PropositionStatement: 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.

The coinvariants functor is right exact

Statement

If $A \to B \to C \to 0$ is exact in left $G$ -modules, then $A_{G} \to B_{G} \to C_{G} \to 0$ is exact.

Proof

Given: An exact sequence $A \to B \to C \to 0$ of left $G$ -modules.

1.1

The induced map $B_{G} \to C_{G}$ is surjective because $B \to C$ is. Let $[b] \in B_{G}$ map to zero. Then the image of $b$ in $C$ is a finite sum $\sum_{j} (g_{j} c_{j} - c_{j})$ . Choose lifts $b_{j} \in B$ of the finitely many $c_{j}$ and put $b^{'} = b - \sum_{j} (g_{j} b_{j} - b_{j})$ . Then $b^{'}$ maps to zero in $C$ , so it lies in the image of $A \to B$ , while $[b^{'}] = [b]$ in $B_{G}$ .

given

2.1

Thus the image of $A_{G} \to B_{G}$ is exactly the kernel of $B_{G} \to C_{G}$ , and the latter map is surjective. This proves right exactness without invoking a commutative-ring tensor theorem for the possibly noncommutative group ring.

step 1.1∎

Depends on

The coinvariants functor

Used by

Group homology as a derived functor Definition

Dependency tree · two levels

3 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, §6.1 (standard reference, not scraped)