DefinitionDefinition: Literature-sourcedProof: Not applicableaudited 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.

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Subcomplex

Definition

Let $C_{∙}$ be a chain complex. A subcomplex of $C$ is a family of subobjects $S_{n} ↪ C_{n}$ such that for every $n$ the differential $d_{n}^{C} : C_{n} \to C_{n - 1}$ restricts to a morphism $d_{n}^{S} : S_{n} \to S_{n - 1}$ making $\begin{array}{ccc} S_{n} & ↪ & C_{n} \\ d_{n}^{S} ↓ & ↓ d_{n}^{C} \\ S_{n - 1} & ↪ & C_{n - 1} \end{array}$ commute.

Because the structure maps of a subobject are monic, each restricted differential is unique when it exists.

Depends on

Used by

Quotient complex Definition
A subcomplex and its quotient complex Example
The differential descends to a quotient complex Lemma
A subcomplex is the kernel of its quotient map Proposition

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)