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 group cohomology via inhomogeneous one-cocycles

Definition

Let $G$ act on an abelian group $A$ . In degree one, the inhomogeneous cochain model takes

$C^{0} (G, A) = A, C^{1} (G, A) = {f : G \to A}, C^{2} (G, A) = {u : G \times G \to A},$

with differentials

$d^{0} (a) (g) = g \cdot a - a,$

and

$d^{1} (f) (g, h) = f (g) + g \cdot f (h) - f (g h) .$

The first cohomology is the cohomology object

$H_{i n h}^{1} (G, A) = \ker d^{1} / im d^{0},$

in the sense of Cohomology object of a cochain complex.

Depends on

Used by

Dependency tree · two levels

6 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)