DefinitionDefinition: Literature-sourcedProof: Not applicablejudge pass (gpt-5.6-terra)audited 2026-09-05

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.

Normalized two-cocycle and two-coboundary

Let $G$ be a group and let $M$ be an abelian $G$ -module, written additively. A function $f : G \times G \to M$ is a normalized two-cocycle when

$g \cdot f (h, k) - f (g h, k) + f (g, h k) - f (g, h) = 0$

for all $g, h, k \in G$ , and

$f (1, g) = f (g, 1) = 0$

for all $g \in G$ .

If $u : G \to M$ is a normalized one-cochain with $u (1) = 0$ , its two-coboundary is

(δ u) (g, h) = g \cdot u (h) - u (g h) + u (g) .

The sets of normalized two-cocycles and two-coboundaries are denoted $Z^{2} (G, M)$ and $B^{2} (G, M)$ .

Used by

Nothing. This result depends on no other item in the library.