PropositionStatement: 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.

Degree-zero group (co)homology

Statement

For every left $G$ -module $M$ , $H^{0} (G; M) = M^{G}$ and $H_{0} (G; M) = M_{G}$ , naturally in $M$ .

Proof

Given: A left $G$ -module $M$ .

1.1given

The zeroth right derived functor of invariants recovers the invariants functor.

2.1step 1.1∎

The zeroth left derived functor of coinvariants recovers coinvariants. Substitute the two definitions of group (co)homology.

Depends on

Used by

Degree-zero invariants and coinvariants Example

Dependency tree · two levels

20 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

Weibel, Definition 6.1.2 (standard reference, not scraped)