Good and geometric quotients for group actions

Assume AC inherited from the quotient suppliers. Let $G$ be a complex affine algebraic group acting algebraically on a classical complex variety $X$ (Classical complex affine algebraic actions and rational modules) and let $Y$ be a $\mathbb{C}$-scheme (A locally ringed space). For the scheme, affine and sheaf clauses below, $X$ denotes its associated reduced finite-type complex scheme, obtained by gluing spectra of its affine coordinate rings; the algebraic action is interpreted on that scheme. Its complex closed points recover the classical variety (Classical algebraic prevarieties, regular maps, and varieties, Classical k-points give closed points over an algebraically closed field). Closed invariant subsets refer to underlying closed subsets with their reduced induced schemes. A morphism $\pi :X\to Y$ of $\mathbb{C}$-schemes, that is, a morphism of locally ringed spaces commuting with the structure morphisms to $\mathrm{Spec}\mathbb{C}$ (Morphisms of locally ringed spaces), is a good quotient of the action if:

(i) $\pi$ is $G$-invariant and surjective;

(ii) $\pi$ is affine, i.e. ${\pi }^{-1}(U)$ is an affine scheme for every affine open $U\subseteq Y$;

(iii) for every open $U\subseteq Y$ the pullback ${\mathcal{O}}_Y(U)\to {\mathcal{O}}_X({\pi }^{-1}U{)}^G$ is an isomorphism onto the $G$-invariant functions;

(iv) for every closed $G$-stable $Z\subseteq X$ the image $\pi (Z)$ is closed in $Y$; and

(v) for disjoint closed $G$-stable $Z_1,Z_2\subseteq X$ one has $\pi (Z_1)\cap \pi (Z_2)=\mathrm{\varnothing }$.

It is a geometric quotient if in addition for every $y\in Y(\mathbb{C})$, the complex points of its fibre ${\pi }^{-1}(y)$ form exactly one $G(\mathbb{C})$-orbit. The fibre condition is stated on complex closed points; it does not identify all scheme points with classical points.

The good-quotient clauses and the single-orbit fibre condition are separate requirements. The projective GIT theorems below establish the good quotient on the semistable locus and guarantee a geometric quotient on the stable locus. A geometric quotient can also occur outside the stable locus, since the single-orbit fibre condition does not require finite stabilizers.

• Relation to the categorical-quotient definition. A good quotient has the categorical universal property for invariant morphisms to classical varieties viewed as their associated schemes; when the target is a classical variety, this is the notion of Categorical and geometric quotients of classical varieties. On complex closed points it gives a geometric quotient there exactly when its fibres are the $G$-orbits; both implications are proved in Good quotients are local on the target and are categorical quotients below. For affine $X$, Finite generation of invariants and the affine categorical quotient supplies the classical closed-point quotient properties. The proof of Affine chart quotients for invariant sections of a linear action additionally verifies the scheme surjectivity, affine and invariant-sheaf clauses, giving good quotients of the affine charts.
• Provenance of the notion. This is Seshadri's notion as presented by Newstead in §1.4 and used by Brion (in the form ${\mathcal{O}}_Y\cong ({\pi }_{\ast }{\mathcal{O}}_X{)}^G$) and by Hoskins in §§3.3-3.4.
• AC. The definition itself uses no choice; the axiom is inherited only through the quotient suppliers that the later theorems invoke, and every theorem that uses such a supplier carries AC explicitly.