Let $R$ be a local ring over a field of zero characteristic. (My typical examples are $k[[x_1,..,x_p]]/I$, $k\{x_1,.,x_p\}/I$, $C^\infty(\Bbb{R}^p,0)$. If it helps one can assume $R$ to be Henselian.)

I'd like to think of the ($k$-linear) ring automorphisms, $Aut(R)$, as the local changes of coordinates in $Spec(R)$. The two objects certainly coincide if $R$ is the localization of an affine ring. I guess they coincide also for the analytic rings. (A reference?)

But for $R=C^\infty(\Bbb{R}^p,0)$ there are automorphisms which do not come from the local change of coordinates. See Page 5.

1. Suppose $R$ is "geometric enough", so that one can speak of $Spec(R)$, its local coordinates, a local change of them. Does every local change of coordinates (that preserves the origin) extend to an automorphism of $R$? This holds if the natural completion map $R\to\hat{R}$ is an embedding, right? What about the more general case?

2. For which rings rings do the two objects coincide? what is the official notation for the "geometric" subgroup of $Aut(R)$? (The notation $Aut(Spec(R))$ is somewhat heavy/lengthy.)

Any paper/review on the state of the art in this direction?