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 restricted BEM complex and the supplied EM finite quotient

PDF p. 6: Singular Beilinson–Lichtenbaum comparison

Statement

Let $k$ have characteristic zero and let $D_{j}$ denote BEM’s cdh-motivic complex restricted to qcqs $Z [1 / p]$ -schemes. On qcqs $k$ -schemes, $j^{*} D_{3}$ is naturally the restriction of the supplied EM cdh-local weight-three complex from qcqs $Q$ -schemes. The identification induces the same natural identification after objectwise derived quotient by $p$ , and evaluation at any $X / k$ computes that quotient as the cofiber of multiplication by $p$ .

Facts & Assumptions

[F1]

External canonical field restriction identifies the BEM and EM cdh models supplies the canonical field-restriction identification, via the same classical smooth- $Q$ complex and its cdh-local left Kan extension.

[F2]

Derived finite coefficients and the Bockstein sequence identifies $C / p$ with the cofiber of $p : C \to C$ and with $C \otimes^{L} Z / p$ .

Proof

Given: $k$ of characteristic zero, any qcqs $X / k$ , and prime $p$ .

Goal: Match the actual finite quotient rather than merely cohomology groups.

1.1F1

Restrict the canonical identification in [F1] first to qcqs $Q$ -schemes and then to $k$ -schemes. Both sides are the indicated EM cdh-motivic target with its actual functorial maps. No finite-type separated site is inserted.

2.1F2step 1.1∎

Restriction and evaluation of complex-valued presheaves are computed objectwise and preserve finite cofibers. Hence they carry the natural cofiber diagram for multiplication by $p$ to the corresponding diagram on the restricted/evaluated complex. Apply [F2] to identify each such cofiber with the specified derived tensor quotient. This proves the full map-level identification.

Depends on

Used by · 1

Singular Beilinson–Lichtenbaum input

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.