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 quasi-isomorphism with no homotopy inverse
Statement refuted
Every quasi-isomorphism is a chain homotopy equivalence.
Facts & Assumptions
Given: The zero map from the acyclic noncontractible complex of An acyclic noncontractible complex from a nonsplit extension to the zero complex.
The source complex is acyclic but not contractible (An acyclic noncontractible complex from a nonsplit extension).
A quasi-isomorphism is a chain map inducing isomorphisms on homology (Quasi-isomorphism).
Every chain homotopy equivalence is a quasi-isomorphism (A chain homotopy equivalence is a quasi-isomorphism).
Counterexample
Because the source and target are both acyclic, the zero map induces isomorphisms on all homology groups. Hence [L2] makes it a quasi-isomorphism.
If this map had a homotopy inverse, the source complex would be homotopy equivalent to the zero complex and therefore contractible, contradicting [L1]. So the map is not a chain homotopy equivalence, and the statement refuted is false. This is consistent with [L3], which gives only the forward implication.
Depends on
Used by
Nothing in the library uses this result yet.
Dependency tree · two levels
10 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)