The following lemma says there is no ambiguity in defining the cohomology of a sheaf $\mathcal{F}$ over an object of the site.
Lemma 21.7.1. Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Let $U$ be an object of $\mathcal{C}$.
If $\mathcal{I}$ is an injective $\mathcal{O}$-module then $\mathcal{I}|_ U$ is an injective $\mathcal{O}_ U$-module.
For any sheaf of $\mathcal{O}$-modules $\mathcal{F}$ we have $H^ p(U, \mathcal{F}) = H^ p(\mathcal{C}/U, \mathcal{F}|_ U)$.
Proof. Recall that the functor $j_ U^{-1}$ of restriction to $U$ is a right adjoint to the functor $j_{U!}$ of extension by $0$, see Modules on Sites, Section 18.19. Moreover, $j_{U!}$ is exact. Hence (1) follows from Homology, Lemma 12.29.1.
By definition $H^ p(U, \mathcal{F}) = H^ p(\mathcal{I}^\bullet (U))$ where $\mathcal{F} \to \mathcal{I}^\bullet $ is an injective resolution in $\textit{Mod}(\mathcal{O})$. By the above we see that $\mathcal{F}|_ U \to \mathcal{I}^\bullet |_ U$ is an injective resolution in $\textit{Mod}(\mathcal{O}_ U)$. Hence $H^ p(U, \mathcal{F}|_ U)$ is equal to $H^ p(\mathcal{I}^\bullet |_ U(U))$. Of course $\mathcal{F}(U) = \mathcal{F}|_ U(U)$ for any sheaf $\mathcal{F}$ on $\mathcal{C}$. Hence the equality in (2). $\square$
The following lemma will be use to see what happens if we change a partial universe, or to compare cohomology of the small and big étale sites.
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 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 in particular $H^ p(U, g^{-1}\mathcal{F}) = H^ p(u(U), \mathcal{F})$. All of these isomorphisms are functorial in $\mathcal{F}$.
\[ R\Gamma (\mathcal{C}, g^{-1}\mathcal{F}) = R\Gamma (\mathcal{D}, \mathcal{F}), \]
\[ R\Gamma (U, g^{-1}\mathcal{F}) = R\Gamma (u(U), \mathcal{F}), \]
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. $\square$
Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Let $\mathcal{F}$ be a sheaf of $\mathcal{O}$-modules. Let $\varphi : U \to V$ be a morphism of $\mathcal{O}$. Then there is a canonical restriction mapping
functorial in $\mathcal{F}$. Namely, choose any injective resolution $\mathcal{F} \to \mathcal{I}^\bullet $. The restriction mappings of the sheaves $\mathcal{I}^ p$ give a morphism of complexes
The LHS is a complex representing $R\Gamma (V, \mathcal{F})$ and the RHS is a complex representing $R\Gamma (U, \mathcal{F})$. We get the map on cohomology groups by applying the functor $H^ n$. As indicated we will use the notation $\xi \mapsto \xi |_ U$ to denote this map. Thus the rule $U \mapsto H^ n(U, \mathcal{F})$ is a presheaf of $\mathcal{O}$-modules. This presheaf is customarily denoted $\underline{H}^ n(\mathcal{F})$. We will give another interpretation of this presheaf in Lemma 21.10.5.
The following lemma says that it is possible to kill higher cohomology classes by going to a covering.
Lemma 21.7.3. Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Let $\mathcal{F}$ be a sheaf of $\mathcal{O}$-modules. Let $U$ be an object of $\mathcal{C}$. Let $n > 0$ and let $\xi \in H^ n(U, \mathcal{F})$. Then there exists a covering $\{ U_ i \to U\} $ of $\mathcal{C}$ such that $\xi |_{U_ i} = 0$ for all $i \in I$.
Proof. Let $\mathcal{F} \to \mathcal{I}^\bullet $ be an injective resolution. Then
Pick an element $\tilde\xi \in \mathcal{I}^ n(U)$ representing the cohomology class in the presentation above. Since $\mathcal{I}^\bullet $ is an injective resolution of $\mathcal{F}$ and $n > 0$ we see that the complex $\mathcal{I}^\bullet $ is exact in degree $n$. Hence $\mathop{\mathrm{Im}}(\mathcal{I}^{n - 1} \to \mathcal{I}^ n) = \mathop{\mathrm{Ker}}(\mathcal{I}^ n \to \mathcal{I}^{n + 1})$ as sheaves. Since $\tilde\xi $ is a section of the kernel sheaf over $U$ we conclude there exists a covering $\{ U_ i \to U\} $ of the site such that $\tilde\xi |_{U_ i}$ is the image under $d$ of a section $\xi _ i \in \mathcal{I}^{n - 1}(U_ i)$. By our definition of the restriction $\xi |_{U_ i}$ as corresponding to the class of $\tilde\xi |_{U_ i}$ we conclude. $\square$
Lemma 21.7.4. Let $f : (\mathcal{C}, \mathcal{O}_\mathcal {C}) \to (\mathcal{D}, \mathcal{O}_\mathcal {D})$ be a morphism of ringed sites corresponding to the continuous functor $u : \mathcal{D} \to \mathcal{C}$. For any $\mathcal{F} \in \mathop{\mathrm{Ob}}\nolimits (\textit{Mod}(\mathcal{O}_\mathcal {C}))$ the sheaf $R^ if_*\mathcal{F}$ is the sheaf associated to the presheaf
\[ V \longmapsto H^ i(u(V), \mathcal{F}) \]
Proof. Let $\mathcal{F} \to \mathcal{I}^\bullet $ be an injective resolution. Then $R^ if_*\mathcal{F}$ is by definition the $i$th cohomology sheaf of the complex
By definition of the abelian category structure on $\mathcal{O}_\mathcal {D}$-modules this cohomology sheaf is the sheaf associated to the presheaf
and this is obviously equal to
which is equal to $H^ i(u(V), \mathcal{F})$ and we win. $\square$
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