}) = H^ p(u(U), \mathcal{F})$. All of these isomorphisms are functorial in $\mathcal{F}$.

Proof. Since it is clear that $\Gamma (\mathcal{C}, g^{-1}\mathcal{F}) = \Gamma (\mathcal{D}, \mathcal{F})$ by hypothesis (e), it suffices to show that $g^{-1}$ transforms injective abelian sheaves into injective abelian sheaves. As usual we use Homology, Lemma 12.29.1 to see this. The left adjoint to $g^{-1}$ is $g_! = f^{-1}$ with the notation of Sites, Lemma 7.21.8 which is an exact functor. Hence the lemma does indeed apply. $\square$

The Stacks project

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