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.
A flat nonprojective module
Example
The -module is flat but not projective. No Axiom of Choice is needed for this example.
Verification
Given: the usual additive group , with its integer scalar action.
No nonzero integer annihilates a nonzero rational number, so is torsion-free.
The specific choice-free counterinstance in Flat modules need not have projective dimension zero proves that is not projective: every homomorphism is zero, so a section of the canonical free cover would have every coordinate zero and could not be a section. Its proof does not assume arbitrary projective abelian groups are free.
By Torsion-free abelian groups are flat, step 1.1 makes flat. Together with step 1.2 this verifies the stated example with both premises supplied choice-free.
Depends on
Used by
Nothing in the library uses this result yet.
Dependency tree · two levels
11 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
- Charles A. Weibel, An Introduction to Homological Algebra (standard reference, not scraped)