TheoremStatement: Literature-sourcedProof: AI-generatedPipeline-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-06‡ rests on unproved material (inherited)

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.

‡ Rests on 3 statements not proved in this library, by way of the results it cites. This item cites no such statement directly; it depends on results that do. The unproved premises it inherits are ‡ Cohen's first model: an infinite Dedekind-finite set of reals, ‡ Sierpiński 1947: the generalised continuum hypothesis implies the Axiom of Choice and ‡ The continuum hypothesis and its generalisation are independent of ZFC. Each is recorded with a citation to the literature and is not established here, because the track that would prove it has not yet been developed in this library. Everything else in this proof is proved here.

Universal coefficients in degree two

Statement

For a trivial $G$ -module $A$ , there is a natural exact sequence

$0 \to {Ext}_{Z}^{1} (G_{a b}, A) \to H^{2} (G; A) \to Hom (M (G), A) \to 0.$

It admits a splitting after choices; no natural splitting is asserted.

Facts & Assumptions

Given: Compute group (co)homology from a free $Z G$ -resolution of $Z$ and then tensor it over $Z G$ with the trivial module $Z$ .

[L1]

The cohomological universal-coefficient theorem gives the natural degree-two short exact sequence for a degreewise free integral chain complex (The universal coefficient theorem for cohomology over a PID).

[L2]

This sequence splits after choices of complements, with no natural splitting asserted (The cohomology universal-coefficient sequence splits nonnaturally).

Proof

technique · direct

1.1

The resulting chain complex is degreewise free over $Z$ , so the cohomological universal-coefficient theorem in degree two gives $0 \to {Ext}_{Z}^{1} (H_{1} (G; Z), A) \to H^{2} (G; A) \to {Hom}_{Z} (H_{2} (G; Z), A) \to 0$ naturally.

L1givenalgebra

2.1

Since $H_{1} (G; Z) = G_{a b}$ and $H_{2} (G; Z) = M (G)$ , this is the displayed sequence. The splitting qualification follows from [L2].

L2step 1.1algebra∎

Depends on

Used by

Central extensions of perfect groups Corollary
Multiplier defined as H²(G,C×) False statement
Universal coefficients split naturally False statement

Dependency tree · two levels

15 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

Clara Löh, Group Cohomology (standard reference, not scraped)