Question about Etale Sheaves

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 $$\mathcal F(\text{Spec }K')$$?

1 Answer

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

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