DefinitionDefinition: Literature-sourcedProof: Not applicablejudge pass (gpt-5.6-terra)audited 2026-08-30

How 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.

Quasi-isomorphism

Definition

A chain map $f : C_{∙} \to D_{∙}$ is a quasi-isomorphism if for every $n \in Z$ the induced map $H_{n} (f) : H_{n} (C) \to H_{n} (D)$ is an isomorphism.

Depends on

A chain map induces a well-defined map on homology

Used by

An exact functor preserves quasi-isomorphisms and reflects them when it is conservative Corollary
A quasi-isomorphism that is not an isomorphism of complexes Counterexample
FALSE: every quasi-isomorphism is an isomorphism of complexes False statement
A chain map is a quasi-isomorphism exactly when its cochain reindexing is Proposition
Isomorphisms of complexes are quasi-isomorphisms Proposition

Dependency tree · two levels

4 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)
The Stacks Project, Definition 12.13.4 (standard reference, not scraped)