I have a question about étale liftings. I take topological rings $A, B, C, D$ such that I have the following commutative diagram :
$\require{AMScd}$ \begin{CD} C @>f>> D\\ @A p AA @AA q A\\ A @>>g> B \end{CD}
All the maps are continuous, $p$ and $q$ are étale. I know that $g$ is an isomorphism (with continuous inverse). Is there any chance for $f$ to be invertible too ?
Thanks a lot for your help !