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.

Exactness of a complex at a degree and acyclic complexes

Definition

Let $C_{∙}$ be a chain complex and let $β_{n} : B_{n} (C) \to Z_{n} (C)$ be the canonical map from boundaries to cycles.

The complex is exact at degree $n$ when $β_{n}$ is an isomorphism. Equivalently, the pair $C_{n + 1} \overset{d_{n + 1}}{\to} C_{n} \overset{d_{n}}{\to} C_{n - 1}$ is exact at $C_{n}$ in the sense of Exactness at a node, so that $im (d_{n + 1})$ and $\ker (d_{n})$ represent the same subobject of $C_{n}$ .

The complex is acyclic if it is exact at every degree.

Depends on

Used by

Dependency tree · two levels

9 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, Section 12.13: Complexes (standard reference, not scraped)