DefinitionDefinition: Literature-sourcedProof: Not applicableaudited 2026-08-30

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Euler characteristic of a finite complex of finite-rank free abelian groups

Definition

Let $C_{∙}$ be a bounded chain complex of abelian groups such that every $C_{n}$ is free of finite rank. Its Euler characteristic is $χ (C) : = \sum_{n \in Z} (- 1)^{n} rank (C_{n}) .$

Boundedness makes the sum finite, and Invariant basis number and the rank of a free module supplies the meaning of $rank$ for each term.

Depends on

Used by

Euler-Poincare formula for finite free complexes Theorem

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

P. Hekmati, Homological Algebra, section 3.1 (standard reference, not scraped)
Charles A. Weibel, Chapter 1 of An Introduction to Homological Algebra (standard reference, not scraped)