SGA 4.5 proof of Hilbert 90 and semilinear Galois action In SGA 4.5's proof of Hilbert 90, proposition 1.5.2(that the inclusion $V'^G \otimes_k k' \rightarrow V'$ is an isomorphism) is deduced from faithfully flat descent as stated in 1.4.5. The way that I interpret the usage of 1.4.5 is that I am looking at a single cover of $Spec\,k$, $Spec\,k'$ and the quasicoherent-sheaf $\tilde{V'}$. 
Question: is the correct way to use it? If so, it doesn't seem that the discussion/definition of semilinear G-action comes up at all - a clarification on this condition will be greatly appreciated!
 A: The semilinear $G$-action on $V'$ is a concrete way of recording the fact
that $\tilde{V}'$ admits descent data from Spec $k$' to Spec $k$.
(Note that $k'\otimes_k k' = \prod_{g \in G} k'$, if $k'/k$ is Galois with group $G$.)
Added in response to comment:
The isomorphism $k'\otimes_k k' \cong \prod_g k'$ of $k'$ algebras is induced by the morphism of $k$-algebras
$$k' \to \prod_g k'$$
given by 
$$a \mapsto (g(a))_{g \in G}.$$
Giving the isomorphism  $k'\otimes_k V' \to V' \otimes_k k'$
then amount to giving a $k'$-linear map $V \to \prod_g V,$
where the $k'$-action on the $g$th factor is twisted by $g$.
Now this is the same as giving a collection of morphisms 
$V \to V$ labelled by $g \in G$ (which you can check will have to be isomorphisms,
if the original morphism is supposed to be one) which twist the
$k'$-action by the corresponding element $g$.  These are the ingredients
for the semi-linear action.  The fact that they form an action will be
forced by the cocycle condition on the descent data.
