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.
Homology is not an exact functor
Statement refuted
Homology sends every short exact sequence of complexes to a short exact sequence in each degree.
Facts & Assumptions
Given: The degreewise split cone sequence of the identity map on .
That sequence has a nonzero connecting morphism on homology (A degreewise split sequence with nonzero connecting map).
Counterexample
The given sequence is short exact term by term, but [L1] shows that its associated homology sequence contains a nonzero connecting map.
A degreewise short exact sequence of homology groups would have zero connecting map. Since the displayed sequence does not, it is a counterexample to exactness of homology.
Depends on
Used by
Nothing in the library uses this result yet.
Dependency tree · two levels
3 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, Chapter 1 of An Introduction to Homological Algebra (standard reference, not scraped)