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 objectwise cofiber bound for the canonical source composite

PDF p. 5: Prime-to-characteristic comparison

Statement

Let F=Q or Fp, let j be a nonnegative integer and let n≥1 be an integer invertible in F. With the preceding source data, the composite Mj/n⟶Z(j)cdh/n→θjLcdhτ≤jBj→ϵjBj is the natural comparison of EM Theorem4.33(1). At every qcqs F-scheme its cofiber has zero cohomology in every degree at most j.

External input

Use the explicit cofiber assertion of Theorem 4.33(1), arXiv:2309.08463v1, printed p. 31 (zero-based PDF page 30). Its short proof calls the last cofiber a fiber. The two differ by a shift; that printed fiber sentence is not used as a local proof of the theorem.

This is the substantial established source range theorem, imported with its actual comparison map. All five local prerequisites isolate the site, source restriction, quotient, map composition and truncation applications; they do not purport to reconstruct the deep theorem.

Facts & Assumptions

Given: The hypotheses in the statement.

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

[F1]

The exact theorem in External input, with the same factorization data as Exact cdh factorization inputs for the source étale map.

Proof

1.1F1

Import the actual map and objectwise cofiber statement of EM Theorem4.33(1), with its exact factorization given in the complete proof on p.31.

2.1F1step 1.1∎

Its n is any positive integer invertible in the prime field, so composite coefficients are included. The explicit source statement is in terms of the cofiber and is the assertion used here. We do not infer it from a sheafified equivalence or replace it by an unverified objectwise exactness claim about localization.

Depends on

Used by · 1

Dependency tree · 2 levels

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 · Theorem4.33(1) and its complete proof, printed p.31; introductory equivalent truncation statement Theorem1.1(3), p.3

For a qcqs F-scheme X and j≥0, n>0 prime to char F, the actual natural comparison M_j(X)/n→RΓ_et(X,μ_n^⊗j) has cofiber supported in cohomological degrees >j. The proof identifies this map through M_j/n→Z(j)_cdh/n≃L_cdh τ≤j B_j→B_j.

Hypotheses: F is a prime field Q or F_p; X is any qcqs F-scheme; j≥0; n>0 invertible in F. No smoothness, regularity, noetherian, finite-type or dimension hypothesis.

Reading limits: The full statement and complete short proof on p.31 were read, together with the exact cdh comparison statements it invokes. The deeper objectwise localization range and the original comparison/descent foundations are imported as this established theorem, not claimed locally reconstructed. The introductory truncation statement was also read. No comparison in degrees above j or with noninvertible n is asserted.

Sources