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

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Homology of a chain-split direct-sum sequence

Statement

If $0 \to A_{∙} \to B_{∙} \to C_{∙} \to 0$ is a chain-split short exact sequence of complexes, then for every $n$ there is an isomorphism $H_{n} (B) ≅ H_{n} (A) \oplus H_{n} (C) .$

Facts & Assumptions

Given: A chain-split short exact sequence $0 \to A_{∙} \to B_{∙} \to C_{∙} \to 0$ of complexes.

[L1]

In a chain-split short exact sequence, every connecting morphism is zero (The connecting morphism vanishes for a chain-split short exact sequence).

[L2]

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

Proof

technique · direct

1.1L1L2givenalgebra

By [L1] and [L2], the long exact homology sequence breaks in each degree into a short exact sequence $0 \to H_{n} (A) \to H_{n} (B) \to H_{n} (C) \to 0.$

2.1step 1.1algebra∎

Because the short exact sequence is chain split, the middle complex is degreewise isomorphic to $A_{∙} \oplus C_{∙}$ with block-diagonal differential. Hence $Z_{n} (B) ≅ Z_{n} (A) \oplus Z_{n} (C), B_{n} (B) ≅ B_{n} (A) \oplus B_{n} (C),$ and quotienting cycles by boundaries gives $H_{n} (B) ≅ H_{n} (A) \oplus H_{n} (C) .$

Depends on

Used by

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Sources

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