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.

The cycle-boundary short exact sequences for a free complex over a PID

Statement

For every chain complex $C$ , the differential and quotient maps give exact sequences $0 \to Z_{n} C \to C_{n} \overset{d_{n}}{\to} B_{n - 1} C \to 0$ and $0 \to B_{n} C \to Z_{n} C \to H_{n} C \to 0$ .

Proof

Given: $Z_{n} C = \ker d_{n}$ , $B_{n} C = im d_{n + 1}$ , and $H_{n} C = Z_{n} C / B_{n} C$ .

1.1given

Corestricting $d_{n} : C_{n} \to C_{n - 1}$ to its image gives a surjection $C_{n} ↠ B_{n - 1} C$ whose kernel is $Z_{n} C$ .

2.1step 1.1algebra∎

Since $d_{n} d_{n + 1} = 0$ , $B_{n} C \subseteq Z_{n} C$ ; the quotient map $Z_{n} C ↠ Z_{n} C / B_{n} C = H_{n} C$ has kernel $B_{n} C$ , proving both sequences.

Depends on

Dependency tree · 2 levels

Sources

Weibel, An Introduction to Homological Algebra