PropositionStatement: Literature-sourcedProof: AI-generatedprecheck passaudited 2026-08-30

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Finite biproducts of complexes are computed degreewise

Statement

Let $A$ be an abelian category. Finite biproducts in $Ch (A)$ are formed degreewise. In particular, $H_{n} (C_{∙} \oplus D_{∙}) ≅ H_{n} (C_{∙}) \oplus H_{n} (D_{∙})$ for every $n$ .

Facts & Assumptions

Given: An abelian category $A$ and chain complexes $C_{∙}$ and $D_{∙}$ in $A$ .

[L1]

$Ch (A)$ is additive, so it has finite biproducts (The category of complexes in an additive category is additive).

[L2]

Homology is an additive functor (Homology is an additive functor).

Proof

technique · direct

1.1L1

The proof of [L1] constructs the biproduct $C_{∙} \oplus D_{∙}$ degreewise, with term $C_{n} \oplus D_{n}$ and differential $d_{n}^{C} \oplus d_{n}^{D}$ . Thus finite biproducts of complexes are formed degreewise.

2.1L2step 1.1∎

Since $H_{n}$ is additive by [L2], it preserves finite biproducts. Applying it to the biproduct from step 1.1 gives $H_{n} (C_{∙} \oplus D_{∙}) ≅ H_{n} (C_{∙}) \oplus H_{n} (D_{∙}) .$

Depends on

Dependency tree · 2 levels

Sources

Charles A. Weibel, Chapter 1 of An Introduction to Homological Algebra