Finite-type field schemes lie in the qcqs source domain

Given: The field $k$, finite-type $k$-scheme $X$, and the Axiom of Choice.

Goal: Verify the qcqs evaluation domain without imposing separatedness.

The Axiom of Choice is assumed for the Noetherian-spectrum and open-subset compactness results below.

Field, Schemes and Morphisms of schemes supply the field and scheme-morphism structures. By Affine schemes and their coordinate rings an affine scheme has a coordinate-ring presentation as a spectrum; The underlying space of an affine spectrum and The prime spectrum and vanishing sets identify its underlying points with prime ideals. An affine open carries the restricted structure sheaf by Affine open subschemes. Finitely generated $k$-algebra means finite type over $k$ in Subalgebra generated by a subset, algebras of finite type, and module-finite algebras.

Locally finite type and finite type morphisms supplies local finite type and quasi-compactness of the structure morphism. By Quasi-compact and quasi-separated morphisms, inverse images of quasi-compact opens are quasi-compact. To check the base, A field has only the zero ideal and itself, hence is Noetherian says the only ideals of $k$ are $(0)$ and $k$. The ideal $(0)$ is proper because $0\mathrm{\ne }1$, and it is prime: if $ab=0$ with $a\mathrm{\ne }0$, multiplication by $a^{-1}$ gives $b=0$. Thus [F0] identifies $\mathrm{Spec}k$ with a one-point space. Any open cover has a member containing that point, so this member is a finite subcover. In particular $X$ is quasi-compact. For a nonempty affine neighbourhood supplied by local finite type, its image lies in a nonempty affine open of this one-point base, necessarily the whole base. Hence its coordinate algebra is finitely generated over $k$, giving the stated affine cover.

A field has only the zero ideal and itself, hence is Noetherian and Every algebra of finite type over a Noetherian ring is a Noetherian ring make those coordinate algebras Noetherian. Under [A1], The spectrum of a Noetherian ring is a Noetherian topological space makes their spectra Noetherian topological spaces.

Noetherian topological spaces via ACC on opens or DCC on closed subsets supplies the increasing-open-sequence definition. Under [A1], Noetherian open subsets are quasi-compact makes every open subset of a Noetherian space quasi-compact. Quasi-compact and quasi-separated schemes supplies the two scheme conditions in the conclusion.

Subspace topology: the traces of the open sets, its closed sets and its bases, the continuity of the inclusion, and the characteristic property of a map into a subspace gives the topology on an open subset $U$ as the traces $V\cap U$ of opens of $X$. In particular the intersections in step 1.1 are open in each $U_i$.