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Finite-type schemes over fields are qcqs

Statement

Let k be a field and X a finite-type k-scheme. Then X is quasi-compact and quasi-separated. No separatedness hypothesis on X is required.

Facts & Assumptions

Given: A field k and a finite-type structure morphism X→Spec⁡k.

Goal: Verify the qcqs domain required by the motivic comparison.

[F1]

The field and structure-morphism conventions are those of Field and Schemes and morphisms over a base. Finite type refers to this morphism, as in Locally finite type and finite type morphisms. Quasi-compactness and quasi-separatedness mean the conditions in Quasi-compact and quasi-separated schemes.

[F2]

A field has only the zero ideal and itself and is Noetherian, by A field has only the zero ideal and itself, hence is Noetherian. Points of Spec⁡k are the prime ideals, by The underlying space of an affine spectrum, with primality as in Prime ideals and maximal ideals in a commutative ring. A Noetherian scheme is a quasi-compact scheme with an affine open cover by spectra of Noetherian rings, by Locally Noetherian and Noetherian schemes.

[F3]

Reference 0 supplies the theorem that a finite-type morphism to a Noetherian scheme has Noetherian source.

[F4]

Reference 1 supplies quasi-separatedness of every locally Noetherian scheme.

[F5]

Here Spec⁡k carries its standard structure sheaf: the resulting spectrum is a locally ringed space by Spec A with its structure sheaf is a locally ringed space, an affine scheme by Affine schemes and their coordinate rings, and a scheme by Schemes, using the whole space as its affine open neighbourhood.

External input

The Noetherian-source theorem in [F3] and the locally-Noetherian quasi-separatedness theorem in [F4] are imported from the cited Stacks statements under their exact hypotheses. The field-base verification and composition below are local arguments, not reconstructions of those imported theorems. This item is a locally proved consequence, rather than an imported claim of a local proof of either theorem.

Proof

1.1givenF1F2F3F5

By [F5], Spec⁡k is an affine scheme with its standard structure sheaf. By [F2], k is Noetherian. Its zero ideal is prime: if ab=0 and a≠0, multiplying by a−1 gives b=0. Since its only other ideal is the improper ideal, Spec⁡k has one point. Any open cover of a one-point space has a single member covering that point, so this space is quasi-compact. Its one-chart cover by Spec⁡k therefore makes it a Noetherian scheme by [F2]. The given finite-type morphism now satisfies [F3], so X is Noetherian.

2.1F1F2F4step 1.1∎

By the Noetherian-scheme definition in [F2], X is quasi-compact and locally Noetherian. Apply [F4] to conclude it is quasi-separated. These are precisely the two conditions in [F1]. This also includes the empty scheme, and no separatedness assumption has been used.

Depends on

Used by · 3

Dependency tree · 2 levels

Each result is drawn at its shortest distance from this item. Arrows run from prerequisites to the items that use them; this item has a heavier outline. External source nodes link directly to the cited text. The view shows up to two dependency steps; open a prerequisite to continue.

External dependencies

The Stacks Project, Morphisms of finite type · Lemma 29.15.6, complete statement and short displayed proof.

A finite-type morphism to a Noetherian scheme has Noetherian source; the corresponding local assertion also holds.

Hypotheses: A morphism of schemes, Noetherian target, and finite-type morphism for the global assertion. Here the target is Spec k, shown Noetherian in step 1.1.

Reading limits: Freshly read the statement and displayed proof at tag 01T6. Import the Noetherian-source theorem; its cited Hilbert basis theorem is not reconstructed. The field-base and qcqs deductions below are local.

The Stacks Project, Noetherian schemes · Definition 28.5.1 and Lemma 28.5.4, with the displayed proof of the latter.

A Noetherian scheme is locally Noetherian and quasi-compact. Every locally Noetherian scheme is quasi-separated.

Hypotheses: A scheme that is locally Noetherian for Lemma 28.5.4. Step 1.1 obtains the stronger Noetherian hypothesis on X.

Reading limits: Freshly read Definition 28.5.1 and the statement and displayed proof of Lemma 28.5.4, including its reduction to intersections of affine opens. Import the quasi-separatedness result; the cited immersion and algebraic-spectrum inputs are not reconstructed.

Sources