Alphabeta Math
TheoremStatement: Literature-sourcedProof: Literature-sourced
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.

Exact source weight-zero and constant-sheaf input

PDF p. 4: Weight zero

Statement

On the qcqs prime-field cdh site specified by EM §3.3, the canonical natural map Z(0)mot,F→Z(0)cdh,F comes from the source cartesian square and its right vertical arrow is an equivalence. Moreover there is the source natural identification Z(0)cdh,F(X)≃RΓcdh(X,Z‾), where Z‾ is the ordinary sheafification of constant integers, in degree zero.

External input

Import the stated cartesian weight-zero data and constant-sheaf identification, under exactly the recorded qcqs hypotheses. The use of ordinary derived evaluation on the chosen site is the meaning of the source’s RΓ with constant sheaf coefficients. No comparison with a separated finite-type big site, a small cdh site, or a big site over k is asserted.

Facts & Assumptions

Given: The hypotheses in the statement.

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

[F1]

The source statements and hypotheses recorded in External input.

Proof

1.1F1

Import precisely the cited weight-zero assertions.

2.1F1step 1.1∎

For F=Q, Example4.11 uses cdh descent of Hodge-completed derived de Rham to identify the right arrow. For F=Fp, Example4.26 uses cdh descent of weight-zero syntomic cohomology, reduced to Fp(0)syn≃RΓet(−,Z/p). The constant-sheaf derived-sections identification is explicitly the right side of Theorem1.1(5) and both examples. These facts are the exact established external inputs, not locally reconstructed descent theorems.

Depends on

Used by · 2

Dependency tree · 1 level

Arrows run from prerequisites to the items that use them. External source nodes link directly to the cited text.

External dependencies

Elden Elmanto and Matthew Morrow, Motivic cohomology of equicharacteristic schemes · Theorem1.1(5) p.3; §3.3 p.14; Example4.11 p.22; Example4.26 p.28

The canonical map Z(0)_mot→Z(0)_cdh is a natural equivalence on every qcqs prime-field scheme; Z(0)_cdh(X)=RΓ_cdh(X,Z) with the ordinary constant integer sheaf on the specified qcqs cdh site. In the defining cartesian square, the right vertical map is an equivalence at weight zero.

Hypotheses: F=Q or F_p; every qcqs F-scheme, with no finite-type, separatedness, regularity or dimension hypothesis. The p.14 cdh squares use finitely presented closed immersions and finitely presented proper maps.

Reading limits: The exact statements, complete short examples and surrounding square statements/proofs were read. The deeper cdh descent of Hodge-completed derived de Rham and étale cohomology invoked in the examples is imported here; no reconstruction or claim of reading its original proofs. The ordinary constant-sheaf interpretation is explicit in Theorem1.1(5).

Sources