PropositionStatement: AI-adaptedProof: AI-generatedPipeline-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-01

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An exact functor carries the long exact homology sequence to the corresponding long exact sequence

Statement

Let $F : A \to B$ be an exact functor between abelian categories. For every short exact sequence of complexes in $A$ , the canonical isomorphisms $F (H_{n} (X)) \tilde{\to} H_{n} (F (X))$ identify the image under $F$ of its long exact homology sequence with the long exact homology sequence of the induced short exact sequence $0 \to F (A_{∙}) \to F (B_{∙}) \to F (C_{∙}) \to 0.$

Facts & Assumptions

Given: An exact functor $F : A \to B$ and a short exact sequence $0 \to A_{∙} \to B_{∙} \to C_{∙} \to 0$ of complexes in $A$ .

[L1]

Exactness means that $F$ preserves the finite limits and colimits relevant to kernels, cokernels, and short exact sequences (Exact functor between abelian categories).

[L2]

Exact functors commute with homology by canonical natural isomorphisms (An exact functor commutes with homology).

[L3]

The long exact homology sequence is natural for morphisms of short exact sequences (The long exact homology sequence is natural).

[L4]

Every short exact sequence of complexes has a long exact homology sequence (The long exact sequence in homology).

Proof

technique · direct

1.1L1L4givenconstruct

By [L1], applying $F$ degreewise to the given short exact sequence of complexes produces another short exact sequence of complexes in $B$ . Applying [L4] to both sequences gives two long exact homology sequences.

2.1L2step 1.1algebra

The canonical isomorphisms from [L2] identify each term $F (H_{n} (X))$ with $H_{n} (F (X))$ , and their naturality identifies the ordinary homology maps on the two sequences.

3.1L1L2L3step 2.1algebra∎

The connecting morphisms are built from kernels and cokernels of the same degreewise diagram, and [L1] preserves those constructions. Therefore the comparison isomorphisms respect the connecting maps as well, so the whole long exact sequence is transported from one side to the other.

Depends on

Used by

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Dependency tree · two levels

16 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)
Romyar Sharifi, Homological Algebra (standard reference, not scraped)