My question refers to some comments occured in following thread:
Galois morphism - group acting on the variety
The setting is that we have a finite Galois morphism $f: X \to S$, where $X$ and $S$ are non-singular and connected projective varieties over $\mathbb{C}$.
Galois means here that if we denote with $G$ the automorphism group of $X$ over $S$ then the quotient $X/G$ exists and the natural morphism $X/G \to S $ is an isomorphism.
While the discussion there occured two points making me curious:
Firstly, @user52991 question if one start with exact sequence
$$0\rightarrow F'\rightarrow F\rightarrow F''\rightarrow 0$$
of coherent sheaves on $X$ and applying the pushforward $f_*$ gives an short exact, (since $f$ finite !) sequence
$$0 \rightarrow f_*F'\rightarrow f_*F\rightarrow f_*F''\rightarrow 0$$
over $S= X/G$.
My first question is how and why does $G$ act on these pushforward sheaves explicitely? The cruical point is how does it act on local sections $f_*F(U)$? And second question:
If we were able to answer that the $G$-action is welldefined on the sheaves above then we can indeed deride the functor of invariants $E \to E^G$. And there ocured the question of this functor is exact.
@Mohan explained that the exactness is based on Hilbert 90 thm. Could anybody explain this argument a bit more how Hilbert 90 is inveolved in the problem?