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 invariants functor is left exact

Statement

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

Proof

Given: The displayed exact sequence of left $G$ -modules.

1.1given

Applying ${Hom}_{Z [G]} (Z, -)$ gives an exact sequence at its first two positions by covariant Hom left exactness.

2.1step 1.1∎

Transport this sequence through the natural isomorphisms of invariants with Hom from the trivial module. This is exactly the asserted sequence.

Depends on

Used by

Group cohomology as a derived functor Definition
Lyndon-Hochschild-Serre spectral sequence Theorem

Dependency tree · two levels

7 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)