Formal-coordinate change group Aut O

Let $\mathcal{O}:=\mathbb{C}\mathrm{⟦}t\mathrm{⟧}$ with its $t$-adic topology, whose neighborhoods of $0$ are the ideals $t^k\mathcal{O}$, and let $\mathfrak{m}=(t)$. The group of formal-coordinate changes is $\mathrm{Aut}\mathcal{O}:=\{\rho (t)=a_{1}t+a_2{t}^2+a_3{t}^3+\cdots :a_i\in \mathbb{C},a_1\mathrm{\ne }0\}\mathrm{.}$ Each series has a unique compositional inverse: after the linear coefficient is inverted, the coefficient of degree $k$ in the inverse-composition equation is $a_1$ times the unknown degree-$k$ coefficient plus terms already determined. We use the right-composition product $({\rho }_1{\rho }_2)(t):={\rho }_2({\rho }_1(t))\mathrm{.}$ It is the opposite of the usual order for composing the displayed functions. The identity is $t$, and $\mathrm{Aut}\mathcal{O}$ is the group of continuous $\mathbb{C}$-algebra automorphisms of $\mathcal{O}$ preserving $\mathfrak{m}$.

For any commutative $\mathbb{C}$-algebra $A$, put ${\mathcal{O}}_A:=A\mathrm{⟦}t\mathrm{⟧}$ with its $t$-adic topology. Write ${\mathrm{Aut}}_A{\mathcal{O}}_A$ for the group of continuous $A$-algebra automorphisms preserving $(t)$; equivalently, its elements are the substitutions $\rho (t)=a_{1}t+a_2{t}^2+\cdots$ with $a_i\in A$ and $a_1\in A^{\times }$, with the same right-composition product. The coefficient recursion for the inverse uses only addition, multiplication, and inversion of $a_1$, so it works over every such $A$.

Write ${\mathrm{\ell }}_n:=-t^{n+1}{\mathrm{\partial }}_t$ for $n\ge 0$. With the right-composition Lie bracket, $[{\mathrm{\ell }}_m,{\mathrm{\ell }}_n]=(m-n){\mathrm{\ell }}_{m+n}$. The subgroup ${\mathrm{Aut}}_{+}\mathcal{O}=\{t+O(t^2)\}$ is tangent to the identity and has Lie algebra $t^2\mathbb{C}\mathrm{⟦}t\mathrm{⟧}{\mathrm{\partial }}_t={\prod }_{n\ge 1}\mathbb{C}{\mathrm{\ell }}_n$, the $t$-adically completed span of the ${\mathrm{\ell }}_n$ for $n\ge 1$. The product means that arbitrary coefficient sequences are allowed; modulo $t^{k+1}{\mathrm{\partial }}_t$, only finitely many coefficients remain.

For a substitution series $\rho$, write ${\varphi }_{\rho }(f(t)):=f(\rho (t))$ for the corresponding continuous algebra automorphism. Then ${\varphi }_{{\rho }_1{\rho }_2}={\varphi }_{{\rho }_1}\circ {\varphi }_{{\rho }_2}$ under the displayed product. A formal coordinate on a $J$-adically complete $\mathbb{C}$-algebra $(R,J)$ is a continuous algebra isomorphism $\tau :R\stackrel{\sim }{\to }\mathcal{O}$ with $\tau (J)=\mathfrak{m}$. Define its right action by $\tau \cdot \rho :={\varphi }_{\rho }^{-1}\circ \tau \mathrm{.}$ More generally, for a commutative $\mathbb{C}$-algebra $A$ and a $J$-adically complete $A$-algebra $R$, a relative formal coordinate is a continuous $A$-algebra isomorphism $\tau :R\stackrel{\sim }{\to }{\mathcal{O}}_A$ with $\tau (J)=t{\mathcal{O}}_A$, with the same inverse-postcomposition action. In either case, $(\tau \cdot {\rho }_1)\cdot {\rho }_2={\varphi }_{{\rho }_2}^{-1}{\varphi }_{{\rho }_1}^{-1}\tau ={\varphi }_{{\rho }_1{\rho }_2}^{-1}\tau =\tau \cdot ({\rho }_1{\rho }_2)\mathrm{.}$ The action is free because $\tau$ is onto; it is transitive on any nonempty coordinate set because the unique series $\rho$ carrying ${\tau }_1$ to ${\tau }_2$ is specified by ${\varphi }_{\rho }={\tau }_1{\tau }_2^{-1}$. If $x:={\tau }^{-1}(t)$ is the corresponding parameter in $J$, then $(\tau \cdot \rho {)}^{-1}(t)=\rho (x)$. Thus the action on parameters is ordinary substitution, while its action on the ring isomorphisms uses the inverse. No formal power series is assumed to converge.