Finite generation of invariants and the affine categorical quotient

Part (i): by [F3] there is a finite-dimensional rational module $W$ and a surjective $G$-algebra map $\mathbb{C}[W^{\ast }]\to A$; by [F1] the invariant algebra $\mathbb{C}[W^{\ast }{]}^G$ is finitely generated over $\mathbb{C}$; by the surjectivity statement of [F2] the induced map $\mathbb{C}[W^{\ast }{]}^G\to A^G$ is surjective. Hence $A^G$ is a finitely generated $\mathbb{C}$-algebra.

The localisation identity: for $f\in A^G$, the algebra $A_f$ is rational, since each fraction $a\mathrm{/}{f}^m$ lies in the image of a finite-dimensional rational submodule containing $a$, divided by the invariant $f^m$. Thus the Reynolds operator exists on $A_f$. Naturality under $A\to A_f$ and linearity over the invariant invertible element $f$ give $R_{A_f}(a\mathrm{/}{f}^m)=R_A(a)\mathrm{/}{f}^m$. If $a\mathrm{/}{f}^m$ is invariant, it is fixed by $R_{A_f}$ and hence has an invariant numerator. Conversely every fraction with invariant numerator is invariant. The natural map $(A^G{)}_f\to A_f$ is injective: if $a\in A^G$ is killed by a power of $f$ in $A$, the same equation holds in the subring $A^G$. Hence $(A_f{)}^G=(A^G{)}_f$, without asserting injectivity of the unlocalized map $A^G\to A_f$.

Parts (ii) and (iii), core: let $I\subseteq A^G$ be a maximal ideal and put $J=IA$. Then $J\cap A^G=R_X(J)=I\mathrm{\ne }{A}^G$ by [F2], so $J$ is a proper ideal of $A$; by [F6] it lies in a maximal ideal $\mathfrak{m}$ of $A$. The contraction $\mathfrak{m}\cap A^G$ is proper and contains $I$, so maximality of $I$ gives $\mathfrak{m}\cap A^G=I$. By the correspondence [F4] the maximal ideal $\mathfrak{m}$ is a point $x\in X$ with $\pi (x)=I$. Thus $\pi :X\to \mathrm{Spec}{A}^G$ is surjective. For a finite generating set $f_1,\dots ,f_n$ of $A^G$, let $P=\mathbb{C}[t_1,\dots ,t_n]$ and map $t_i\mapsto f_i$; this is surjective onto $A^G$. The subring $A^G$ of the reduced ring $A$ is reduced, so this kernel is radical; [F4] identifies its zero locus with the coordinate-ring model, giving a closed embedding $\mathrm{Spec}{A}^G\hookrightarrow {\mathbb{C}}^n$. The morphism in (ii) is the composite of the surjective $\pi$ with this closed embedding, so its image is exactly that closed affine subvariety. If a different generating set is chosen, both closed images represent $\mathrm{Spec}{A}^G$ via their coordinate-ring maps, and [F5] gives the canonical isomorphism between them.

Part (vi): if $X$ is irreducible then $A$ is a domain by [F4], and $A^G$ is a subring of a domain, hence a domain, so $X\mathrm{/}\text{\hspace{0.17em}⁣}\mathrm{/}G=\mathrm{Spec}{A}^G$ is irreducible. If in addition $X$ is normal, then $A$ is integrally closed in $F(A)$; let $u\in \mathrm{Frac}(A^G)$ be integral over $A^G$. Since $\mathrm{Frac}(A^G)\subseteq \mathrm{Frac}(A{)}^G$, the element $u$ lies in $\mathrm{Frac}(A)$ and is integral over $A$, so $u\in A$; being fixed by $G$, it lies in $A^G$. Hence $A^G$ is integrally closed in its fraction field, and by [F9] the variety $X\mathrm{/}\text{\hspace{0.17em}⁣}\mathrm{/}G$ is normal.

Part (iv): let $Y\subseteq X$ be closed and $G$-stable with radical ideal $I_Y\subseteq A$. The quotient map $A\to A\mathrm{/}{I}_Y$ is surjective and $G$-equivariant, so by [F2] the induced map $A^G\to (A\mathrm{/}{I}_Y{)}^G$ is surjective with kernel $I_Y\cap A^G$; by [F5] the corresponding morphism $Y\mathrm{/}\text{\hspace{0.17em}⁣}\mathrm{/}G=\mathrm{Spec}(A\mathrm{/}{I}_Y{)}^G\to \mathrm{Spec}{A}^G=X\mathrm{/}\text{\hspace{0.17em}⁣}\mathrm{/}G$ is a closed immersion. For closed $G$-stable $Y,Y^{\mathrm{\prime }}$ and a point $q\in X\mathrm{/}\text{\hspace{0.17em}⁣}\mathrm{/}G$ with maximal ideal $\mathfrak{m}\subseteq A^G$, one has $q\in \pi (Y)\cap \pi (Y^{\mathrm{\prime }})$ exactly when both $I_Y+\mathfrak{m}A$ and $I_{Y^{\mathrm{\prime }}}+\mathfrak{m}A$ are proper. If they are, then $I_Y+I_{Y^{\mathrm{\prime }}}+\mathfrak{m}A$ is proper: otherwise $1=a+b+c$ with $a\in I_Y$, $b\in I_{Y^{\mathrm{\prime }}}$, $c\in \mathfrak{m}A$, and applying $R_X$, which maps $I_Y$ into $I_Y\cap A^G\subseteq \mathfrak{m}$, $I_{Y^{\mathrm{\prime }}}$ into $\mathfrak{m}$ and $\mathfrak{m}A$ onto $\mathfrak{m}$, would give $1\in \mathfrak{m}$. A maximal ideal containing this proper ideal is a point of $Y\cap Y^{\mathrm{\prime }}$ mapping to $q$, so $q\in \pi (Y\cap Y^{\mathrm{\prime }})$; the reverse inclusion is immediate.

Part (v): each fibre ${\pi }^{-1}(q)$ is nonempty by surjectivity of step 2.2, closed and $G$-stable; it contains a closed orbit by [F7]. If it contained two distinct closed orbits $Gx,G{x}^{\mathrm{\prime }}$, then applying part (iv) of step 3.1 to the closed $G$-stable sets $Gx$ and $G{x}^{\mathrm{\prime }}$ gives $\pi (Gx)\cap \pi (G{x}^{\mathrm{\prime }})=\pi (Gx\cap G{x}^{\mathrm{\prime }})=\mathrm{\varnothing }$, contradicting that both contain $q$. Hence each fibre contains exactly one closed orbit.

Part (iii), full universality: let $h:X\to Z$ be a $G$-invariant morphism to a separated classical variety $Z$. First, $h$ is constant on every fibre of $\pi$: for $y$ in a fibre, $h$ is constant on the closure of the orbit $Gy$, because the preimage of the value $h(y)$ is closed in $X$ and contains $Gy$; that closure lies in the fibre, which is closed and $G$-stable, and contains the unique closed orbit of the fibre by step 4.1, so $h(y)$ equals the value on that orbit, the same value on every point of the fibre. Write $\bar{h}$ for the induced map on $X\mathrm{/}\text{\hspace{0.17em}⁣}\mathrm{/}G$. For $q\in X\mathrm{/}\text{\hspace{0.17em}⁣}\mathrm{/}G$ choose an affine chart $V\subseteq Z$ containing $\bar{h}(q)$; the closed $G$-stable set $C=h^{-1}(Z\setminus V)$ has closed image $\pi (C)$ by step 3.1, and $q\mathrm{\notin }\pi (C)$ because the whole fibre of $q$ maps into $V$. A regular function $g\in A^G$ vanishes on $\pi (C)$ and is nonzero at $q$ by the correspondence [F4]; then $h$ maps the principal open $X_g={\pi }^{-1}(D(g))$ into $V$, each coordinate of $h{\mathrm{\mid }}_{X_g}$ is an invariant regular function on the affine open $X_g$, hence lies in $(A_g{)}^G=(A^G{)}_g$ by [F10] and step 2.1, and the dictionary [F5] produces a morphism $D(g)\to V$ inducing $\bar{h}$. These local morphisms agree on overlaps, since $\pi$ is surjective, so they glue to a morphism $\bar{h}:X\mathrm{/}\text{\hspace{0.17em}⁣}\mathrm{/}G\to Z$ with $\bar{h}\circ \pi =h$; uniqueness is surjectivity of $\pi$. Thus $\pi$ is a categorical quotient, and (ii) follows from the same dictionary because any two finite generating sets present $\mathrm{Spec}{A}^G$ canonically.

Assembly and conventions: (i) is step 1.1, (ii) and (iii) are steps 2.2 and 5.1, (iv) is step 3.1, (v) is step 4.1 and (vi) is step 2.3. In the classical register, $\mathrm{Spec}$ of a finitely generated reduced complex algebra denotes the affine variety of its complex closed points with the classical regular-function sheaf, and no identification with the space of all scheme primes is used. If $X=\mathrm{\varnothing }$ then $A=0$, the invariant algebra is zero, finite generation is immediate, $X\mathrm{/}\text{\hspace{0.17em}⁣}\mathrm{/}G=\mathrm{\varnothing }$, and the fibre assertions are vacuous; the maximal-ideal argument above concerns nonempty $X$. The Axiom of Choice is inherited from the embedding, Reynolds, Nullstellensatz and normality suppliers named above. This proves all six clauses.