Good quotients are local on the target and are categorical quotients

Clauses (i)-(v) are local on the target. Assume every restriction ${\pi }_i:{\pi }^{-1}(U_i)\to U_i$ is a good quotient. Then $\pi$ is $G$-invariant because the $U_i$ cover $Y$ and invariance is a local condition on the target, and $\pi$ is surjective because each ${\pi }_i$ is. Affineness follows from Affineness is local on the target: refine the $U_i$ by affine opens; their preimages are affine since each ${\pi }_i$ is affine. For clause (iii), each restriction ${\mathcal{O}}_Y(U_i)\to {\mathcal{O}}_X({\pi }^{-1}{U}_i{)}^G$ is an isomorphism, and for arbitrary open $U\subseteq Y$ the maps over $U\cap U_i$ agree on overlaps because they are determined by restriction of regular functions; since both ${\mathcal{O}}_Y$ and the invariant-function presheaf are sheaves, the map ${\mathcal{O}}_Y(U)\to {\mathcal{O}}_X({\pi }^{-1}U{)}^G$ is an isomorphism. For clause (iv), if $Z\subseteq X$ is closed and $G$-stable, then $\pi (Z)\cap U_i={\pi }_i(Z\cap {\pi }^{-1}(U_i))$ is closed in $U_i$ because $Z\cap {\pi }^{-1}(U_i)$ is closed and $G$-stable there; a subset of $Y$ whose traces on all $U_i$ are closed is closed. Clause (v) is checked the same way: $\pi (Z_1)\cap \pi (Z_2)\cap U_i={\pi }_i(Z_1\cap {\pi }^{-1}{U}_i)\cap {\pi }_i(Z_2\cap {\pi }^{-1}{U}_i)=\mathrm{\varnothing }$. Hence $\pi$ is a good quotient.

Constancy on complex-point fibres. Let $h:X\to Z$ be invariant, with $Z$ the associated separated scheme of a classical variety. If complex points $x,x^{\mathrm{\prime }}$ have the same image under $\pi$ but distinct images $z,z^{\mathrm{\prime }}$ under $h$, the disjoint closed invariant subsets $h^{-1}(z),h^{-1}(z^{\mathrm{\prime }})$ have intersecting images under $\pi$, contradicting clause (v). Thus $h$ is constant on complex-point fibres. A complex point maps to a closed point of any complex scheme: in each affine open its map to $\mathbb{C}$ is surjective with maximal kernel.

The geometric property is local. Suppose each restriction is a geometric quotient. A good quotient has orbit fibres exactly when every restriction has orbit fibres, since the fibres of $\pi$ over $U_i$ are the fibres of ${\pi }_i$; by step 1.1 and [F1] this is exactly the geometric-quotient property. Conversely if $\pi$ is a good quotient with orbit fibres then each restriction is one.

Constancy on all topological fibres. Since $X$ is quasi-compact and $\pi$ is surjective, $Y$ is quasi-compact. Choose a finite affine cover $V_i=\mathrm{Spec}{B}_i$ of $Y$. Affineness gives ${\pi }^{-1}(V_i)=\mathrm{Spec}{A}_i$, an affine open of finite-type $X$, so $A_i$ is a finite-type complex algebra. Therefore $R=X{\times }_{Y}X$ is finite type over $\mathbb{C}$: over $V_i$ its ring $A_i{\otimes }_{B_i}{A}_i$ is a quotient of $A_i{\otimes }_{\mathbb{C}}{A}_i$. The closed equalizer in $R$ of $h{\mathrm{pr}}_1,h{\mathrm{pr}}_2$ contains every complex closed point by step 1.2, hence has underlying set all of $R$ by [F4]. For points $x,x^{\mathrm{\prime }}$ over $y$, the tensor product $\kappa (x){\otimes }_{\kappa (y)}\kappa (x^{\mathrm{\prime }})$ is nonzero (tensoring field extensions over a field preserves nonzero injections). A prime of it gives a common field-valued point of $R$ dominating $x,x^{\mathrm{\prime }}$, so the equalizer condition on underlying points forces $h(x)=h(x^{\mathrm{\prime }})$. Thus $h$ is constant on every topological fibre.

Open charts and descent. For an affine chart $V=\mathrm{Spec}A$ of $Z$, put $U=h^{-1}(V)$. Constancy on fibres makes $U$ saturated. Consequently $W=\pi (U)=Y\setminus \pi (X\setminus U)$ is open by clause (iv), and $U={\pi }^{-1}(W)$. Every coordinate in $A$ pulls back to an invariant regular function on $U$, which descends uniquely to $W$ by clause (iii); these descended functions respect sums, products and all relations because pullback is an isomorphism. They define a morphism $W\to \mathrm{Spec}A$ by Morphisms to an affine scheme and global sections. On overlaps the underlying maps agree by surjectivity of $\pi$ and the coordinate pullbacks agree by the same sheaf isomorphism, so they glue to $\phi :Y\to Z$ with $\phi \pi =h$. The same arguments give uniqueness of both its underlying map and sheaf map. This is the categorical universal property.

Geometric criterion and classical topology. The geometric criterion is precisely the complex-point fibre condition in [F1]. For its relation with the classical convention, the induced map $X(\mathbb{C})\to Y(\mathbb{C})$ is surjective: every fibre over a complex point is nonempty by scheme surjectivity and finite type over $\mathbb{C}$ by the affine chart description in step 2.2, so [F4] supplies a complex point in it. If $U\subseteq Y(\mathbb{C})$ has open preimage in $X(\mathbb{C})$ and the fibres are orbits, its complementary preimage is a closed invariant classical subset, hence the complex points of a closed reduced subset $C\subseteq X$. Clause (iv) makes $\pi (C)$ closed in $Y$, and $\pi (C)\cap Y(\mathbb{C})=Y(\mathbb{C})\setminus U$: a fibre of $C$ over a complex point is nonempty precisely when it has a complex point, again by [F4]. Thus $U$ is open in the induced classical topology. The converse follows by continuity, and the invariant-function condition is inherited from clause (iii). When $Y$ is a classical variety through its associated scheme, these are exactly the classical geometric-quotient conditions.

Steps 1.1 and 2.1 prove locality on the target for the good and geometric properties, step 3.1 proves the categorical universal property of a good quotient, and step 4.1 proves that a good quotient has orbit fibres exactly when it is geometric, which is assertion (ii). The Axiom of Choice is used through the closed-point and maximal-ideal suppliers as recorded in [F3].