TheoremStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedprecheck passaudited 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.

The cohomology universal-coefficient sequence splits nonnaturally

Statement

Assume the Axiom of Choice. Let $R$ be a PID, $C$ a chain complex of free $R$ -modules, $G$ an $R$ -module, and $n \in Z$ . The cohomological UCT sequence $0 \to {Ext}_{R}^{1} (H_{n - 1} C, G) \to H^{n} {Hom}_{R} (C, G) \overset{{ev}_{n}}{\to} {Hom}_{R} (H_{n} C, G) \to 0$ splits after choosing a complement of $Z_{n} C$ in $C_{n}$ . No splitting natural in the complex $C$ is asserted.

Proof

Given: $R, C, G, n$ as stated and the UCT sequence of The universal coefficient theorem for cohomology over a PID.

1.1

The exact sequence $0 \to Z_{n} C \to C_{n} \overset{d_{n}}{\to} B_{n - 1} C \to 0$ comes from The cycle-boundary short exact sequences for a free complex over a PID. Under AC, Boundaries and cycles in a free complex over a PID are free makes $B_{n - 1} C$ free and Free modules are projective, with the exact choice boundary makes it projective. Lift its identity through $d_{n}$ to choose a section $s$ . Then $1 - s d_{n}$ factors through the inclusion $Z_{n} C \to C_{n}$ as a projection $π : C_{n} \to Z_{n} C$ restricting to the identity on cycles.

givenconstruct

2.1

Write $q : Z_{n} C \to H_{n} C$ for the quotient. For $f : H_{n} C \to G$ , define $σ (f) = [f q π]$ . The map $f q π : C_{n} \to G$ is a cocycle: $d_{n + 1}$ lands in $B_{n} C \subseteq Z_{n} C$ , where $π$ is the identity and $q$ is zero. This assignment is $R$ -linear in $f$ , so it defines a homomorphism into cohomology.

step 1.1algebra

3.1

For a cycle $z$ , $(f q π) (z) = f (q z)$ , so ${ev}_{n} σ (f) = f$ . Thus $σ$ is a section of the surjection in the exact UCT sequence. Its construction uses the chosen projection; it supplies existence without asserting naturality in $C$ .

step 2.1algebra∎

Depends on

Used by

Universal coefficients in degree two Theorem

Dependency tree · two levels

14 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, An Introduction to Homological Algebra (standard reference, not scraped)