I have trouble showing the following in the proof of Prop. 2 in Abelian Varieties (pg.70 my edition, Chapter about quotients by finite groups):
Suppose you have a Noetherian ring $B=A^G$ as invariants of some action of a finite group on a Noetherian ring $A$. Also $A$ is finite over $B$ as module. The following map, which I suppose is multiplication is claimed to be an iso:
$N^G\otimes_B A \to N$ for $N$ an $A$-module with $G$-action.
My problem is that I don't understand how to prove the following special case:
Let $A\simeq B^k$ and suppose $G$ acts simply transitive by permutations. Then Mumford says it's obvious that the map is an iso. I guess I understand the thinking behind it, i.e. recovering the orbits of the non-invariant elements in $N$, but I have trouble to write down a map which does that.