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

How statement and proof provenance work

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Bounded, bounded below, and bounded above complexes

Definition

A chain complex $C_{∙}$ is:

bounded below if $C_{n} = 0$ for all $n ≪ 0$ ;
bounded above if $C_{n} = 0$ for all $n ≫ 0$ ;
bounded if $C_{n} = 0$ outside some finite interval of integers.

For a cochain complex $C^{∙}$ the same words mean:

$C^{n} = 0$ for all $n ≪ 0$ ;
$C^{n} = 0$ for all $n ≫ 0$ ;
$C^{n} = 0$ outside a finite interval.

Under the reindexing convention of Cochain complex in an abelian category, bounded below and bounded above exchange roles, while boundedness is unchanged.

Depends on

Used by

Euler characteristic of a finite complex of finite-rank free abelian groups Definition
The category of chain complexes Definition

Dependency tree · two levels

3 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)
Romyar Sharifi, Homological Algebra, Definition 2.7.6 (standard reference, not scraped)