I'm reading the book "Galois Theory of $p$-Extensions" by Helmut Koch. And I can't understand the Theorem 7.2 of his book.
The assumptions on the theorem is as follows :
$G$ is a profinite group, $R$ is a compact commutative ring with identity.
The Theorem states that if $A$ is a compact $R$-algebra, then every (continuous) morphism from $G$ into the unit group of $A$ can be extended uniquely to a morphism from $R[[G]]$ to $A$.
The uniqueness of the extension is easy to see as $R[G]$ is dense in $R[[G]]$. But I don't understand the existence of the morphism if there is no condition that $A$ is complete.