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Constant cdh sections on the two article sites

PDF p. 10: Lemma 5.1, weight-zero restriction

Statement

Assume the Axiom of Choice and choose the site universes specified below. Let $k$ be a field with prime subfield $F$ , and let $X$ be a separated finite-type $k$ -scheme. Evaluation at $X$ of the constant integer cdh sheaf on the specified qcqs $F$ -scheme site is canonically isomorphic to evaluation at $X$ of the constant integer cdh sheaf on the site of separated finite-type $k$ -schemes. The isomorphism is natural for morphisms between such $k$ -schemes.

Here qcqs means quasi-compact and quasi-separated. The first site is the small skeleton of qcqs $F$ -schemes in the chosen universe, with topology generated by Nisnevich covers and abstract blowup families whose closed immersion and proper map are finitely presented. The second site consists of separated finite-type $k$ -schemes with the cdh topology generated by Nisnevich covers and abstract blowup families. An abstract blowup family is ${Z \to T, T^{'} \to T}$ , where $Z \to T$ is a closed immersion, $T^{'} \to T$ is proper, and $T^{'} ∖ (T^{'} \times_{T} Z) \to T ∖ Z$ is an isomorphism. The constant integer sheaf is the sheafification of the presheaf with value $Z$ and identity restriction maps.

Facts & Assumptions

Given: Choice, the two specified sites and a separated finite-type $k$ -scheme $X$ .

Goal: Compare the actual constant-sheaf evaluations and their restriction maps.

[F1]

The generators and universe convention of the qcqs prime-field site are fixed by The source qcqs cdh site. The finite-type interface Completely decomposed proper covers and the cdh topology defines cdh using Nisnevich covers and completely decomposed proper maps.

[F4]

On separated finite-type $k$ -schemes that topology is also generated by Nisnevich covers and actual abstract blowup families, including for nonreduced schemes. Completely decomposed proper maps are abstract-blowup covers

[F2]

Sheafification is obtained by applying the plus construction twice; plus sections are matching families on covering sieves modulo agreement on a common covering refinement. Scheme sites and sheafification

[F3]

Every separated finite-type scheme over a field is noetherian. A finite-type morphism between noetherian schemes is finitely presented; an étale morphism is locally of finite presentation. A quasi-compact open subspace of a noetherian scheme is noetherian. These standard finite-presentation facts follow on affine charts from finite generation of ideals in noetherian rings.

Proof

1.1F1F3F4given

By [F4], the finite-type topology is generated equally by Nisnevich covers and actual abstract blowup families. Every generator in this presentation, viewed over $F$ , is a cover in the qcqs site: its closed immersion and proper map are finitely presented by [F3], and its Nisnevich family is a Nisnevich cover in the larger site. The same holds for pullbacks and finite composites of generators. Thus this comparison includes the completely decomposed proper generators of the specified finite-type interface.

1.2F1F3construct

Conversely, consider a generating cover of a separated finite-type $k$ -scheme $T$ in the qcqs site. In an abstract blowup family both members are separated and of finite type over $T$ , hence are separated finite-type $k$ -schemes. For a Nisnevich cover, refine its members by affine open subschemes and use quasi-compactness of $T$ to choose finitely many of these opens still covering $T$ in the Nisnevich sense: use the assumed Axiom of Choice to select, for each point, a residue-field-preserving lift and an affine neighbourhood of that lift (and representatives in the fixed small skeleton); the resulting Nisnevich family admits a finite subfamily on a noetherian base by noetherian induction on the closed complement of an open over which a chosen étale morphism has a section. Each chosen affine is of finite type over $T$ because it is quasi-compact and étale, hence finitely presented over $T$ ; it is separated. Thus the refined family belongs to the finite-type site. The same argument applies after each base change to a finite-type object, so finite composites of generating covers in the larger site admit refinements in the smaller site.

2.1F2step 1.1step 1.2

Consequently, on each separated finite-type $k$ -scheme $T$ , finite-type covering families are cofinal among the generating-basis refinements used in the larger topology. This comparison is made in the slices over $T$ : every object mapping to $T$ inherits its $k$ -structure, and every arrow over $T$ respects that structure. Thus the different base fields introduce no additional arrows in the matching diagrams on these families. Their iterated fiber products over $T$ also belong to the finite-type site. For a presheaf obtained by restricting a larger-site presheaf, matching families and their common-refinement equivalence may therefore be computed using those same families on either site. Applying this observation first to the constant presheaf and then to its plus construction identifies the two double-plus values. The identification respects restriction because pullback and refinement of the same families are used in both constructions.

3.1F2step 2.1∎

The double-plus values are the stated constant-sheaf sections. Step 2.1 gives the canonical natural isomorphism, including the empty scheme.

Depends on

Used by · 1

Zeroth projection is rank

Dependency tree · 3 levels

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