Alphabeta MathHow 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 is a PID and every is a free -module, then and are free for every .
Proof
Given: and inside the free module .
Both and are -submodules of .
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
Used by
- The cohomological universal-coefficient extension map Lemma
- The homological universal-coefficient Tor obstruction map Lemma
- The Kunneth Tor map Lemma
- The cohomology universal-coefficient sequence splits nonnaturally Theorem
- The Kunneth theorem for free complexes over a PID Theorem
- The universal coefficient theorem for cohomology over a PID Theorem
- The universal coefficient theorem for homology over a PID Theorem
Dependency tree · two levels
7 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)