I would like to see explicitly on the level of completed local rings, how for curves Geometric Frobenius looks differently than a separable cover ramified at a point.
In other words, let $k$ be an algebraically closed field of characteristic $2$ and consider the following inclusion maps:
$k[x] \to k[x,y]/(y^2-x)$
$k[x] \to k[x,y]/(y^2-x+yx)$
After localizing at $(x)$ and $(x,y)$ respectively, how can I compute the induced maps of completed local rings and see that these maps are non-isomorphic but are isomorphic if the characteristic of $k$ is different from $2$? This must hold due to algebraic geometry but I would like to see this result more elementary.
I know that the completion will always be of the form $k[[z]]$, but I cannot keep track of the morphisms. Is there an algorithm of computing completed local rings and induced maps between them in general? At least in the version that I know it, Cohen Structure Theorem is not constructive enough for explicit computations.