$l$-adic cohomology: Lecture 4

References. The following are the references I use for writing this posting:

• Freitag and Kiehl,
• Milne,
• Hochster,
• Vakil,
• Stacks Project (which will be referred as "SP").

Of course, I may cite more references as I go.

Goal. The goal is to follow the first chapter of the first reference.

Presheaves/sheaves on a site (valued in abelian groups). Let $\mathcal{C}$ be any category. A presheaf (valued in the category $\textbf{Ab}$ of abelian groups) on $\mathcal{C}$ is a functor $\mathscr{F} : \mathcal{C}^{\mathrm{op}} \rightarrow \textbf{Ab}.$

Now, suppose that $\mathcal{C}$ is equipped with a Grothendieck topology (i.e., $\mathcal{C}$ is a site). A presheaf $\mathscr{F}$ on $\mathcal{C}$ is called a sheaf if for any object $U$ of $\mathcal{C}$ and a cover $\{U_{i} \rightarrow U\}_{i \in I}$ of $U,$ the induced diagram $$\mathscr{F}(U) \rightarrow \prod_{i \in I}\mathscr{F}(U_{i}) \rightrightarrows \prod_{(i, j) \in I^{2}} \mathscr{F}(U_{i} \times_{U} U_{j})$$ is an equilizer diagram in $\textbf{Ab}.$

How is the above diagram defined? The first map is obtined by applying $\mathscr{F}$ to maps $U_{i} \rightarrow U$ and then using the universal property of the product. There are two maps from the second term to the third. The first one comes from the maps of the form $U_{i} \times_{U} U_{j} \rightarrow U_{i},$ and the second one comes from the maps of the form $U_{i} \times_{U} U_{j} \rightarrow U_{j}.$

Notation. If $\mathcal{C}$ is a site, for any map $U \rightarrow V$ in $\mathcal{C},$ we describe the induced map $\mathscr{F}(V) \rightarrow \mathscr{F}(U)$ as $s \mapsto s|_{U}.$ This notation does not mean that there is a unique map from $U$ to $V,$ but we will use it as long as it does not create any confusion.

Remark. Let $\mathcal{C}$ be a site. Note that a presheaf $\mathscr{F}$ is a sheaf if and only if for every cover $\mathscr{U} = \{U_{i} \rightarrow U\}_{i \in I}$ of each object $U$ in $\mathcal{C},$ the following conditions are satisfied :

• for each $s, t \in \mathscr{F}(U),$ if $s|_{U_{i}} = t|_{U_{i}}$ for every $U_{i} \rightarrow U$ in $\mathscr{U},$ then $s = t$.
• given any $(s_{i})_{i \in I} \in \prod_{i \in I}\mathscr{F}(U_{i}),$ if $s_{i}|_{U_{i} \times_{U} U_{j}} = s_{i}|_{U_{i} \times_{U} U_{j}}$ for every $U_{i} \rightarrow U$ and $U_{j} \rightarrow U$ in $\mathscr{U}$ (including the case $i = j$), then there is $s \in \mathscr{F}(U)$ such that $s|_{U_{i}} = s_{i}$ for every $U_{i} \rightarrow U$ in $\mathscr{U}.$

We want to define cohomology of a sheaf on the étale site $X_{ét}$ of a scheme $X.$ The following facts let us do this:

Fact 1 (SP03NT). The category of sheaves $\textbf{Sh}(X_{ét})$ is an abelian category that has enough injectives.

A part of showing the above fact also yields the following:

Theorem. The global section functor functor $\Gamma : \textbf{Sh}(X_{ét}) \rightarrow \textbf{Ab}$ given by $\mathscr{F} \mapsto \Gamma(X, \mathscr{F}) := \mathscr{F}(X)$ (with morphisms are mapped to their restrictions to the global sections) is left-exact.

Proof. (cf. SP03CN). Let $$0 \rightarrow \mathscr{F} \xrightarrow{\phi} \mathscr{G} \xrightarrow{\psi} \mathscr{H}$$ be an exact sequence in $\textbf{Sh}(X_{ét}).$ We want to show that the induced sequence $$0 \rightarrow \mathscr{F}(X) \xrightarrow{\phi} \mathscr{G}(X) \xrightarrow{\psi} \mathscr{H}(X)$$ is exact.

To show the exactness at $\mathscr{F}(X),$ it is enough to show the following:

Lemma 1. Let $\mathscr{F} \xrightarrow{\phi} \mathscr{G}$ be any morphism of sheaves in $\textbf{Sh}(X_{ét}).$ Then $\ker(\phi_{U}) = \ker(\phi)(U)$ for every $U$ in $\textbf{Sh}(X_{ét}).$

Proof of Lemma 1. We construct a presheaf $\mathscr{K}$ on $X_{ét}$ by defining $$\mathscr{K}(U) := \ker(\phi_{U}) \subset \mathscr{F}(U).$$ Given any map $U \rightarrow V$ of $X_{ét},$ the restriction map $\mathscr{K}(V) \rightarrow \mathscr{K}(U)$ is given by restricting the map $\mathscr{F}(V) \rightarrow \mathscr{F}(U).$ This defines a functor $\mathscr{K} : X_{ét}^{\mathrm{op}} \rightarrow \textbf{Ab},$ because $\mathscr{F}$ is a functor.

We first show that $\mathscr{K}$ is a sheaf. Let $s \in \mathscr{K}(U)$ and say $\{U_{i} \rightarrow U\}_{i \in I}$ is a cover of $U.$ If $s|_{U_{i}} = 0 \in \mathscr{K}(U) \subset \mathscr{F}(U)$ for each $i,$ then $s = 0$ in $\mathscr{F}(U),$ so $s = 0$ in $\mathscr{K}(U).$ Next, suppose that we have $s_{i} \in \mathscr{K}(U_{i})$ for each $i \in I$ such that $$s_{i}|_{U_{i} \times_{U} U_{j}} = s_{j}|_{U_{i} \times_{U} U_{j}} \in \mathscr{K}(U_{i} \times_{U} U_{j}) \subset \mathscr{F}(U_{i} \times_{U} U_{j})$$ for all $i, j \in I.$ Since $\mathscr{F}$ is a sheaf, we may find $s \in \mathscr{F}(U)$ such that $$s|_{U_{i}} = s_{i} \in \mathscr{F}(U_{i}).$$ For all $i,$ we have $$\phi_{U}(s)|_{U_{i}} = \phi_{U_{i}}(s_{i}) = 0$$ because $s_{i} \in \mathscr{K}(U_{i}) = \ker(\phi_{U_{i}}).$ This implies that $\phi_{U}(s) = 0,$ so $s \in \mathscr{K}(U).$ This establishes the fact that $\mathscr{K}$ is a sheaf.

We now claim that $\mathscr{K} = \ker(\phi).$ The inclusions $\mathscr{K}(U) \hookrightarrow \mathscr{F}(U)$ build into a map $\iota : \mathscr{K} \rightarrow \mathscr{F}$ of presheaves. It is immediate that the composition $\mathscr{K} \xrightarrow{\iota} \mathscr{F} \xrightarrow{\phi} \mathscr{G}$ is the zero map (i.e., $\phi \circ \iota = 0$), so to finish the check, we fix any other sheaf map $j : \mathscr{K}' \rightarrow \mathscr{F}$ such that $\phi \circ j = 0.$ We want to show that there exists a unique map $\eta : \mathscr{K}' \rightarrow \mathscr{K}$ such that $j = \iota \circ \eta.$ This requirement forces $j_{U} = \iota_{U} \circ \eta_{U},$ so given any object $U$ of $X_{ét},$ the map $\eta_{U} : \mathscr{K}'(U) \rightarrow \mathscr{K}(U)$ has to be given by the usual property of kernels in $\textbf{Ab}.$ More explicitly, since the image of $j_{U}$ lies in $\ker(\phi_{U}) = \mathscr{K}(U),$ it factors as $$\mathscr{K}'(U) \rightarrow \mathscr{K}(U) \hookrightarrow \mathscr{F}(U).$$ It is immediate that these map build a map $\eta : \mathscr{K}' \rightarrow \mathscr{K}$ of sheaves. Hence, we have proved that $\mathscr{K} = \ker(\phi).$ This finishes the proof (of Lemma 1). $\Box$

Going back to the proof of Theorem, it remains to show the exactness at $\mathscr{G}(X).$ That is, we need to show that $\mathrm{im}(\phi_{X}) = \ker(\psi_{X}).$ We have $$\ker(\mathrm{coker}(\phi)) = \mathrm{im}(\phi) = \ker(\psi),$$ the second of which uses the exactness at $\mathscr{G}.$ Thus, by Lemma 1, we have $$\ker(\mathrm{coker}(\phi)(X)) = \ker(\psi_{X}).$$ The inclusion $$\mathrm{im}(\phi_{X}) = \ker(\mathrm{coker}(\phi_{X})) \subset \ker(\mathrm{coker}(\phi)(X)) = \ker(\psi_{X})$$ follows immediately from diagram chasing.

To show the reverse inclusion, we actually need to understand what $\mathrm{coker}(\phi)$ is. It is the "sheafification" of the presheaf on $X_{ét},$ which we briefly discuss now:

Sheafification. Given a presheaf $\mathscr{F}$ on $X_{ét},$ a sheafification $\tilde{\mathscr{F}}$ of $\mathscr{F}$ is a sheaf on $X_{ét}$ together with a presheaf map $\mathscr{F} \rightarrow \tilde{\mathscr{F}}$ satisfying the following property: for any sheaf $\mathscr{G}$ on $X_{ét}$ and a presheaf map $\phi : \mathscr{F} \rightarrow \mathscr{G},$ there is a unique sheaf map $\tilde{\mathscr{F}} \rightarrow \mathscr{G}$ that $\phi$ factors through the sheafification.

Remark. Note that if a sheafification of $\mathscr{F}$ exists, it is unique up to a unique isomorphism. It is also immediate that taking sheafification defines a functor $\textbf{Psh}(X_{ét}) \rightarrow \textbf{Sh}(X_{ét})$ from the category of presheaves over $X_{ét}$ to that of sheaves.

The sheafification always exists, but we will not discuss the proof of it but cite a reference instead. At this point, if we denote by $\mathrm{coker}(\phi_{-})$ the presheaf given by $U \mapsto \mathrm{coker}(\phi_{U}),$ one can check that the cokernel $\mathscr{G} \rightarrow \mathrm{coker}(\phi)$ of the sheaf map $\phi : \mathscr{F} \rightarrow \mathscr{G}$ can be constructed as the following composition: $$\mathscr{G} \rightarrow \mathrm{coker}(\phi_{-}) \rightarrow \widetilde{\mathrm{coker}(\phi_{-})} =: \mathrm{coker}(\phi).$$ Note that the usual proof for the small Zariski site of $X$ (Vakil 2.6.1) works here without any change.

We need a little bit more to finish our proof:

Fact 2 (Vistoli Theorem 2.64). The desired sheafification can be constructed so that it satisfies the following two properties:

1. for any $s \in \mathscr{F}(U),$ if its image in $\tilde{\mathscr{F}}(U)$ is zero, then there is a cover $\{U_{i} \rightarrow U\}_{i \in I}$ such that $s|_{U_{i}} = 0$ for each $i.$
2. for any $t \in \tilde{\mathscr{F}}(U),$ there is a cover $\{U_{i} \rightarrow U\}_{i \in I}$ such that $t|_{U_{i}}$ is in the image of $\mathscr{F}(U_{i}) \rightarrow \tilde{\mathscr{F}}(U_{i})$ for each $i.$

Remark. Note that if we have any sheaf $\tilde{\mathscr{F}}$ and a presheaf map $\mathscr{F} \rightarrow \tilde{\mathscr{F}}$ satisfying the above two properties, then $\mathscr{F} \rightarrow \tilde{\mathscr{F}}$ is necessarily a sheafification.

We are now ready to finish our proof of Theorem. Denote by $\eta : \mathscr{G} \rightarrow \mathrm{coker}(\phi)$ the map comes with $\mathrm{coker}(\phi).$ Consider $\eta_{X} : \mathscr{G}(X) \rightarrow \mathrm{coker}(\phi)(X).$ We know $\ker(\psi_{X}) = \ker(\eta_{X}),$ so what remains for us to show is $\ker(\eta_{X}) \subset \mathrm{im}(\phi_{X}).$ 

Fix any $s \in \ker(\eta_{X}) \subset \mathscr{G}(X).$ We want to show that $s \in \mathrm{im}(\phi_{X}).$ Since $\eta_{X}(s) = 0 \in \mathrm{coker}(\phi)(X),$ we have a map $\mathrm{coker}(\phi_{X}) \rightarrow \mathrm{coker}(\phi)(X)$ given by the universal property of the cokernel (source) such that $\bar{s} \mapsto \eta_{X}(s) = 0.$ Note that this is exactly the sheafification map, so by Fact 2-1, we can find a cover $\{U_{i} \rightarrow U\}_{i \in I}$ such that $\overline{s|_{U_{i}}} = 0$ in $\mathrm{coker}(\phi_{U_{i}})$ (i.e., $s|_{U_{i}} \in \mathrm{im}(\phi_{U_{i}})$) for each $i.$ Hence, we can write $s|_{U_{i}} = \phi_{U_{i}}(t_{i})$ for some (necessarily unique) $t_{i} \in \mathscr{F}(U_{i})$ for each $i.$ Since $\phi_{U_{i} \times_{X} U_{j}} : \mathscr{F}(U_{i} \times_{X} U_{j}) \rightarrow \mathscr{G}(U_{i} \times_{X} U_{j})$ is injective, we have $$t_{i} |_{U_{i} \times_{X} U_{j}} = t_{j} |_{U_{i} \times_{X} U_{j}}$$ for all $i, j \in I.$ As $\mathscr{F}$ is a sheaf, this implies that we have $t \in \mathscr{F}(X)$ such that $t|_{U_{i}} = t_{i}$ for all $i.$ Since $\mathscr{G}$ is a sheaf, it follows that $\phi_{X}(t) = s,$ so $s \in \mathrm{im}(\phi_{X}),$ as desired. This finishes the proof (of Theorem). $\Box$

Definition of étale cohomology group. Using the facts above, by looking at any sheaf $\mathscr{F}$ on $X_{ét}$ as an object of $\textbf{Sh}(X_{ét}),$ one writes $H^{i}(X_{ét}, -)$ to be the $i$-th right derived functor of the global section functor. Given any sheaf $\mathscr{F}$ on $X_{ét},$ the abelian group $H^{i}(X_{ét}, \mathscr{F})$ is called the $i$-th étale cohomology group of the sheaf $\mathscr{F}.$

Remark. It is important to note that our discussion works even if we used the category $\textbf{Mod}_{R}$ of $R$-modules instead of $\textbf{Ab}$ for any given ring $R.$