LemmaStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedprecheck passjudge 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.

Boundaries and cycles in a free complex over a PID are free

Statement

Assume the Axiom of Choice. If $R$ is a PID and every $C_{n}$ is a free $R$ -module, then $B_{n} C \subseteq C_{n}$ and $Z_{n} C \subseteq C_{n}$ are free for every $n$ .

Proof

Given: $B_{n} C = im d_{n + 1}$ and $Z_{n} C = \ker d_{n}$ inside the free module $C_{n}$ .

1.1given

Both $B_{n} C$ and $Z_{n} C$ are $R$ -submodules of $C_{n}$ .

2.1step 1.1∎

Applying A submodule of an arbitrary-rank free module over a PID is free under the stated Choice hypothesis separately to these two inclusions proves that both modules are free.

Depends on

Dependency tree · 2 levels

Sources

Weibel, An Introduction to Homological Algebra