LemmaStatement: AI-generatedProof: AI-generated

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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.

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Cofiber bounds imply low-degree cohomology isomorphisms

PDF p. 4: equivalence in degrees at most three

Statement

Let $f : A \to B$ be a morphism in $D (Z)$ and let $C$ be its cofiber, represented by a mapping cone. If $H^{j} (C) = 0$ for every $j \leq n$ , then $H^{i} (f)$ is an isomorphism for every $i \leq n$ and is injective for $i = n + 1$ . If, in addition, $H^{n + 1} (C) = 0$ , it is an isomorphism also for $i = n + 1$ .

Facts & Assumptions

Given: A derived-category morphism and a cone with the stated vanishing.

Goal: Establish the stated low-degree consequences.

[F1]

Cohomology is a functor on $D (Z)$ (Cohomology descends to the derived category).

Proof

1.1F1construct

Represent the derived morphism by a chain map after replacing its source by a quasi-isomorphic complex. Its mapping cone fits in a degreewise split short exact sequence of complexes $0 \to B \to Cone (f) \to A [1] \to 0$ . Taking cycles and boundaries gives the long exact sequence $H^{i - 1} (C) \to H^{i} (A) \to H^{i} (B) \to H^{i} (C)$ . The result is independent of representatives by [F1].

2.1step 1.1algebra∎

For $i \leq n$ , both adjacent cone groups in step 1.1 vanish, giving an isomorphism. For $i = n + 1$ , the left group $H^{n} (C)$ vanishes, giving injectivity; if also $H^{n + 1} (C) = 0$ , the right group vanishes and the map is surjective as well. This proves every claimed range without a condition on $H^{n + 2} (C)$ .

Depends on

Cohomology descends to the derived category

Used by · 1

Exact range of a cofiber connectivity bound

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.