ExampleConstruction: AI-adaptedVerification: AI-generatedPipeline-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-01

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.

The connecting map for a short exact sequence of two-term complexes

Example

Fix a nonzero integer $m$ . Consider the chain map $\times m : Z [0] \to Z [0]$ and its canonical short exact cone sequence $0 \to Z [0] \to Cone (\times m) \to Z [1] \to 0.$ The middle term is a two-term complex, and its connecting morphism $\partial_{1} : H_{1} (Z [1]) \to H_{0} (Z [0])$ is multiplication by $m$ .

Facts & Assumptions

Given: A nonzero integer $m$ .

[L1]

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

Verification

technique · direct

1.1L1givenconstruct

A class in $H_{1} (Z [1])$ is represented by an integer $x \in Z = C_{0}$ . In the cone sequence, $(0, x)$ is a lift of that cycle to degree $1$ of $Cone (\times m)$ .

2.1L1step 1.1algebra∎

The cone differential sends $(0, x)$ to $(m x, 0)$ . Therefore [L1] gives $\partial_{1} ([x]) = [m x] \in H_{0} (Z [0]) = Z .$ So the connecting morphism is multiplication by $m$ .

Depends on

Elementwise formula for the connecting map in module categories

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

4 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)