Affine chart quotients for invariant sections of a linear action

Affineness and stability. If $X=\mathrm{\varnothing }$, its homogeneous coordinate ring is zero, the invariant ring and every localized chart ring are zero, both loci and every quotient target are empty, and all conclusions hold with the empty morphisms. Assume henceforth $X\mathrm{\ne }\mathrm{\varnothing }$. The sheaf $\mathcal{O}(1){\mathrm{\mid }}_X$ is ample and linearized, and $f$ is an invariant global section of $\mathcal{O}(d){\mathrm{\mid }}_X$ (homogeneous coordinate ring, affine cone projective set), and [F2] identifies $X_f$ with the affine chart $D_{+}(f)$ of $\mathrm{Proj}R$. Its $G$-stability follows from [F1].

Finite generation of the invariant ring of the chart. $R^G$ is a finitely generated graded $\mathbb{C}$-algebra with $(R^G{)}_0=\mathbb{C}$ by [F4], so $(R^G{)}_{(f)}$ is a finitely generated $\mathbb{C}$-algebra by the second part of [F4]: indeed, if $R^G=\mathbb{C}[u_1,\dots ,u_r]$ with $u_i$ homogeneous of degrees $e_i$ and $d=\mathrm{deg}f$, then $(R^G{)}_{(f)}$ is generated by the finitely many elements $u_i^d\mathrm{/}{f}^{e_i}$ together with the elements $({\prod }_i{u}_i^{{\epsilon }_i})\mathrm{/}{f}^l$ for all exponent vectors $0\le {\epsilon }_i<d$ with $\sum {\epsilon }_i{e}_i=dl$, because every monomial $\prod u_i^{a_i}$ of degree divisible by $d$ splits as ${\prod }_i(u_i^d{)}^{q_i}\cdot {\prod }_i{u}_i^{{\epsilon }_i}$ with ${\epsilon }_i<d$ and the remainder of degree divisible by $d$.

The invariant coordinate ring of the chart. By [F2] the chart $D_{+}(f)$ of $\mathrm{Proj}R$ has coordinate ring $R_{(f)}$, and the affine chart $X_f$ has coordinate ring $\mathcal{O}(X_f)=R_{(f)}$. By [F3] applied to the graded rational $G$-algebra $R$ and the homogeneous invariant $f$, $(R_{(f)}{)}^G=((R_f{)}_0{)}^G=(R^G{)}_{(f)}$; this proves (ii).

The chart quotient is a scheme good quotient. Put $C=\mathcal{O}(X_f)$ and $B=C^G=(R^G{)}_{(f)}$ by step 2.1. The affine scheme morphism ${\pi }_f:\mathrm{Spec}C\to \mathrm{Spec}B=D_{+}(f)$ is invariant and affine. It is surjective at every scheme point: the $B$-linear Reynolds retraction splits $B\hookrightarrow C$, so for every prime $\mathfrak{p}\subset B$ tensoring gives an injection $\kappa (\mathfrak{p})\hookrightarrow C{\otimes }_B\kappa (\mathfrak{p})$. This nonzero fibre algebra has a prime by [F5]. On each principal target open $D(b)$, the rational-localized Reynolds computation gives $(C_b{)}^G=B_b$, so the sheaf clause holds on a basis and hence on all opens. For a closed invariant subset with radical stable ideal $I\subset C$, invariant exactness gives $(C\mathrm{/}I{)}^G=B\mathrm{/}(I\cap B)$. Applying the same Reynolds-splitting fibre argument to $C\mathrm{/}I$ shows its image is exactly the scheme closed subset $V(I\cap B)$. If two such subsets are disjoint, their ideals satisfy $I+J=C$; write $1=a+b$, apply Reynolds, and use preservation of stable ideals to obtain $1\in (I\cap B)+(J\cap B)$, so their scheme images are disjoint. These are all good-quotient clauses, proving (iii); the target is $D_{+}(f)$, so ${\pi }_f^{-1}(D_{+}(f))=X_f$.

Compatibility. For invariant homogeneous $f,g$ of positive degrees the charts satisfy $D_{+}(fg)=D_{+}(f)\cap D_{+}(g)$ and $X_{fg}=X_f\cap X_g$, and localizing the identifications of step 2.1 at $h=g^{\mathrm{deg}f}\mathrm{/}{f}^{\mathrm{deg}g}$ gives $\mathcal{O}(X_{fg})=(\mathcal{O}(X_f){)}_h$ with invariant ring $((R^G{)}_{(f)}{)}_h=(R^G{)}_{(fg)}$; both ${\pi }_f$ and ${\pi }_g$ restrict on $X_{fg}$ to the affine quotient morphism with target $\mathrm{Spec}(R^G{)}_{(fg)}=D_{+}(fg)$, because the corresponding ring maps are the canonical localizations of $\mathcal{O}(X_f{)}^G\to \mathcal{O}(X_f)$ and of $\mathcal{O}(X_g{)}^G\to \mathcal{O}(X_g)$ at the same localization. This is assertion (iv) and completes the proof; no new choice is made, the Axiom of Choice being inherited from the invariant-theory and quotient suppliers [F6].