Alphabeta Math
CorollaryStatement: Literature-sourcedProof: AI-adaptedprecheck passaudited 2026-09-04
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.

For a trivial action, first cohomology is Hom

Statement

If G acts trivially on an abelian group A, then

H1(G,A)≅Hom⁡(G,A).

Facts & Assumptions

Given: A trivial action of G on an abelian group A.

[L1]

The crossed-homomorphism model computes H1(G,A) (The inhomogeneous one-cocycle model agrees with crossed homomorphisms in degree one).

[L2]

A group homomorphism f:G→A is characterized by f(gh)=f(g)+f(h) in additive notation (Monoid homomorphism and group homomorphism).

Proof

technique · direct
1.1givenL1L2

Under the trivial action, the crossed-homomorphism identity becomes z(gh)=z(g)+z(h), which is exactly the homomorphism law of [L2]. So crossed homomorphisms are precisely the homomorphisms G→A.

2.1L1step 1.1algebra∎

Principal crossed homomorphisms are all zero, because g⋅a−a=a−a=0 for every g and a. Therefore the quotient H1(G,A) is just the group of homomorphisms from step 1.1.

Depends on

Used by

Dependency tree · two levels

8 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