LemmaStatement: AI-generatedProof: AI-generated

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

The source qcqs cdh site

PDF p. 4: Weight zero

Statement

Let $F = Q$ or $F_{p}$ . Choose universes large enough to contain the schemes in use, and a suitably small skeleton $C_{F}$ of the qcqs $F$ -schemes in that universe. On $C_{F}$ use the topology generated by Nisnevich covers and families ${Y \to X, X^{'} \to X}$ from cartesian squares with $Y \to X$ a finitely presented closed immersion, $X^{'} \to X$ finitely presented and proper, and $X^{'} ∖ (X^{'} \times_{X} Y) ≅ X ∖ Y$ . This is the precise general cdh site used here. No separated finite-type hypothesis over $F$ or $k$ is imposed. Its evaluation object may be any qcqs $k$ -scheme, viewed over its prime field $F$ .

Facts & Assumptions

Given: The hypotheses in the statement.

Goal: Establish the displayed construction or conclusion with its actual maps.

[F1]

The universe and generated-topology constructions are those of Scheme sites and sheafification. EM §3.3 p.14 specifies the displayed generators.

Proof

1.1F1construct

A topology with these generating sieves exists: intersect all Grothendieck topologies containing them. The topology axioms survive intersection, and the maximal topology belongs to the collection. This constructs the least such topology on the chosen small category.

2.1givenstep 1.1∎

These are exactly the square and covering conventions specified in EM §3.3 p.14: in particular both finiteness conditions on the square are retained. A qcqs $k$ -scheme is an object after composing its structure morphism with $Spec k \to Spec F$ ; quasi-compactness and quasi-separatedness here are properties of the underlying scheme. Thus the site supplies every evaluation object required. The previously completed separated finite-type site is a different specified site and is left unchanged; no equivalence between their cohomologies is asserted.

Depends on

Scheme sites and sheafification

Used by · 13

Dependency tree · 3 levels

Arrows run from prerequisites to the items that use them. External source nodes link directly to the cited text. The view shows up to three dependency steps; open a prerequisite to continue.