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.

Cycle and boundary subobjects of a complex

Definition

Let $C_{∙}$ be a chain complex in an abelian category and fix $n \in Z$ .

The $n$ th cycle subobject of $C$ is $Z_{n} (C) : = \ker (d_{n}) ↪ C_{n} .$

The $n$ th boundary subobject of $C$ is $B_{n} (C) : = im (d_{n + 1}) ↪ C_{n} .$

Thus both are subobjects of $C_{n}$ , one defined by the kernel of $d_{n}$ and the other by the image of $d_{n + 1}$ .

Depends on

Used by

Exactness of a complex at a degree and acyclic complexes Definition
Homology object of a chain complex Definition
A two-term complex has kernel and cokernel homology Example
FALSE: the boundaries of a complex are a quotient of its cycles False statement
A chain map carries cycles to cycles and boundaries to boundaries Lemma
The boundary subobject factors through the cycle subobject Lemma

Dependency tree · two levels

6 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)