The Stacks project

21.7 Locality of cohomology

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 $\mathcal{C}$.

  1. If $\mathcal{I}$ is an injective $\mathcal{O}$-module then $\mathcal{I}|_ U$ is an injective $\mathcal{O}_ U$-module.

  2. 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 ├ętale 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

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

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. $\square$

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
\begin{equation} \label{sites-cohomology-equation-restriction-mapping} H^ n(V, \mathcal{F}) \longrightarrow H^ n(U, \mathcal{F}), \quad \xi \longmapsto \xi |_ U \end{equation}

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

\[ \Gamma (V, \mathcal{I}^\bullet ) \longrightarrow \Gamma (U, \mathcal{I}^\bullet ) \]

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

\[ H^ n(U, \mathcal{F}) = \frac{\mathop{\mathrm{Ker}}(\mathcal{I}^ n(U) \to \mathcal{I}^{n + 1}(U))}{\mathop{\mathrm{Im}}(\mathcal{I}^{n - 1}(U) \to \mathcal{I}^ n(U))}. \]

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

\[ f_*\mathcal{I}^0 \to f_*\mathcal{I}^1 \to f_*\mathcal{I}^2 \to \ldots \]

By definition of the abelian category structure on $\mathcal{O}_\mathcal {D}$-modules this cohomology sheaf is the sheaf associated to the presheaf

\[ V \longmapsto \frac{\mathop{\mathrm{Ker}}(f_*\mathcal{I}^ i(V) \to f_*\mathcal{I}^{i + 1}(V))}{\mathop{\mathrm{Im}}(f_*\mathcal{I}^{i - 1}(V) \to f_*\mathcal{I}^ i(V))} \]

and this is obviously equal to

\[ \frac{\mathop{\mathrm{Ker}}(\mathcal{I}^ i(u(V)) \to \mathcal{I}^{i + 1}(u(V)))}{\mathop{\mathrm{Im}}(\mathcal{I}^{i - 1}(u(V)) \to \mathcal{I}^ i(u(V)))} \]

which is equal to $H^ i(u(V), \mathcal{F})$ and we win. $\square$

Comments (0)

Post a comment

Your email address will not be published. Required fields are marked.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.

In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 01FU. Beware of the difference between the letter 'O' and the digit '0'.