Alphabeta Math
TheoremStatement: Literature-sourcedProof: AI-generatedPipeline-generatedaudited 2026-09-06
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 bar cochain complex computes group cohomology

Statement

For every left G-module M, the cohomology of (C∙(G,M),d) is naturally Hn(G;M).

Proof

Given: The free bar resolution P→Z and an injective resolution M→I.

1.1given

Form the finite-diagonal double complex Hom⁡Z[G](P,I). Projectivity of each Pi makes the columns exact away from Hom⁡(P,M).

1.2given

Injectivity of each Ij makes the rows exact away from Hom⁡(Z,I). The two acyclic-assembly comparisons identify the cohomology of these edge complexes.

2.1step 1.1step 1.2∎

The right edge computes Ext⁡Z[G]n(Z,M)=Hn(G;M), while the left edge is homogeneous bar cochains; the homogeneous--inhomogeneous isomorphism identifies it with C∙(G,M).

Depends on

Used by

Dependency tree · two levels

25 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