Alphabeta Math
ExampleConstruction: AI-adaptedVerification: AI-generatedPipeline-generatedprecheck passaudited 2026-08-31
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.

Relative homology of a map between stalk complexes

Example

For the stalk-complex map ×m:Z[0]→Z[0] with m≠0, the relative homology objects are H0(Z,Z;×m)≅Z/mZ,Hn(Z,Z;×m)=0 for n≠0.

Facts & Assumptions

Given: A nonzero integer m.

[L1]

Relative homology is the homology of the mapping cone (The relative homology of a chain map).

[L2]

The cone of multiplication by m on Z[0] has homology H0≅Z/mZ and Hn=0 for n≠0 (The cone of multiplication by m on the integers).

Verification

technique · direct
1.1L1givenalgebra

By [L1], Hn(Z,Z;×m)=Hn(Cone⁡(×m)).

2.1L2step 1.1algebra∎

Substituting the explicit cone homology from [L2] gives the displayed relative homology groups.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

12 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