PropositionStatement: AI-adaptedProof: AI-generatedPipeline-generatedprecheck passaudited 2026-09-01‡ rests on unproved material (inherited)

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.

‡ Rests on 3 statements not proved in this library, by way of the results it cites. This item cites no such statement directly; it depends on results that do. The unproved premises it inherits are ‡ Cohen's first model: an infinite Dedekind-finite set of reals, ‡ Sierpiński 1947: the generalised continuum hypothesis implies the Axiom of Choice and ‡ The continuum hypothesis and its generalisation are independent of ZFC. Each is recorded with a citation to the literature and is not established here, because the track that would prove it has not yet been developed in this library. Everything else in this proof is proved here.

The cone connecting map agrees with the shifted identity up to the declared sign

Statement

For the canonical short exact sequence $0 \to D_{∙} \to Cone (f)_{∙} \to C [1]_{∙} \to 0$ of a chain map $f : C_{∙} \to D_{∙}$ in a module category, the connecting morphism $\partial_{n} : H_{n} (C [1]) \to H_{n - 1} (D)$ corresponds under the shift isomorphism $H_{n} (C [1]) \tilde{\to} H_{n - 1} (C)$ to the homology map $H_{n - 1} (f)$ . In particular, when $f = 1_{C}$ , the connecting morphism is the shifted identity up to the sign built into the shift convention.

Facts & Assumptions

Given: A chain map $f : C_{∙} \to D_{∙}$ of module complexes and an integer $n$ .

[L1]

The cone long exact sequence is obtained from the canonical short exact cone sequence (The cone long exact sequence).

[L2]

In module categories, the connecting map is computed by lifting a cycle and taking its boundary class (Elementwise formula for the connecting map in module categories).

[L3]

The shift isomorphism identifies $H_{n} (C [1])$ with $H_{n - 1} (C)$ using the sign convention fixed for shifts (Homology of a shift is shifted homology).

Proof

technique · direct

1.1

A class in $H_{n} (C [1])$ is represented by a cycle $x \in C_{n - 1}$ . In the canonical cone sequence, the element $(0, x) \in Cone (f)_{n}$ lifts that class, and its boundary is $(f_{n - 1} (x), 0) .$ By [L2], the connecting morphism sends $[x]$ to $[f_{n - 1} (x)] \in H_{n - 1} (D)$ .

L2L3givenalgebra

2.1

Step 1.1 is exactly the formula for $H_{n - 1} (f)$ after identifying $H_{n} (C [1])$ with $H_{n - 1} (C)$ via [L3]. Hence the connecting map of the cone sequence agrees with $H_{n - 1} (f)$ under that shift identification. When $f = 1_{C}$ , this becomes the shifted identity with precisely the sign encoded in [L3].

L1L3step 1.1algebra∎

Depends on

Used by

A degreewise split sequence with nonzero connecting map Example
FALSE: a degreewise split short exact sequence of complexes has zero connecting map False statement
FALSE: the homology functor is exact on short exact sequences of complexes False statement

Dependency tree · two levels

13 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Charles A. Weibel, Chapter 1 of An Introduction to Homological Algebra (standard reference, not scraped)