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.

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