TheoremStatement: Literature-sourcedProof: AI-generatedPipeline-generatedjudge pass (gpt-5.6-terra)audited 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.

Invariants are Hom from the trivial module

Statement

For every left $G$ -module $M$ , evaluation at $1$ is a natural isomorphism ${Hom}_{Z [G]} (Z, M) ≅ M^{G}$ .

Proof

Given: A left $G$ -module $M$ with the trivial-module convention.

1.1given

If $f$ is $Z [G]$ -linear, then $g f (1) = f (g \cdot 1) = f (1)$ , so evaluation lands in $M^{G}$ .

2.1step 1.1∎

For $m \in M^{G}$ , define $f_{m} (z) = z m$ . Then $f_{m} (g \cdot z) = z m = g f_{m} (z)$ , hence $f_{m}$ is $Z [G]$ -linear; evaluation sends it to $m$ , and a homomorphism from $Z$ is determined by $1$ . The two constructions commute with maps $M \to N$ .

Depends on

Used by

Group cohomology as a derived functor Definition
The invariants functor is left exact Proposition

Dependency tree · two levels

5 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, Lemma 6.1.1 (standard reference, not scraped)