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.
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These labels describe origin, not correctness: citations and verification chips remain separate evidence.
FALSE: every quasi-isomorphism is an isomorphism of complexes
Statement
Every quasi-isomorphism of chain complexes is an isomorphism in .
Facts & Assumptions
Given: The zero map from the acyclic complex to the zero complex.
A quasi-isomorphism is a chain map inducing isomorphisms on homology (Quasi-isomorphism).
is an abelian category (Abelian groups form an abelian category).
Refutation
As in the standard identity-complex computation, the source complex is acyclic and the target zero complex is also acyclic, so every homology object on both sides is zero. Therefore the zero chain map induces isomorphisms on all homology objects and is a quasi-isomorphism by [L1].
The source complex is nonzero while the target is zero, so no inverse chain map can exist. Thus this quasi-isomorphism is not an isomorphism of complexes.
Depends on
Used by
- A quasi-isomorphism that is not an isomorphism of complexes Counterexample
Dependency tree · two levels
12 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)