DefinitionDefinition: Literature-sourcedProof: Not applicablePipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-01

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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AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
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A morphism of short exact sequences of complexes

Definition

A morphism of short exact sequences of complexes is a commutative ladder $\begin{array}{ccccccccc} 0 & \to & A_{∙} & \to & B_{∙} & \to & C_{∙} & \to & 0 \\ ↓ & ↓ & ↓ \\ 0 & \to & A_{∙}^{'} & \to & B_{∙}^{'} & \to & C_{∙}^{'} & \to & 0 \end{array}$ in which each row is a short exact sequence of complexes and each vertical map is a chain map.

Equivalently, it is a triple of chain maps $a : A_{∙} \to A_{∙}^{'}$ , $b : B_{∙} \to B_{∙}^{'}$ , and $c : C_{∙} \to C_{∙}^{'}$ for which the obvious squares with the inclusion and projection maps commute in every degree.

Depends on

Used by

Naturality of the homology connecting morphism Theorem

Dependency tree · two levels

5 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)