A nonsingular fixed point is a nonsingular point of the fixed point set

Question

Let $G$ be a finite group and let $X$ be a complex algebraic set equipped with an action of $G$ via biregular isomorphisms $\alpha_g$, $g \in G$. Let $x$ be a point of $X$ which is fixed by the action of $G$. I want to understand the proof of the following fact : if $x$ is a nonsingular point of $X$ then $x$ is a nonsingular point of the fixed point set $X^G$.

Actually, I found a proof in the book Algebraic Groups: The Theory of Group Schemes of Finite Type over a Field of J. S. Milne (Theorem 13.7), which follows the proof of the article A Fixed Point Formula for Action of Tori on Algebraic Varieties by B. Iversen. I well understand the most part of the proof of Milne's book, however I do not understand the proof of Lemma 13.4 : why is the local ring of $X^G$ at $x$ the quotient of the local ring of $X$ at $x$ by the ideal $\mathfrak{a} := \{f - f \circ \alpha_g~|~f \in m, g \in G\} (= \{f - f \circ \alpha_g~|~f \in \mathcal{O}_{X,x}, g \in G\})$ ?

More precisely, I well understand why, for any local algebra $R$, $Hom(\mathcal{O}_{X,x}/\mathfrak{a},R)$ is isomorphic to $Hom(\mathcal{O}_{X,x},R)^G$ but I do not understand why this fact implies the equality between $\mathcal{O}_{X^G,x}$ and $\mathcal{O}_{X,x}/\mathfrak{a}$.

Thanks for any help !

Answer 1

Thanks to Asvin's comments, I managed to get to a complete answer to my question : I will try to write it below.

First, Milne's book proof lies on Yoneda Lemma applied to the category of local algebras over $\mathbb{C}$ (see above's comments). Now, let $\varphi : \mathcal{O}_{X^G,x} = \mathbb{C}[X^G]_{m_x} \rightarrow R$ be a morphism of local $\mathbb{C}$-algebras. It corresponds to a morphism of $\mathbb{C}$-algebras $\psi : \mathbb{C}[X^G] \rightarrow R$ with $Ker\, \psi \subset m_x$ and $\psi(m_x) \subset m$, if $m$ denotes the unique maximal ideal of $R$. The morphism $\psi$ corresponds to a $R$-point $a$ of $X^G$, i.e. a point with coordinates in $R$ which verifies the equations of $X^G$, that is the equations of $X$ and the equations $\alpha_g - id = 0$, $g \in G$. The $R$-point $a$ then corresponds to a morphism of $\mathbb{C}$-algebras $\rho : \mathbb{C}[X] \rightarrow R$ which is fixed by the induced action of $G$ given by right composition with the $\alpha_g$'s (recall that the $\alpha_g$'s are regular isomorphism) and verifies $Ker \, \rho \subset m_x$ (here $m_x$ denotes the maximal ideal of polynomial functions of $\mathbb{C}[X]$ vanishing at $x$) and $\rho(m_x) \subset m$. Therefore $\rho$ corresponds to a morphism of local $\mathbb{C}$-algebras $\mathcal{O}_{X,x} = \mathbb{C}[X]_{m_x} \rightarrow R$, which is fixed by the induced action of $G$.

This proves the equality $Hom(\mathcal{O}_{X^G,x}, R) = Hom(\mathcal{O}_{X,x},R)^G$. Since we also have $Hom(\mathcal{O}_{X,x}/\mathfrak{a},R) = Hom(\mathcal{O}_{X,x},R)^G$, Yoneda lemma shows that $\mathcal{O}_{X,x}/\mathfrak{a}$ is isomorphic, as a local $\mathbb{C}$-algebra, to $\mathcal{O}_{X^G,x}$.