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 actual big cdh slice over a field

PDF p. 6: Singular Beilinson–Lichtenbaum comparison

Statement

Assume AC and compatible universe bounds. Fix a prime $p$ , $S = Spec Z [1 / p]$ , and let $C = {S c h}_{S}^{q c q s}$ with BEM’s big cdh topology: finite Nisnevich covers and finitely presented abstract blowup covers generate it. For a field $k$ with $p$ invertible, put $U = Spec k$ . The slice $C / U$ is exactly the category of qcqs $k$ -schemes with the restricted BEM big cdh topology. The covering sieves of an object $(V, f)$ correspond to the covering sieves of $V$ in $C$ . For presheaves with values in $D (Z)$ , both restriction $j^{*}$ along $j : C / U \to C$ and the functor $j_{*} Q (V) = Q (V \times_{S} U \to U)$ preserve cdh sheaves.

Facts & Assumptions

[F1]

External BEM big-site finite Beilinson–Lichtenbaum comparison specifies BEM’s big site and cover generators. Scheme sites and sheafification fixes the size convention and meaning of a site. All schemes and necessary products lie in the chosen enlarged universe.

[F2]

A complex-valued sheaf here means a presheaf satisfying derived Čech descent for every covering family (limits in $D (Z)$ ); $L_{C}$ below is its reflective sheafification. These are BEM’s ordinary sheaf conventions, not an unasserted identification with the separately fixed hypercomplete finite-type site.

Proof

Given: $S, k, U, C$ as stated.

Goal: Identify the slice site and prove both descent-preservation assertions.

1.1F1construct

A map $V \to U$ in $C$ is precisely a qcqs $k$ -scheme. Every arrow $W \to V$ acquires the unique structure map $W \to V \to U$ , and any composite into $(V, f)$ remains over $U$ . Thus $(C / U) / (V, f) ≅ C / V$ , identifying their sieves. Each Nisnevich or abstract blowup covering of $V$ has this unique lift over $U$ ; conversely forgetting a lifted generator gives such a generator. The generated topologies therefore agree under that identification. This establishes the asserted big-site convention.

2.1F1F2step 1.1∎

A sheaf $P$ on $C$ satisfies descent for each covering Čech diagram over $V$ . That same diagram is the underlying diagram for a cover of $(V, f)$ , so $j^{*} P$ is a sheaf. For a sheaf $Q$ on the slice, a cover of $V$ pulls back to a cover of $V \times_{S} U$ , with its Čech nerve equal to the pulled-back nerve. Evaluating $Q$ on this nerve proves descent for $j_{*} Q$ . Both functors are the displayed actual functors; this does not compare with the older separated finite-type site.

Depends on

Used by · 4

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.