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Let $R$ be a local (commutative, associative) 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, "$Aut(Spec(R))$". The two objects certainly coincide if $R$ is the localization of an affine ring.
More generally, let $S=k[x_1,..,x_p]/I$, let $S\subseteq R\subseteq \hat{S}$, the completion with respect to $(x_1,..,x_p)$. Then the two objects coincide for $R$.

But for $R=C^\infty(\Bbb{R}^p,0)$ there are endomorphisms which do not come from the local maps of coordinates. See Page 5. (In this particular example one has an endomorphism, not an automorphism. Still, it is not clear that here $Aut(R)=``Aut(Spec(R))"=Aut(\Bbb{R}^p,0)$, the later is the group of germs of local diffeomorphisms).

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 $R$?

2. For which "geometric" rings rings $Aut(R)=``Aut(Spec(R))"$? (i.e. the group of all the automorphisms of $R$ vs the group of the local coordinate changes in $Spec(R)$.) what is the official notation for the "geometric" subgroup $``Aut(Spec(R))"$ of $Aut(R)$? (The notation $Aut(Spec(R))$ is somewhat heavy/lengthy.)

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

Let $R$ be a local (commutative, associative) 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 ring automorphisms, $Aut(R)$, (those that act on the field as identity) as the local changes of coordinates, "$Aut(Spec(R))$". The two objects certainly coincide if $R$ is the localization of an affine ring.
More generally, let $S=k[x_1,..,x_p]/I$, let $S\subseteq R\subseteq \hat{S}$, the completion with respect to $(x_1,..,x_p)$. Then the two objects coincide for $R$.

But for $R=C^\infty(\Bbb{R}^p,0)$ there are endomorphisms which do not come from the local maps of coordinates. See Page 5. (In this particular example one has an endomorphism, not an automorphism. Still, it is not clear that here $Aut(R)=``Aut(Spec(R))"=Aut(\Bbb{R}^p,0)$, the later is the group of germs of local diffeomorphisms).

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 $R$?

2. For which "geometric" rings rings $Aut(R)=``Aut(Spec(R))"$? (i.e. the group of all the automorphisms of $R$ vs the group of the local coordinate changes in $Spec(R)$.) what is the official notation for the "geometric" subgroup $``Aut(Spec(R))"$ of $Aut(R)$? (The notation $Aut(Spec(R))$ is somewhat heavy/lengthy.)

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

Let $R$ be a local (commutative, associative) 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, "$Aut(Spec(R))$". The two objects certainly coincide if $R$ is the localization of an affine ring.
More generally, let $S=k[x_1,..,x_p]/I$, let $S\subseteq R\subseteq \hat{S}$, the completion with respect to $(x_1,..,x_p)$. Then the two objects coincide for $R$.

But for $R=C^\infty(\Bbb{R}^p,0)$ there are endomorphisms which do not come from the local maps of coordinates. See Page 5. (In this particular one has endomorphism, not an automorphism. Still, it is not clear that here $Aut(R)=``Aut(Spec(R))"=Aut(\Bbb{R}^p,0)$, the later is the group of germs of local diffeomorphisms).

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 $R$?

2. For which "geometric" rings rings $Aut(R)=``Aut(Spec(R))"$? (i.e. the group of all the automorphisms of $R$ vs the group of the local coordinate changes in $Spec(R)$.) what is the official notation for the "geometric" subgroup $``Aut(Spec(R))"$ of $Aut(R)$? (The notation $Aut(Spec(R))$ is somewhat heavy/lengthy.)

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

Let $R$ be a local (commutative, associative) 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, "$Aut(Spec(R))$". The two objects certainly coincide if $R$ is the localization of an affine ring.
More generally, let $S=k[x_1,..,x_p]/I$, let $S\subseteq R\subseteq \hat{S}$, the completion with respect to $(x_1,..,x_p)$. Then the two objects coincide for $R$.

But for $R=C^\infty(\Bbb{R}^p,0)$ there are endomorphisms which do not come from the local maps of coordinates. See Page 5. (In this particular example one has an endomorphism, not an automorphism. Still, it is not clear that here $Aut(R)=``Aut(Spec(R))"=Aut(\Bbb{R}^p,0)$, the later is the group of germs of local diffeomorphisms).

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 $R$?

2. For which "geometric" rings rings $Aut(R)=``Aut(Spec(R))"$? (i.e. the group of all the automorphisms of $R$ vs the group of the local coordinate changes in $Spec(R)$.) what is the official notation for the "geometric" subgroup $``Aut(Spec(R))"$ of $Aut(R)$? (The notation $Aut(Spec(R))$ is somewhat heavy/lengthy.)

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

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? (i.e. the group of all the automorphisms of $R$ vs the group of the local coordinate changes in $Spec(R)$.) 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?

Let $R$ be a local (commutative, associative) 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, "$Aut(Spec(R))$". The two objects certainly coincide if $R$ is the localization of an affine ring.
More generally, let $S=k[x_1,..,x_p]/I$, let $S\subseteq R\subseteq \hat{S}$, the completion with respect to $(x_1,..,x_p)$. Then the two objects coincide for $R$.

But for $R=C^\infty(\Bbb{R}^p,0)$ there are endomorphisms which do not come from the local maps of coordinates. See Page 5. (In this particular one has endomorphism, not an automorphism. Still, it is not clear that here $Aut(R)=``Aut(Spec(R))"=Aut(\Bbb{R}^p,0)$, the later is the group of germs of local diffeomorphisms).

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 $R$?

2. For which "geometric" rings rings $Aut(R)=``Aut(Spec(R))"$? (i.e. the group of all the automorphisms of $R$ vs the group of the local coordinate changes in $Spec(R)$.) what is the official notation for the "geometric" subgroup $``Aut(Spec(R))"$ of $Aut(R)$? (The notation $Aut(Spec(R))$ is somewhat heavy/lengthy.)

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