Example 59.59.3 (03QV)—The Stacks project

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:

\begin{align*} H_{\acute{e}tale}^0(S_{\acute{e}tale}, \mathbf{G}_ m) & = \Gamma (S, \mathcal{O}_ S^*) \\ H_{\acute{e}tale}^1(S_{\acute{e}tale}, \mathbf{G}_ m) & = H_{Zar}^1(S, \mathcal{O}_ S^*) = \mathop{\mathrm{Pic}}\nolimits (S) \\ H_{\acute{e}tale}^ i(S_{\acute{e}tale}, \mathbf{G}_ a) & = H_{Zar}^ i(S, \mathcal{O}_ S) \end{align*}

Also, for any quasi-coherent sheaf $\mathcal{F}$ on $S_{\acute{e}tale}$ we have

\[ H^ i(S_{\acute{e}tale}, \mathcal{F}) = H_{Zar}^ i(S, \mathcal{F}), \]

see Theorem 59.22.4. In particular, this gives the following sequence of equalities

\[ 0 = \mathop{\mathrm{Pic}}\nolimits (\mathop{\mathrm{Spec}}(K)) = H_{\acute{e}tale}^1(\mathop{\mathrm{Spec}}(K)_{\acute{e}tale}, \mathbf{G}_ m) = H^1(G, (K^{sep})^*) \]

which is none other than Hilbert's 90 theorem. Similarly, for $i \geq 1$,

\[ 0 = H^ i(\mathop{\mathrm{Spec}}(K), \mathcal{O}) = H_{\acute{e}tale}^ i(\mathop{\mathrm{Spec}}(K)_{\acute{e}tale}, \mathbf{G}_ a) = H^ i(G, K^{sep}) \]

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 Stacks project

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