TheoremStatement: Literature-sourcedProof: AI-generatedPipeline-generatedprecheck passverified 2026-09-24 (gpt-6-sol)

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Universal coefficients in degree two

Statement

Assume the Axiom of Choice and supplied projective and injective resolution data for group (co)homology. 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: The choice and resolution hypotheses above. Use the explicit free bar resolution $P_{∙} \to Z$ over $Z G$ , and put $C_{∙} = Z \otimes_{Z G} P_{∙}$ . The bar-comparison theorem identifies ${Hom}_{Z G} (P_{∙}, A)$ with the derived cohomology computed from the supplied injective resolution.

[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

1.1L1givenalgebra

Each $C_{n}$ is free abelian on $G^{n}$ . Because $A$ has trivial $G$ -action, the tensor-Hom adjunction gives a natural isomorphism ${Hom}_{Z} (C_{∙}, A) ≅ {Hom}_{Z G} (P_{∙}, A)$ . The latter cochain complex computes $H^{*} (G; A)$ by The bar cochain complex computes group cohomology, while $H_{*} (C_{∙})$ computes integral group homology under the supplied resolution data. Choice permits [L1] to be applied to the free integral complex $C_{∙}$ . In degree two it gives $0 \to {Ext}_{Z}^{1} (H_{1} (G; Z), A) \to H^{2} (G; A) \to {Hom}_{Z} (H_{2} (G; Z), A) \to 0$ .

2.1L2step 1.1algebra∎

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].

Depends on

Used by

Central extensions of perfect groups Corollary
Universal coefficients split naturally False statement

Dependency tree · two levels

31 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)