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

Used by

The cone of the zero map is the direct sum with a shift Proposition
The full subcategory of acyclic complexes is thick in the homotopy category Proposition
The quotient of the mapping cylinder by its source is the mapping cone Proposition
The canonical mapping-cone sequence is degreewise split short exact Theorem
The homotopy category is additive Theorem
The mapping cylinder factors a chain map Theorem

Dependency tree · two levels

10 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)