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.

Gluing upper bounds through an actual fiber sequence

PDF p. 4: Motivic dimension vanishing

Statement

Let $A \to B \to C$ be a fiber sequence in $D (Z)$ , and let $N \in Z$ . If $A$ is supported in cohomological degrees $\leq N$ , then $H^{N} (B) \to H^{N} (C)$ is surjective and $H^{i} (B) \to H^{i} (C)$ is an isomorphism for every $i > N$ . If in addition $C$ is supported in degrees $\leq N$ , then so is $B$ .

Facts & Assumptions

Given: The specified fiber sequence and integer $N$ .

Goal: Transfer the two outer bounds to the middle term with the exact degree endpoints.

[F1]

Cohomology descends to the derived category supplies functorial cohomology on $D (Z)$ .

[F2]

In the standard derived triangulated structure, distinguished triangles are the triangles isomorphic to localized mapping-cone triangles, and each gives the long exact cohomology sequence. Use the usual fixed-universe size convention for complexes of abelian groups. The derived category inherits a triangulated structure

Proof

1.1F1F2given

The given fiber sequence extends to its distinguished triangle $A \to B \to C \to A [1]$ in the standard derived structure. Applying [F2] and the cohomology functors [F1], its long exact sequence contains $H^{i} (A) \to H^{i} (B) \to H^{i} (C) \to H^{i + 1} (A)$ . At $i = N$ the last group is zero, which gives the stated surjectivity. At $i > N$ both outer groups are zero, so the middle arrow is an isomorphism.

2.1F1step 1.1∎

If $H^{i} (C) = 0$ for $i > N$ , the isomorphism in step 1.1 gives $H^{i} (B) = 0$ for those same degrees. This proves the middle upper bound without any splitting or identification of unrelated groups.

Depends on

Used by · 2

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