DefinitionDefinition: Literature-sourcedProof: Not applicablejudge pass (gpt-5.6-terra)audited 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.

First nonabelian cohomology as a pointed set

Definition

Let $G$ act on a group $M$ by automorphisms. The set of nonabelian $1$ -cocycles is

$Z_{n a b}^{1} (G, M) : = {z : G \to M : z (g h) = z (g) (g \cdot z (h))} .$

The group $M$ acts on this set by

$(a * z) (g) : = a z (g) (g \cdot a)^{- 1} .$

The first nonabelian cohomology set is the orbit set

$H_{n a b}^{1} (G, M) : = Z_{n a b}^{1} (G, M) / M,$

pointed by the orbit of the trivial cocycle $g \mapsto 1$ .

Depends on

Crossed homomorphism for a G-group

Used by

Dependency tree · two levels

2 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

Chaoli Li, Class field theory: proofs (standard reference, not scraped)