Question about Etale Sheaves

Question

Let consider the one point field scheme $Spec(K)$ and denote by $G:=\text{Gal}(\overline{K}/K)$ the corresponding Galois group.

We consider by $\mathbf{Sh}\big(\text{Spec }k)_{\text{et}}$ the category of etale sheaves on $Spec(k)$. Then there is a wll known theorem that the exist a cetegory equivalence

$$ \mathbf{Sh}\big(\text{Spec }K)_{\text{et}} \cong\{\text{discrete abelian group with continuous }G_K\text{-action}\} $$

given explicitely by

$$\mathcal F \mapsto \varinjlim_{K'/K\text{ finite separable extension}} \mathcal F(\text{Spec }K') $$

with inverse $$\big(\text{Spec }k'\mapsto M^{\text{Gal}(\overline K/K')}\big) \leftarrow M $$

My question is how $G_K$ acts explicitely on $\varinjlim_{K'/K\text{ finite separable extension}} \mathcal F(\text{Spec }K')$?

So in other words if we take a separable field extension $K \subset K'$ what does $G_K$ with $\mathcal F(\text{Spec }K')$?

Answer 1

Given $\sigma \in Gal(K^s/K)$, let $\sigma_{K'} \in Gal(K'/K)$ be restriction of $\sigma$ to $K'$. The map $\sigma_{K'}:K' \to K'$ gives rise to $F(\sigma_{K'}):F(K') \to F(K')$ (by definition of a presheaf as a covariant functor on category of $K$-algebras) and so for $a_{K'} \in F(K')$, define $$\sigma_{K'} \cdot a_{K'} = F(\sigma)(a_{K'}) \in F(K')$$

We have to check this gives rise to well defined action of $G_K$ on $\mathrm{colim}_{K'} F(K')$. If $K \subset K' \subset K''$, we have map $F(K') \to F(K'')$ and apply $F$ to the commutative diagram

\begin{array}{ccc} K' & \to & K'' \\ \downarrow \sigma_{K'}& & \downarrow \sigma_{K''}\\ K & \to & K'' \end{array}