Reflection basic invariants form a regular sequence

Choose coordinates $x_1,\dots ,x_n$. For each coordinate $x_j$, the polynomial ${\prod }_{g\in G}(T-g{x}_j)$ is monic of degree $q=\mathrm{\mid }G\mathrm{\mid }$, has invariant coefficients, and vanishes at $T=x_j$. Its coefficient of $T^{q-a}$ is homogeneous of degree $a$. Reduce powers $x_j^q$ using these equations. Induction on the sum of exponents shows that the monomials $x_1^{a_1}\cdots x_n^{a_n}$ with $0\le a_j<q$ generate $S$ over $R$; the reductions preserve total degree. Thus $S\mathrm{/}I$ is finite dimensional. If all $f_i(v)$ vanish, every invariant takes its constant value at $v$ by F2. F1 applied to $Gv$ and $\{0\}$ forces $v=0$. Conversely positive-degree polynomials vanish at zero.

F2 identifies $R$ with a polynomial algebra in $m$ variables of positive weights $d_i$. The number of its monomials of weighted degree at most $N$ grows between positive constants times $(N+1{)}^m$: when $m>0$, all exponents at most $\lfloor N\mathrm{/}(m\max d_i)\rfloor$ give the lower bound, and all exponents are at most $N$ for the upper bound; for $m=0$ the count is one. The analogous count for $S$ grows as $(N+1{)}^n$, by the same coordinate argument. Since $R_{\le N}\subseteq S_{\le N}$, the lower bound forces $m\le n$. The finite homogeneous spanning family of 1.1 gives a graded surjection from finitely many degree shifts of $R$ onto $S$, so the upper bound forces $n\le m$. Hence $m=n$, without a transcendence-degree or Nullstellensatz appeal.

Put $\mathfrak{m}=(x_1,\dots ,x_n)$ and $A=S_{\mathfrak{m}}$. This is a nonzero Noetherian local ring: F8 applies over the field, and any ideal in a localization is generated by the images of generators of its contraction. The coordinate tuple is regular on $A$: after quotienting by its first $j$ terms the ring is the localization at the origin of the polynomial domain in the remaining coordinates, and multiplication by the next coordinate is injective. The terminal quotient is $\mathbb{C}$. Therefore F4 gives $\mathrm{depth}A\ge n$, F6 gives $\dim A\le n$, and F5 forces equality. Thus $A$ is Cohen--Macaulay. The quotient $A\mathrm{/}IA$ is nonzero, since $I\subseteq \mathfrak{m}$, and finite dimensional over $\mathbb{C}$ by 1.1. It has finite length as an $A$-module: every strict module chain is a strict complex-subspace chain. Now 2.1 and F6 make $(f_1,\dots ,f_n)$ a parameter system, so F7 makes it regular on $A$.

Regularity descends here to the graded ring $S$. Indeed, for a homogeneous ideal $J$ and nonzero homogeneous class $u\in S\mathrm{/}J$, localization at $\mathfrak{m}$ cannot kill $u$: if $su=0$ with $s\mathrm{\notin }\mathfrak{m}$, the lowest-degree component is $s(0)u\mathrm{\ne }0$. If multiplication by a homogeneous $f_j$ had a kernel on $S\mathrm{/}(f_1,\dots ,f_{j-1})$, one of the homogeneous components of a nonzero kernel element would therefore give a nonzero kernel after localization, contradicting 3.1. Every quotient remains nonzero because the ideals have positive-degree generators. F3 proves global regularity.

Let $H_M(t)={\sum }_{a\ge 0}{\dim }_{\mathbb{C}}{M}_a{t}^a$ for these graded spaces, whose components are finite dimensional. The injective multiplication by $f_j$ on the preceding quotient gives, degree by degree, $H_{S\mathrm{/}(f_1,\dots ,f_j)}(t)=(1-t^{d_j})H_{S\mathrm{/}(f_1,\dots ,f_{j-1})}(t)$. Counting polynomial monomials gives $H_S(t)=(1-t{)}^{-n}$ and, by F2, $H_R(t)={\prod }_j(1-t^{d_j}{)}^{-1}$. Thus $H_{S\mathrm{/}I}(t)H_R(t)=H_S(t)$.

Choose any homogeneous basis ${\bar{b}}_1,\dots ,{\bar{b}}_l$ of the finite-dimensional graded quotient and homogeneous lifts $b_a$. They span $S$ over $R$ by degree induction: for homogeneous $s$ subtract the combination of lifts representing its quotient class; the remainder is ${\sum }_j{f}_j{s}_j$ with $\mathrm{deg}{s}_j<\mathrm{deg}s$, so apply induction. The resulting graded surjection ${⨁}_{a}R(-\mathrm{deg}{b}_a)⟶S$ has equal source and target dimensions in every degree by 5.1; its degreewise kernels are zero. Hence it is an isomorphism and the lifts are a free basis. For $n=0$, $S=R=\mathbb{C}$, $I=0$, the empty regular sequence has nonzero terminal quotient and the basis is any nonzero constant. Degree-one invariants and fixed directions are allowed throughout. All basis/lift selections concern one finite-dimensional space; no AC is used.