Alphabeta Math
CounterexampleConstruction: AI-adaptedVerification: AI-generatedprecheck passaudited 2026-08-30
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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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Two distinct chain maps inducing the same homology map

Statement refuted

Two chain maps with the same induced maps on homology must be equal.

Facts & Assumptions

Given: The complex 0→Z→1ZZ→0 and its two endomorphisms f=1C and g=0C.

[L1]

Ab is an abelian category (Abelian groups form an abelian category).

[L2]

The corresponding A-page false statement is false (FALSE: a chain map is determined by its maps on homology).

Counterexample

technique · direct
1.1L1givenalgebra

The complex is acyclic, so every homology group is zero. The maps f and g are distinct because their degree-1 components are 1Z and 0.

2.1L2step 1.1∎

Since all homology groups vanish, both induced homology maps are zero in every degree. Therefore f≠g but they induce the same homology map, as claimed in [L2].

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

9 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