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Noetherian schemes belong to the qcqs domain

PDF p. 4: Motivic dimension vanishing: Noetherian domain membership

Statement

Assume the Axiom of Choice. Let $X$ be a Noetherian scheme. Every open subset of its underlying space $∣ X ∣$ , with the subspace topology, is quasi-compact. Consequently $X$ is quasi-compact and quasi-separated (qcqs): its underlying space is quasi-compact, and the intersection of every two affine open subschemes is quasi-compact. Quasi-compact means that every open cover has a finite subcover, without any Hausdorff condition. The empty scheme and empty open subsets are included.

Statement references

Locally Noetherian and Noetherian schemes

Quasi-compact and quasi-separated schemes

Affine open subschemes

The spectrum of a Noetherian ring is a Noetherian topological space

Noetherian open subsets are quasi-compact

Open cover, subcover, and compact topological space; a compact subset is a subspace that is compact in its own right

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

The Axiom of Choice

Schemes

Affine schemes and their coordinate rings

The underlying space of an affine spectrum

Noetherian commutative rings and modules

Noetherian topological spaces via ACC on opens or DCC on closed subsets

Topology on a set, open and closed sets, clopen sets, the closed-set axiomatisation, and the coarser/finer comparison

Morphisms of locally ringed spaces

Morphisms of ringed spaces

Homeomorphism, open map, closed map, embedding, and what it means for a property to be topological

Facts & Assumptions

Given: The Axiom of Choice and a Noetherian scheme $X$ .

Goal: Establish quasi-compactness of every open subset and hence the qcqs conditions.

[A1]

The Axiom of Choice is assumed to meet the explicit hypotheses of both spectrum and compact-open suppliers in [F2]; it also permits the finite selections of subcovers below. The Axiom of Choice

[F1]

A Noetherian scheme has a finite affine open cover $X_{1}, \dots, X_{m}$ , with each $X_{a}$ isomorphic to the spectrum of a Noetherian ring; for $X = \emptyset$ the empty cover suffices. Affine opens carry the restricted structure sheaf and their underlying topology. Locally Noetherian and Noetherian schemes Affine open subschemes Schemes Affine schemes and their coordinate rings The underlying space of an affine spectrum Noetherian commutative rings and modules

[F2]

Under Choice, the spectrum of a Noetherian ring is a Noetherian topological space, and every open subset of a Noetherian topological space is quasi-compact, including the empty open subset. The spectrum of a Noetherian ring is a Noetherian topological space Noetherian open subsets are quasi-compact Noetherian topological spaces via ACC on opens or DCC on closed subsets

[F3]

Quasi-compactness means the intrinsic finite-subcover property. Relative open sets are traces of ambient open sets, and tracing twice gives the same subspace topology. Quasi-separatedness of a scheme means quasi-compactness of intersections of two affine opens. Open cover, subcover, and compact topological space; a compact subset is a subspace that is compact in its own right 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 Quasi-compact and quasi-separated schemes The whole space, the empty set and intersections of two open sets are open. Topology on a set, open and closed sets, clopen sets, the closed-set axiomatisation, and the coarser/finer comparison

[F4]

Affine identifications are isomorphisms of locally ringed spaces: here an isomorphism means a morphism with a two-sided inverse. Morphisms of locally ringed spaces are morphisms of ringed spaces, whose underlying maps are continuous and whose composition composes those maps. A homeomorphism is a continuous bijection with continuous inverse; it carries open sets bijectively to open sets. Morphisms of locally ringed spaces Morphisms of ringed spaces Homeomorphism, open map, closed map, embedding, and what it means for a property to be topological

Proof

1.1A1F1F2F3F4choose

Choose the finite cover of [F1]. For any open subset $O \subseteq ∣ X ∣$ , the subset $O \cap X_{a}$ is open in $X_{a}$ . By [F4] the affine identification and its inverse induce continuous maps whose composites are identity maps, hence a homeomorphism. Therefore the inverse image of $O \cap X_{a}$ is open in the spectrum of the Noetherian coordinate ring. By [A1] and [F2] that inverse image is quasi-compact. Pulling any open cover back along this homeomorphism and pushing a finite subcover forward proves that $O \cap X_{a}$ is quasi-compact in its subspace topology. The successive subspace topologies agree by [F3].

2.1A1F1F3step 1.1∎

Let $U$ be any open cover of the space $O$ . For each $a$ , its traces on $O \cap X_{a}$ form an open cover of that space. Step 1.1 gives finitely many such traces covering it; choose their corresponding members of $U$ . There are only finitely many $a$ , so the union of these finite lists is a finite subfamily of $U$ and covers $O = ⋃_{a = 1}^{m} (O \cap X_{a})$ . If $O$ is empty, take the empty subfamily. Hence every open $O$ is quasi-compact. Taking $O = ∣ X ∣$ proves quasi-compactness of $X$ , and taking $O$ to be the intersection of any two affine opens proves quasi-separatedness by [F3]. Thus $X$ is qcqs.

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