As a consequence of the discussion in the preceding sections we obtain the equivalence of étale cohomology of the spectrum of a field with Galois cohomology.
Lemma 59.59.1. Let $S = \mathop{\mathrm{Spec}}(K)$ with $K$ a field. Let $\overline{s}$ be a geometric point of $S$. Let $G = \text{Gal}_{\kappa (s)}$ denote the absolute Galois group. The stalk functor induces an equivalence of categories
\[ \textit{Ab}(S_{\acute{e}tale}) \longrightarrow \text{Mod}_ G, \quad \mathcal{F} \longmapsto \mathcal{F}_{\overline{s}}. \]
Proof. In Theorem 59.56.3 we have seen the equivalence between sheaves of sets and $G$-sets. The current lemma follows formally from this as an abelian sheaf is just a sheaf of sets endowed with a commutative group law, and a $G$-module is just a $G$-set endowed with a commutative group law. $\square$
Lemma 59.59.2. Notation and assumptions as in Lemma 59.59.1. Let $\mathcal{F}$ be an abelian sheaf on $\mathop{\mathrm{Spec}}(K)_{\acute{e}tale}$ which corresponds to the $G$-module $M$. Then
in $D(\textit{Ab})$ we have a canonical isomorphism $R\Gamma (S, \mathcal{F}) = R\Gamma _ G(M)$,
$H_{\acute{e}tale}^0(S, \mathcal{F}) = M^ G$, and
$H_{\acute{e}tale}^ q(S, \mathcal{F}) = H^ q(G, M)$.
Example 59.59.3. Sheaves on $\mathop{\mathrm{Spec}}(K)_{\acute{e}tale}$. Let $G = \text{Gal}(K^{sep}/K)$ be the absolute Galois group of $K$.
The constant sheaf $\underline{\mathbf{Z}/n\mathbf{Z}}$ corresponds to the module $\mathbf{Z}/n\mathbf{Z}$ with trivial $G$-action,
the sheaf $\mathbf{G}_ m|_{\mathop{\mathrm{Spec}}(K)_{\acute{e}tale}}$ corresponds to $(K^{sep})^*$ with its $G$-action,
the sheaf $\mathbf{G}_ a|_{\mathop{\mathrm{Spec}}(K^{sep})}$ corresponds to $(K^{sep}, +)$ with its $G$-action, and
the sheaf $\mu _ n|_{\mathop{\mathrm{Spec}}(K^{sep})}$ corresponds to $\mu _ n(K^{sep})$ with its $G$-action.
By Remark 59.23.4 and Theorem 59.24.1 we have the following identifications for cohomology groups:
Also, for any quasi-coherent sheaf $\mathcal{F}$ on $S_{\acute{e}tale}$ we have
see Theorem 59.22.4. In particular, this gives the following sequence of equalities
which is none other than Hilbert's 90 theorem. Similarly, for $i \geq 1$,
where the $K^{sep}$ indicates $K^{sep}$ as a Galois module with addition as group law. In this way we may consider the work we have done so far as a complicated way of computing Galois cohomology groups.
The following result is a curiosity and should be skipped on a first reading.
Lemma 59.59.4. Let $R$ be a local ring of dimension $0$. Let $S = \mathop{\mathrm{Spec}}(R)$. Then every $\mathcal{O}_ S$-module on $S_{\acute{e}tale}$ is quasi-coherent.
Proof. Let $\mathcal{F}$ be an $\mathcal{O}_ S$-module on $S_{\acute{e}tale}$. We have to show that $\mathcal{F}$ is determined by the $R$-module $M = \Gamma (S, \mathcal{F})$. More precisely, if $\pi : X \to S$ is étale we have to show that $\Gamma (X, \mathcal{F}) = \Gamma (X, \pi ^*\widetilde{M})$.
Let $\mathfrak m \subset R$ be the maximal ideal and let $\kappa $ be the residue field. By Algebra, Lemma 10.153.10 the local ring $R$ is henselian. If $X \to S$ is étale, then the underlying topological space of $X$ is discrete by Morphisms, Lemma 29.36.7 and hence $X$ is a disjoint union of affine schemes each having one point. Moreover, if $X = \mathop{\mathrm{Spec}}(A)$ is affine and has one point, then $R \to A$ is finite étale by Algebra, Lemma 10.153.5. We have to show that $\Gamma (X, \mathcal{F}) = M \otimes _ R A$ in this case.
The functor $A \mapsto A/\mathfrak m A$ defines an equivalence of the category of finite étale $R$-algebras with the category of finite separable $\kappa $-algebras by Algebra, Lemma 10.153.7. Let us first consider the case where $A/\mathfrak m A$ is a Galois extension of $\kappa $ with Galois group $G$. For each $\sigma \in G$ let $\sigma : A \to A$ denote the corresponding automorphism of $A$ over $R$. Let $N = \Gamma (X, \mathcal{F})$. Then $\mathop{\mathrm{Spec}}(\sigma ) : X \to X$ is an automorphism over $S$ and hence pullback by this defines a map $\sigma : N \to N$ which is a $\sigma $-linear map: $\sigma (an) = \sigma (a) \sigma (n)$ for $a \in A$ and $n \in N$. We will apply Galois descent to the quasi-coherent module $\widetilde{N}$ on $X$ endowed with the isomorphisms coming from the action on $\sigma $ on $N$. See Descent, Lemma 35.6.2. This lemma tells us there is an isomorphism $N = N^ G \otimes _ R A$. On the other hand, it is clear that $N^ G = M$ by the sheaf property for $\mathcal{F}$. Thus the required isomorphism holds.
The general case (with $A$ local and finite étale over $R$) is deduced from the Galois case as follows. Choose $A \to B$ finite étale such that $B$ is local with residue field Galois over $\kappa $. Let $G = \text{Aut}(B/R) = \text{Gal}(\kappa _ B/\kappa )$. Let $H \subset G$ be the Galois group corresponding to the Galois extension $\kappa _ B/\kappa _ A$. Then as above one shows that $\Gamma (X, \mathcal{F}) = \Gamma (\mathop{\mathrm{Spec}}(B), \mathcal{F})^ H$. By the result for Galois extensions (used twice) we get
as desired. $\square$
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).
All contributions are licensed under the GNU Free Documentation License.
Comments (0)