Let $X$ be a $K$-scheme and $L \vert K$ be a Galois extension with Galois group $G= Gal(L,K)$.
Let consider the base change $X_L := X \otimes_K L:= X \times_{Spec(K)} Spec(L)$. Since $X_L$ is a $L$-scheme $G$ can act on it.
My problems are following: I see some ways $G$ acting on $X_L$ and on it's structure sheaf but I'm not sure if they all coinside/corelate to each other and why?
- Let $g \in G$ then it induce an automorphism $g: Spec(L) \to Spec(L)$ and one can define the action of $g$ on $X_L$ via commutative diagram
$$ \require{AMScd} \begin{CD} X_L @>{\bar{g}} >> X_L \\ @VVprV @VVprV \\ Spec(L) @>{g}>> Spec(L); \end{CD} $$
or sugestively $\bar{g}: id_X \times g$
- Let $\mathcal{F}$ be a $\mathcal{O}_{X}$-module. I heard that $G$ can induce canonically an "$\mathcal{O}_{X_L}$-linear-action" on tnduced sheaf $\mathscr{F}\otimes \mathcal{O}_L$ acting on local sections $\mathcal{F}(U)$ for open $U$.
How concretely this action is described? Comes it from the same action as in case 1.?
In the sense of local action $(\mathcal{F}(U) \otimes_K L) ^g = \mathcal{F}(U) \otimes_K L^g$ ? So only on second summand? Or are these two actions different?
Espesially I don't see how could $G$ act on local sections of an arbitrary $\mathcal{O}_{X}$-module $\mathcal{F}$.
Futhermore this concept allows to define the so called sub-$\mathcal{O}_{X_L}$-module $\mathcal{F}^G \subset \mathcal{F}$ of invariants. But with respect to which action of $G$?
So the main point of my question is if 1. and 2. "generate" the same action and how this action extends to $\mathcal{O}_{X_L}$-modules.
Is this exactly THE canonical Galois action on a scheme which in the literature often mentioned but nowhere explicitely described?